# Appraisal and Simulation on Codynamics of Pneumonia and Meningitis with Vaccination Intervention: From a Mathematical Model Perspective

Authors: Belela Samuel Kotola; Dawit Melese Gebru; Haileyesus Tessema Alemneh
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

The membranes that encompass the brain and spinal cord become inflamed by the potentially fatal infectious disease called pneumococcal meningitis. Pneumonia and meningitis “coinfection” refers to the presence of both conditions in a single host. In this work, we accounted for the dynamics of pneumonia and meningitis coinfection in communities by erroneously using a compartment model to analyze and suggest management techniques to stakeholders. We have used the next generation matrix approach and derived the effective reproduction numbers. When the reproduction number is less than one, the constructed model yields a locally asymptotically stable disease-free equilibrium point. Additionally, we conducted a sensitivity analysis to determine how different factors affected the incidence and transmission rate, which revealed that both the pneumonia and meningitis transmission rates are extremely sensitive. The performance of our numerical simulation demonstrates that the endemic equilibrium point of the pneumonia and meningitis coinfection model is locally asymptotically stable whenmaxR1,R2>1. Finally, as preventative and control measures for the coinfection of pneumonia and meningitis illness, the stakeholders must concentrate on reducing the transmission rates, reducing vaccination wane rates, and boosting the portion of vaccination rates for both pneumonia and meningitis.


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## Body

## 1. Introduction

The word “epidemiology” is derived from the Greek term “demos,” which means “people,” and “logos,” which means “the study of”. In other words, the term “epidemiology” has its roots in the study of the experiences of a population. Despite the fact that many definitions have been provided, the one that best encapsulates the fundamental ideas and public health perspective of epidemiology is: “Epidemiology is the study of the prevalence and causes of health-related conditions or incidents in particular populations, as well as the application of this information to the prevention or treatment of health issues” [1, 2]. In epidemiology, the frequency and distribution of health events in a population are studied.By the term “frequency,” we not only mean the number of health events, such as the number of cases of meningitis or diabetes in a population, but also the correlation between that number and the size of the population [2, 3].The underlying premise of epidemiology is that disease does not develop in a community at random but rather develops only when an individual has the proper confluence of risk factors or determinants. Individuals are the “patients” of clinicians, whereas communities are the “patients” of epidemiologists. In light of this, while dealing with a patient who is unwell, the clinician and the epidemiologist have separate duties. When a patient with diarrheal illness first shows up, for instance, both parties are concerned with making the right diagnosis [3–5].A potentially fatal infection called pneumococcal meningitis inflames the membranes that cover the brain and spinal cord. The meninges, which are these layers, serve to shield the brain from damage and infection [5]. Millions of people have died as a result of pneumonia, an airborne disease that is caused by breathing harmful organisms, primarily Streptococcus pneumonia. Other illnesses, including meningitis, ear infections, and sinus infections, are also brought on by these bacteria [5–7]. Moreover, these illnesses can afflict people of all ages, from infants to the elderly. Pneumonia is particularly hazardous when the immune system is weakened, as in infants or the elderly, or when it is concomitant with another illness like meningitis [7, 8]. Pneumonia is a common coinfection that occurs at the time of admission in cases of bacterial meningitis and is independently linked to a poor prognosis and death [9].The most frequent pneumococcal infection in children are caused by 13 different varieties of pneumococcal bacteria. There are vaccines named PCV13, which can protect against these types, and PPSV23, which can protect against 23 other types. For the sake of this investigation, we have studied prior work by other researchers who used mathematical modeling to explain the transmission and spread of coinfections with pneumonia and meningitis, such as [7, 10–13]. The majority of these investigations were carried out to identify community-level infectious disease control methods. To our knowledge, no one has created and examined the three kinds of vaccine independently in a mathematical model of meningitis and pneumonia coinfection in a specific community. As a result, this recently proposed study considers the dynamics of meningitis and pneumonia in communities, utilizing a deterministic compartmental model to analyze and recommend appropriate management techniques to actors. Therefore, we are driven and intrigued to investigate the three vaccine kinds for meningitis and pneumonia coinfection in this work by developing a mathematical model of meningitis and pneumonia coinfection combined with regulating techniques. We have laid up the basic framework for this investigation as follows: in Section 2, we outline and develop the compartmental mathematical model of coinfection with meningitis and pneumonia. The model analysis includes the equilibrium points, fundamental reproduction numbers, and stability analysis of the submodels and the main model also presented in Section 2. Numerical simulations and sensitivity analysis are presented in Section 3. The study’s discussion and conclusion were then finalized.


## 2. Mathematical Model Formulations and Its Qualitative Analysis

In this section we have proposed a mathematical model which depend upon the assumption and present the qualitative properties of the constructed model.


### 2.1. Baseline Model Formulation and Assumptions

We have developed the new model by expanding the model which was developed previously by another researcher based on the following hypotheses. Under this study, we consider homogenous population and that the factors such as sex, social status, and race do not affect the probability of being infected. The model subdivides the human total populationNt into nine mutually-exclusive compartments, namely, susceptible population St, pneumonia-only infectious PIt, meningitis-only infectious MIt, meningitis and pneumonia coinfectious MPIt, treated class Tt, meningitis vaccinated MV(t), pneumonia vaccinated PV(t), class of people who take both vaccines of pneumonia and meningitis (PCV13 Pneumococcal conjugate vaccine) group MPVt, and recovered class R. The recovery from natural immunity and the effects of vertical transmission to pneumonia and meningitis were assumed to be insignificant in this study. Epidemiologically, individuals in the removed/recovered compartment Rt do not attain permanent immunity so that we are assigned such case by the parameterY.In this study the mass action-incidence rate of new infections are used, and the modification parametersω1 and ω2 are the factors that describe the fact of how infectiousness level of pneumonia increases the susceptibility level to meningitis disease and vice versa, respectively. The meningitis disease is assumed to be transmitted after effective contact between the susceptible and meningitis infectious classes with effective contact rate a, where a is a composite parameter that measures the meningitis contact rate κ and the probability of transmission upon contact q. Additionally, pneumonia disease is assumed to be transmitted after effective contact between the susceptible and pneumonia infectious classes with effective contact rate b, where b is a composite parameter which measure the pneumonia infectious contact rate c and the probability of transmission upon contact p. Individuals can get meningitis by contact rate a from a meningitis-only infected or coinfected person with force of infection of meningitis α1=aMI+ℯ1MPI and join MI compartment where ℯ1 is the modification parameter.An individual can get pneumonia with contact rate ofb from a pneumonia-only infected or coinfected person with force of infection of pneumonia α2=bPI+ℯ2MPI and join the PI compartment with modification parameter ℯ2. Pneumonia-only infected individuals also can get an additional meningitis infection with force of infection and modification parameter ω1α1 and join coinfected compartment MPI. The coinfected compartment increases because of individuals that come from meningitis-only infected compartment are infected by pneumonia with force of infection and modification parameter ω2α2. Since the coinfected individuals are aware of the disease, they remain there in the treated compartment even if they are free from either pneumonia or meningitis until they are free from all the diseases. The parameters used in the model are described in Table 1.Table 1
Descriptions of parameters of the model.
ParameterDescriptionaMeningitis effective contact ratebPneumonia effective contact rateτ1The rate at which meningitis-infected individuals are treated enter to and treated classτ2The rate at which pneumonia-infected individuals are treated and enter to treated classτ3The rate at which meningitis and pneumonia coinfected individuals are treated and enter to treated classβThe rate at which treated class recoveredμNatural death rateδ1Meningitis-only caused death rateδ2Pneumonia-only caused death rateδ3Meningitis and pneumonia coinfection caused death rate.ω1Modification parameter andω1≥1ω2The modification parameter andω2≥1YRate of loss of immunityπThe portion of vaccinated new bornλRecruitment rateε2The rate at which a pneumonia vaccinated individual takes meningitis vaccinationε1The rate at which a meningitis vaccinated individual takes pneumonia vaccinationϕPneumonia vaccine wanes rateφMeningitis vaccine wanes rateρThe portion in which pneumonia vaccine is givenUsing the above assumptions and parameters, we have constructed the following schematic diagram that is given in Figure1.Figure 1
Schematic diagram of full pneumonia and meningitis codynamics.From the diagram given in Figure1, the corresponding dynamical systems will be as follows from the assumptions of the model and using the above basic model assumption we have the following flow chart.
(1)dSdt=1−πλ+ϕPV+φMV+YR–α1+α2+μS,dMVdt=1−ρπλ−μ+φ+ε1MV,dPVdt=ρπλ−μ+ϕ+ε2PV,dMPVdt=ε1MV+ε2PV−μMPV,dMIdt=α1S–ω2α2+τ1+δ1+μMI,dPIdt=α2S−ω1α1+τ2+δ2+μPI,dMPIdt=ω1α1PI+ω2α2MI−τ3+δ3+μMPI,dTdt=τ1MI+τ2PI+τ3MPI−β+μT,dRdt=βT−Y+μR.This system of differential equation is the mathematical representation of full meningitis and pneumonia model which is the combination of the two diseases. In the next section, we have studied the qualitative behavior of the constructed model. For simplification of our work, we split the full meningitis-pneumonia coinfection model into submodels, which are meningitis-only and pneumonia-only models. First, we will study the qualitative behavior of the submodel and then qualitative behavior of the full model is followed.


### 2.2. Positivity of Solutions and Invariant Region of the Only Pneumonia-Infected Model

In this subsection, we have considered the model of pneumonia only in the absence of meningitis disease. This procedure will help us to summarize and conclude some properties of the full coinfected model depending on the properties of sub models. To gate this submodel from the full model, we set,MI=MPI=0, and we have the following dynamical system.The corresponding dynamical systems are as follow.(2)dSdt=1−πλ+ϕPV+YR–α2+μS,dPIdt=α2S−τ2+δ2+μPI,dPVdt=πλ−μ+ϕPV,dTdt=τ2PI−β+μT,dRdt=βT−Y+μR.For the dynamical systems to be epidemiologically meaningful as well as well-posed, we need to prove that all the state variables of dynamical systems are nonnegative.Theorem 1.
All the populations of the system with positive initial conditions are positive.
Proof: assumeS0>0,PI0>0,PV0>0,T0>0andR0>0 are positive for time t>0 and for all nonnegative parameters.
First, let us takeT=supt>0 such that S t′>0,PIt′>0,PVt′>0,Tt′>0 and Rt′>0,t′∈0,t}.
From the first equation of system (2), we do have dS/dt=1−πλ+ϕPV+YR–α2+μS⇒St=S0e−∫0Tα2+μdt′+e−∫0Tα2+μdt′∫0Te∫0Tα2+μdt′1−πλ+YR+ϕPVdt>0⇒St>0.
There,St is positive. Following the same procedure, all the remaining state variables are nonnegative. Therefore, from proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.Theorem 2.
The total human population of the dynamical system (2) is positively closed in the closed invariant setΩ1=S,PI,PV,T,Rϵℝ+4:N1≤λ/μ.
Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [14].
Proof: assume the total population of the model isN1. To get an invariant region, which shows boundedness of solution, it can be obtained as follows.
(3)N1=S+PI+PV+T+R⇒dN1dt=λ−μS−δ2PI−μPI−μPV−μT−μR,(4)dN1dt=λ−μN1−δ2PI⇒dN1dt+μN1≤λ,(5)N1t≤N0e−μt+λμ⇒limt⟶∞N1t≤λμ,
⇒0≤N1≤λ/μ .
Therefore, the dynamical system that we do have is bounded.


### 2.3. Existence and Stability of Disease-Free Equilibrium Point

The disease-free equilibrium point is obtained by making all the equations equal to zero, provided thatPI=0 and the obtained disease-free equilibrium point is given by
(6)Ep0=SO,PV,OPIO,T0,RO=λμ1−πμ+ϕ+πϕμ+ϕ,πλμ+ϕ,0,0,0.


#### 2.3.1. Effective Reproduction Number

The reproduction number is the number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [15–17].Using the next generation matrix method, we have obtained the effective reproduction number of pneumonia-infected-only submodel, which isRefp=λb/μ1−πμ+ϕ+πϕ/μ+ϕτ2+δ2+μ.Theorem 3.
The disease-free equilibrium pointEp0 of the model in system (2) is locally asymptotically stable if the effective reproduction number Refp<1 and is unstable ifRefp>1.
Proof: from the Jacobean matrixJEp0 of the model (2), with respect to S,PV,PI,T,R at the disease-free equilibrium point, we have the following characteristics equation.
(7)–μ−λ1r1−r2−λ2−μ+ϕ−λ3−β+μ−λ4−Y+μ−λ5=0
Wherer1=Refpr2 and r2=τ2+δ2+μ,(8)⇒λ1=–μ,λ3=−μ+ϕ,λ4=−β+μ,λ5=−Y+μ.
Hence, all the parameters are nonnegative, and all the eigenvalues of the corresponding Jacobean matrix are negative. Butλ2=r1−r2=Refpr2−r2=r2Refp−1,(9)⇒λ2=r2Refp−1,⇒λ2<0iffRefp<1.Therefore, the disease-free equilibrium point is locally asymptotically stable if and only if Refp<1, otherwise it is unstable, that is, ifRefp>1.


#### 2.3.2. Global Stability of Disease-Free Equilibrium Point of the Model

To verify the global stability of the disease-free equilibrium point of the pneumonia monoinfection model, we have used an adopted method of Castillo-Chavez et al. used by others scholar such as [18, 19].Lemma 4.
If the pneumonia monoinfection model can be written as(10)dYdt=GY,W,dZdt=HY,W,HY,0=0,
whereY∈ℝm be the components of noninfected individuals and W∈ℝn be the components of infected individuals including treated class and Ep0=Y∗,0 denotes the disease-free equilibrium point of dynamical system (2).
Assume(i)
FordY/dt=GY,0, Y∗ is globally asymptotically stable (GAS)(ii)
HY,W=BW−HˇY,W, HˇY,W≥0 for Y,W∈Ω1 where B=DWHY∗,0 is an M-matrix, i.e., the off diagonal elements of B are nonnegative and Ω1 is the region in which the system makes biological sense. Then the fixed point Ep0=Y∗,0 is globally asymptotically stable equilibrium point of the system (2) wheneverRefp<1.Lemma 5.
The disease-free equilibrium pointEp0 of the pneumonia monoinfection model (2) is globally asymptotically stable if Refp<1 and the two sufficient conditions given in Lemma 4 are satisfied.
Proof: here we are applying Lemma5 on the pneumonia monoinfection model (2) and we have gotten the following matrices.
(11)dYdt=GY,W=1−πλ+ϕPV+YR–α2+μSπλ−μ+ϕPV,dWdt=HY,W=α2S−τ2+δ2+μPIτ2PI−β+μTβT−Y+μR,GY,0=1−πλ+ϕPV−μSπλ−μ+ϕPV,and
(12)HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=bλμ1−πμ+ϕ+πϕμ+ϕPI−α2S00,HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=S0−SbPI00.
SinceS≤S0, we have Hˇ1Y,W≥0, thus, the disease-free equilibrium point Ep0 is globally asymptotically stable if Refp<1. Biologically, whenever Refp<1, the only pneumonia infection disease dies out while the total population increases [18].


### 2.4. The Existence and Stability of Endemic Equilibrium Point

The endemic equilibrium point of the dynamical system of (2) is obtained by making the right side of the system equal to zero, providing that PI≠0. We have supposed that the endemic equilibrium point of the model is denoted by Ep∗=S∗,PI∗,Pv∗,T∗,R∗ and the corresponding force of infection isα2t=bPI∗t.For simplification of algebraic manipulation, we have assumed the parameters in the model by another variable as follows,k1=1−πλ,k2=τ2+μ,k3=τ2+δ2+μ,k4=πλ,k5=ϕ+μ,k6=β+μ,k7=Y+μ,k8=γβτ2,k9=k3k6k7, k10=k1+ϕk4/k5, k11=k9k10, k12=k2k9k13=k3k12 and k14=k8k3. Now the equation of force of infection can be rearranged as
(13)α2∗k14α2∗+bk11−k13=0⇒α2∗=0ork14α2∗+bk11−k13=0,⇒α2∗=k13−bk11/k14 but α2∗=k13−bk11/k14=β+μY+μτ2+δ2+μτ2+μϕ+μ−bλϕ+μ1−π−ϕπ/γβτ2ϕ+μ=β+μY+μτ2+δ2+μτ2+μϕ+μ/γβτ2ϕ+μ−λbϕ+μ1−π+ϕπ/γβτ2ϕ+μ,α2∗=β+μY+μτ2+δ2+μτ2+μϕ+μ/γβτ2λb/μ1−πμ+ϕ+πϕ/μ+ϕτ2+δ2+μ−1⇒α2∗=β+μY+μτ2+δ2+μτ2+μϕ+μ/γβτ2Rp−1⇒α2∗>0 if Refp>1.Therefore, there is a unique endemic equilibrium point for pneumonia monoinfected model as given byEp∗=S∗,PI∗,Pv∗,T∗,R∗where
(14)S∗=τ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗,PI∗=τ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗α2∗τ2+δ2+μ,Pv∗=πλϕ+μ,T∗=α2∗τ2τ2+δ2+μβ+μτ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗,R∗=τ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗βτ2α2∗τ2+δ2+μβ+μY+μ.Theorem 6.
The endemic equilibrium point of system (2) Ep∗=S∗,PI∗,Pv∗,T∗,R∗ is locally asymptotically stable for the reproduction number Reffp>1.
Proof: to show that the local stability of the endemic equilibrium point, we have used the method of the Jacobian matrix and the Routh Hurwitz stability criteria.
Then the corresponding characteristic equation is obtained from the determinant of(15)A−λBC0DEF−λ00000G−λ000H0I−λ0000JK−λ=0,
whereA=–α2Ip∗+μ, B=−α2S∗, C=ϕ, D=Y, E=α2Ip∗, F=α2S∗−τ2+δ2+μ,
(16)G=−μ+ϕ,H=τ2,I=−β+μ,J=βandk=−Y+μ,⇒a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5=0,
wherea0=1,−I+A+F+G+K=a1, IA+IF+AF+IG+AG+FG+IK+AK+FK+GK=a2, −−B+IAF+IAG+IFG+AFG+IAK+IFK+AFK+IGK+AGK+FGK=a3, −IB−IAFG+DEHJ+BK−IAFK−IAGK−IFGK−AFGK=a4, a5=DEGHJ+IBK−IAFGK.To apply the Routh-Hurwitz stability criteria, it is obligatory to check if the necessary condition of all the coefficients have the same sign or not. Sincea0=1 is positive in sign, all a1, a2,a3, a4, and a5 should be positives in sign. All the coefficients of the characteristic’s polynomial are positives whenever Reffp>1. We have observed that the first column of the Routh Hurwitz array has no sign change, thus the root of the characteristics equation of the dynamical system are negative. Hence, the endemic equilibrium point of the dynamical system is locally asymptotically stable.


### 2.5. Positivity of Solutions and Invariant Region of the Only Meningitis-Infected Model

We have madePI=MPI=0 from the full pneumonia and meningitis coinfection model to obtain this submeningitis-only model, and got the following dynamical system.
(17)dSdt=1−πλ+φMV+YR–α1+μS,dMIdt=α1S–τ1+δ1+μMI,dMVdt=πλ−μ+φMV,dTdt=τ1MI−β+μT,dRdt=βT−Y+μR.The above dynamical systems are needed to be epidemiologically meaningful as well as well-posed. To prove that, we have intimated that all the state variables of dynamical systems are nonnegative.Theorem 7.
All the populations of the system with positive initial conditions are positive.
Proof: assumeS0>0,MI0>0,MV0>0,T0>0 and R0>0 are positive for time t>0 and for all nonnegative. First let us take T=supt>0suchthatSt′>0,MIt′>0,MVt′>0,Tt′>0andRt′>0,t′∈0,t.
From the first equation of system (17), we do have
(18)dSdt=1−πλ+φMV+YR–α1+μS,St=S0e−∫0Tα1+μdt′+e−∫0Tα1+μdt′∫0Te∫0Tα1+μdt′1−πλ+YR+φMVdt>0,⇒St>0.
Therefore,St is positive. Subsequent to the same procedure, the remaining state variables are nonnegative. Therefore, from the stated proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.Theorem 8.
All the populations of the system with positive initial conditions are nonnegative
The total human population of the dynamical system (17) is positively closed in the closed invariant set Ω2=S,MI,MV,T,Rϵℝ+5:N2≤λ/μ. Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [14].
Proof: to get an invariant region, boundedness of solution is obtained as follow.(19)N2=S+MI+MV+T+R,dN2dt=λ−μN2−δ1MI⇒0≤N2≤λμ.
Therefore, the dynamical system that we do have is bounded.


### 2.6. Existence and Stability of Disease-Free Equilibrium Point

The disease-free equilibrium point is obtained by making all the equations in the system equal to zero, provided that providing thatMI=0. Therefore, the disease-free equilibrium point is
(20)Ep0=SO,MVO,MIO,T0,RO=λμ1−πμ+φ+πφμ+φ,πλμ+φ,0,0,0.


#### 2.6.1. Effective Reproduction Number

The reproduction number can be defined as a number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [16, 17, 20].To compute the reproduction number, first distinguishing the new infected from all other changes in the host population is mandatory as follows.LetFix:be the rate of appearance of new infected in compartmenti,V+ix:be the rate of transfer of individuals in to compartment i,V−ix:be the rate of transfer of individuals out of compartmenti.And thenVix=V−ix−V+i but F=∂Fi/∂XjXo and V=∂Vi/∂XjXo, where F and V are mxm matrix with m is number of infected compartment. Fv−1 is the next generation matrix, and the spectral radius of next generation matrix is needed for the reproduction number we are seeking for.Thus∂FiX/∂Xj=aS and ∂ViX/∂Xj and FV−1=λa/μ1−πμ+φ+πφ/μ+φτ1+δ1+μTherefore, the effective reproduction number of meningitis monoinfected submodel is(21)Refm=λaμ1−πμ+φ+πφμ+φτ1+δ1+μ.Theorem 9.
The disease-free equilibrium pointEm0 of the model in system (17) is locally asymptotically stable if the effective reproduction number Refm<1, and it is unstable if Refm>1.
Proof:
Using the Jacobean matrixJEp0 of the model (17) with respect to S,MV,MI,T,R at the disease-free equilibrium point, we have the characteristic equation
(22)⇒–μ−λ1t1−t2−λ2−μ+φ−λ3−β+μ−λ4−Y+μ−λ5=0wheret1=Refmt2andt2=τ1+δ1+μ,⇒λ1=–μ,λ3=−μ+φ,λ4=−β+μ,λ5=−Y+μ.
Hence, all the parameters are nonnegative, all the eigenvalues of the corresponding Jacobean matrix are negative other thanλ2.
Forλ2,λ2=t2Reffm−1⇒λ2=t2Reffm−1,(23)⇒λ2<0iffRefm<1.
Therefore, the disease-free equilibrium point of the meningitis monoinfected model is locally asymptotically stable if the effective reproduction numberRefm<1 and is unstable if Refm>1.


#### 2.6.2. Global Stability of Disease-Free Equilibrium Point

We utilized the approach developed by Castillo-Chavez et al. and used it to confirm the overall stability of the disease-free equilibrium point of the meningitis monoinfection model [18, 19].Lemma 10.
If the pneumonia monoinfection model can be written as(24)dYdt=GY,W,dZdt=HY,W,HY,0=0,
whereY∈ℝm be the components of noninfected individuals and W∈ℝn be the components of infected individuals including the treated class, and Em0=Y∗,0 denotes the disease-free equilibrium point of the dynamical system (3).
Assume(i)
FordY/dt=GY,0,Y∗ is globally asymptotically stable (GAS)(ii)
HY,W=BW−HˇY,W, HˇY,W≥0 for Y,W∈Ω1 where B=DWHY∗,0 is an M-matrix, i.e., the off diagonal elements of B are nonnegative and Ω2 is the region in which the system makes biological sense
Then the fixed pointEm0=Y∗,0 is globally asymptotically stable equilibrium point of the system (17) whenever Refm<1.Lemma 11.
The disease-free equilibrium pointEm0 of the pneumonia monoinfection model (17) is globally asymptotically stable if Refm<1 and the two sufficient conditions given in Lemma 10 are satisfied.
Proof: here we are applying Lemma11 on the meningitis monoinfection model (17) and we have gotten the following matrices dY/dt=GY,W=1−πλ+φMV+YR–α1+μSπλ−μ+φMV,(25)dWdt=HY,W=aλμ1−πμ+ϕ+πϕμ+ϕ−τ1+δ1+μ00τ1−β+μ00β−Y+μ,GY,0=1−πλ+φMV−μSπλ−μ+φMV.
Here after some steps of calculations, we have determined that(26)HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=S0−SaMI00.
SinceS≤S0, we have Hˇ1Y,W≥0, thus, the disease-free equilibrium point Em0 of model (17) is globally asymptotically stable if Refm<1. Biologically, whenever Refm<1, the meningitis monoinfection disease dies out while the total population increases [18].


### 2.7. Existence and Stability of Endemic Equilibrium Point

The endemic equilibrium point of the dynamical system of (3) is obtained by making the right side of the system equal to zero, providing that MI≠0. We have supposed that the endemic equilibrium point of the model is denoted by Em∗=S∗,MI∗,Mv∗,T∗,R∗ and the corresponding force of infection is α1t=bMI∗t. For simplification of algebraic manipulation, we have assumed the parameters in the model by another variable as follows.
(27)n1=1−πλ,n2=τ1+μ,n3=τ1+δ1+μ,n4=πλ,n5=μ+φ,n6=β+μ,n7=Y+μ,n8=γβτ1,n9=n3n6n7,n10=n1+φn4/n5,n11=n9n10,n12=n2n9n13=n3n12 and n14=n8n3 . Now the equation of force of infection can be rearranged as
(28)α1∗n14α1∗+an11−n13=0⇒α1∗=0orn14α1∗+an11−n13=0⇒α1∗=n13−an11n14,butα1∗=n13−an11n14=β+μY+μτ1+δ1+μτ1+μφ+μ−aλφ+μ1−π−φπγβτ1φ+μ=β+μY+μτ1+δ1+μτ1+μφ+μγβτ2ϕ+μ−λaφ+μ1−π+φπγβτ1φ+μ,α1∗=β+μY+μτ1+δ1+μτ1+μφ+μγβτ1λaμ1−πφ+μ+πφφ+μτ1+δ1+μ−1,⇒α1∗=β+μY+μτ1+δ1+μτ1+μφ+μγβτ1Reffm−1⇒α1∗>0ifReffm>1.Therefore, the unique endemic equilibrium point for the meningitis monoinfected model is given byEm∗=S∗,MI∗,Mv∗,T∗,R∗, where
(29)S∗=τ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ1+μτ1+δ1+μβ+μY+μ−γβτ1α1∗,MI∗=τ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ1+μτ1+δ1+μβ+μY+μ−γβτ1α1∗α1∗τ1+δ1+μ,Mv∗=πλφ+μ,T∗=α1∗τ1τ1+δ1+μβ+μτ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ1+μτ1+δ1+μβ+μY+μ−γβτ1α1∗,R∗=τ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ2+μτ1+δ1+μβ+μY+μ−γβτ2α2∗βτ2α2∗τ1+δ1+μβ+μY+μ.Theorem 12.
The endemic equilibrium point of system (17) Em∗=S∗,MI∗,Mv∗,T∗,R∗ is locally asymptotically stable for the reproduction number Reffm>1.
Proof: To show the local stability of the endemic equilibrium point we have used the method of Jacobian matrix and Routh Hurwitz stability criteria.
From Jacobian matrix, we have obtained the following characteristic equation(30)a−λf−λg−λi−λk−λ−bi−λk−λ−deg−λhj=0
Wherea=–α1Im∗+μ, b=−α1S∗, c=φ, d=Y, e=α1Im∗, f=α1S∗−τ1+δ1+μ,
(31)g=−μ+φ,h=τ1,i=−β+μ,j=βandk=−Y+μ.⇒a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5=0
Wherea0=1, −i+a+f+g+k=a1, ai+if+af+ig+ag+fg+ik+ak+fk+g=a2,−−b+aif+aig+ifg+afg+aik+ifk+afk+gik+agk+fgk=a3, −bi−aifg+dehj+bk−aifk−aigk−ifgk−afgk=a4
a5=aeghj+bik−aifgk.To apply Routh-Hurwitz stability criteria, it is the must to check the necessary condition of all the coefficients have the same sign or not. Sincea0=1 is positive in sign, all a1, a2,a3, a4 and a5 should be positives in sign. All the coefficients of the characteristic’s polynomial are positives wheneverReffp>1.We have observed that the first column of the Routh Hurwitz array has no sign change, thus the root of the characteristics equation of the dynamical systema0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5=0 are negative. Hence, the endemic equilibrium point of the dynamical system is locally asymptotically stable.


### 2.8. Positivity and Boundedness of Full Pneumonia and Meningitis Coinfected Model

The corresponding dynamical system of the full pneumonia and meningitis coinfection model is given in Equation (1).The constructed model is expected to be meaningful epidemiologically as well as well-posed. We need to prove that all the state variables of the dynamical system are positive.Theorem 13.
All the population of the system with positive initial conditions are nonnegative
Proof: AssumeS0>0,MV0>0,PV0>0,MPV0>0, MI0>0,PI0>0,MPI0>0T0>0, and R0>0 are positive for time t>0 and for all nonnegative parameters. Let us define τ=supt>0suchthatSt>0,MVt>0,PVt>0,MPVt>0,MIt>0,PIt>0,MPIt>0,Tt>0,Rt>0andt∈0,t.
Since allSt,MVt,PVt,MPVt,MIt,PIt,MPIt,TtandRt are continuous, we can say for τ>0. If τ=+∞, then positivity holds.
Nevertheless, if0<τ<+∞, then all the state variables are zeros.
From the first equation of system (1) we do have
(32)dSdt=1−πλ+ϕPV+φMV+YR–α1+α2+μSSt=S0e−∫0τα1+α2+μdt+e−∫0τα1+α2+μdt∫0τe∫0τα1+α2+μdt1−πλ+ϕPV+φMV+YRdt>0⇒St>0
Following same procedure, all the remaining state variables are nonnegative.
Therefore, from proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.Theorem 14.
The total human population is assumed to beN and the dynamical system (1) is positively invariant in the closed invariant set Ω=S,MV,PV,MPV,MI,PI,MPI,T,Rϵℝ+9:N≤λ/μ. Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [24]. Proof: to get an invariant region, which shows boundedness of solution, is obtained as follow.
(33)dNdt=dSdt+dMVdt+dPVdt+dMPVdt+dMIdt+dPIdt+dMPIdt+dTdt+dRdt,(34)⇒dNdt=λ−μN−δ2PI−δ1MI−δ3MPI,(35)⇒Nt≤N0e−μt+λμ⇒0≤N≤λμ.
Therefore, the dynamical system that we have constructed is bounded.


### 2.9. Disease-Free Equilibrium Point and Its Stability

The disease-free equilibrium point of full pneumonia and meningitis coinfection modelEpmO is obtained by making all the right-hand-side of equation in system (1), providing that all the infectious classes are equal to zero.
(36)SO,MV,OPVO,MPV0,MI0,PI0,MPI0,T0,RO=1−πλα1+α2+μ+πλα1+α2+μϕρμ+ϕ+ε2+1−ρφμ+φ+ε1,1−ρπλμ+φ+ε1,ρπλμ+ϕ+ε2,1−ρπλε1μμ+φ+ε1+ρπλε2μμ+ϕ+ε2,0,0,0,0,0


#### 2.9.1. Effective Reproduction Number

The reproduction number is the average number of people that become infected because of the entry of one infectious person into a completely susceptible population in the absence of intervention. Moreover, reproduction number is utilized to determine the effect of the control measures and to understand the capability of the spread of the infection to disseminate in the entire community when the control strategies are applied [15, 17, 21].The reproduction number of pneumonia and meningitis confection model denoted byReff, which is manipulated by the Van den Driesch, Pauline, and James Warmouth next generation matrix approach [20], is the largest eigenvalue of the next generation matrix FV−1=∂FiErO/∂xj∂νiErO/∂xj−1, where Fi is the rate of appearance of new infection in compartment i, νi is the transfer of infections from one compartment i to another, and Epm0 is the disease-free equilibrium point.The(37)Fix=α1SOα2SO0000000,νi=π−1λ−ϕPVO−φMVO−YR0+μSOμ+φ+ε1MVO−1−ρπλμ+ϕ+ε2PVO−ρπλμMPV0−ε1MVO−ε2PVOω2α2+τ1+δ1+μMI0ω1α1+τ2+δ2+μPI0τ3+μMPI0−ω1α1PI−ω2α2MI0β+μT−τ1MI0−τ2PI−τ3MPI0Y+μR0−βT0.Then(38)F=1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1a0a0bb000,and
(39)V=τ1+δ1+μ000τ2+δ2+μ000τ3+μ⇒FV−1=aτ1+δ1+μ0aτ3+μ0bτ2+δ2+μbτ3+μ000.The eigenvalues of the next generation matrixF.V−1are
(40)0,aλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ.Therefore, the effective reproduction number of full meningitis and pneumonia model is(41)Reff=maxaλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ,Reff=maxR1,R2 where R1=aλ1−π/τ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2/μ+ϕ+ε2μπλa/τ1+δ1+μ, and R2=bλ1−π/τ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2/μ+ϕ+ε2μπλb/τ2+δ2+μ.Theorem 15.
The disease-free equilibrium pointEpmO of the model in system (17) is locally asymptotically stable if the effective reproduction number Reff<1 and is unstable ifReff>1.
Proof:
The Jacobean matrixJEpmO of the model (1) with respect to S,MV,PV,MPV,MI,PI,MPI,T,R at the disease-free equilibrium point is the following:
(42)JEpmO=–μφϕ0r1r2r30Y0r4000000000r50000000ε1ε2−μ000000000r6000000000r7000000000r8000000τ1τ2τ3r900000000βr10,
where(43)r1=−a1−πλ/μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r2=−b1−πλ/α1+α2+μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r3=−aℯ1+bℯ21−πλ/μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r4=−μ+φ+ε1,r5=−μ+ϕ+ε2,r6=a1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1–τ1+δ1+μ,r7=b1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1−τ2+δ2+μ,r8=−τ3+μ,r9=−β+μandr10=−Y+μ⇒λ1+μλ4+μ−λ2+r4−λ3+r5−λ5+r6−λ6+r7−λ7+r8−λ8+r9−λ9+r10⇒λ1=−μ,λ4=−μ,λ2=r4=−μ+φ+ε1,λ3=r5=−μ+ϕ+ε2,λ7=r8=−τ3+μ,λ8=r9=−β+μ,λ5=τ1+δ1+μaλμ+φμ1−πμ+φ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1aλ1−πμ+φ+πφμμ+φτ1+δ1+μ−1⇒λ5=τ1+δ1+μaλμ+φμ1−πμ+φ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1Reffm−1.λ6=τ2+δ2+μbλμ+ϕμ1−πμ+ϕ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1λbμ1−πμ+ϕ+πϕμ+ϕτ2+δ2+μ−1⇒λ6=τ2+δ2+μbλμ+ϕμ1−πμ+ϕ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1Reffp−1andλ9=r10=−Y+μ.Hence, all the parameters are nonnegative as well as all the eigenvalues of the corresponding Jacobean matrix are negative exceptλ5 andλ6.The sign of eigenvaluesλ5 and λ6 depends on the values of Reffm and Reffp, respectively. Moreover, λ5<0 whenever Reffm<1 and λ6<0 if and only if Reffp<1.Therefore, the disease-free equilibrium point of the full meningitis and pneumonia coinfection model is local asymptotically stable only ifReffm<1 and Reffp<1, otherwise it is unstable.


#### 2.9.2. Global Stability of Disease-Free Equilibrium Point

The disease-free equilibriums of the meningitis monoinfection model and the pneumonia monoinfection model are both globally asymptotically stable whenever their corresponding effective reproduction numbers values are smaller than unity, as demonstrated in Section3 Figure 2 and 3, respectively. In light of this conclusion, the disease-free equilibrium point of the pneumonia and meningitis coinfection model (1) is globally asymptotically stable if Reff=maxR1,R2<1.Figure 2
Sensitivity indices forR2.Figure 3
Stability of endemic equilibrium whenReff=maxR1,R2>1.


### 2.10. Endemic Equilibrium Point and Its Stability

The endemic equilibrium point of the dynamical system (1) is obtained by making the right side of the system equal to zero providing that MI≠0, PI≠0 and MPI≠0. We have supposed the endemic equilibrium point of the model is denoted by Epm∗=S∗,Mv∗,Pv∗,MPv∗,MI∗,PI∗,MPI∗,T∗,R∗ and the corresponding forces of infection are
(44)α1∗=aMI∗+ℯ1MPI∗andα2∗=bPI∗+ℯ2MPI∗,S∗=1α1∗+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,Mv∗=1−ρπλμ+φ+ε1,Pv∗=ρπλμ+ϕ+ε2,MPv∗=ε1μ1−ρπλμ+φ+ε1+ε2μρπλμ+ϕ+ε2,MI∗=α1ω2α2∗+τ1+δ1+μ1α1∗+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,PI∗=α2ω1α1∗+τ2+δ2+μ1α1+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,MPI∗=1τ3+μω1α1∗α2ω1α1∗+τ2+δ2+μ1α1∗+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ+α1ω2α2∗ω2α1∗+τ1+δ1+μα1∗+α2∗+μτ3+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,T∗=τ1MI∗+τ2PI∗+τ3MPI∗β+μ,R∗=βT∗Y+μ.Theorem 16.
The endemic equilibrium pointEpm∗ is locally asymptotically stable if the Reff>1, otherwise it is unstable.
Proof: the local stability of the endemic equilibrium point of the full model is verified by using a numerical simulation in Section4 Figure 3.


## 2.1. Baseline Model Formulation and Assumptions

We have developed the new model by expanding the model which was developed previously by another researcher based on the following hypotheses. Under this study, we consider homogenous population and that the factors such as sex, social status, and race do not affect the probability of being infected. The model subdivides the human total populationNt into nine mutually-exclusive compartments, namely, susceptible population St, pneumonia-only infectious PIt, meningitis-only infectious MIt, meningitis and pneumonia coinfectious MPIt, treated class Tt, meningitis vaccinated MV(t), pneumonia vaccinated PV(t), class of people who take both vaccines of pneumonia and meningitis (PCV13 Pneumococcal conjugate vaccine) group MPVt, and recovered class R. The recovery from natural immunity and the effects of vertical transmission to pneumonia and meningitis were assumed to be insignificant in this study. Epidemiologically, individuals in the removed/recovered compartment Rt do not attain permanent immunity so that we are assigned such case by the parameterY.In this study the mass action-incidence rate of new infections are used, and the modification parametersω1 and ω2 are the factors that describe the fact of how infectiousness level of pneumonia increases the susceptibility level to meningitis disease and vice versa, respectively. The meningitis disease is assumed to be transmitted after effective contact between the susceptible and meningitis infectious classes with effective contact rate a, where a is a composite parameter that measures the meningitis contact rate κ and the probability of transmission upon contact q. Additionally, pneumonia disease is assumed to be transmitted after effective contact between the susceptible and pneumonia infectious classes with effective contact rate b, where b is a composite parameter which measure the pneumonia infectious contact rate c and the probability of transmission upon contact p. Individuals can get meningitis by contact rate a from a meningitis-only infected or coinfected person with force of infection of meningitis α1=aMI+ℯ1MPI and join MI compartment where ℯ1 is the modification parameter.An individual can get pneumonia with contact rate ofb from a pneumonia-only infected or coinfected person with force of infection of pneumonia α2=bPI+ℯ2MPI and join the PI compartment with modification parameter ℯ2. Pneumonia-only infected individuals also can get an additional meningitis infection with force of infection and modification parameter ω1α1 and join coinfected compartment MPI. The coinfected compartment increases because of individuals that come from meningitis-only infected compartment are infected by pneumonia with force of infection and modification parameter ω2α2. Since the coinfected individuals are aware of the disease, they remain there in the treated compartment even if they are free from either pneumonia or meningitis until they are free from all the diseases. The parameters used in the model are described in Table 1.Table 1
Descriptions of parameters of the model.
ParameterDescriptionaMeningitis effective contact ratebPneumonia effective contact rateτ1The rate at which meningitis-infected individuals are treated enter to and treated classτ2The rate at which pneumonia-infected individuals are treated and enter to treated classτ3The rate at which meningitis and pneumonia coinfected individuals are treated and enter to treated classβThe rate at which treated class recoveredμNatural death rateδ1Meningitis-only caused death rateδ2Pneumonia-only caused death rateδ3Meningitis and pneumonia coinfection caused death rate.ω1Modification parameter andω1≥1ω2The modification parameter andω2≥1YRate of loss of immunityπThe portion of vaccinated new bornλRecruitment rateε2The rate at which a pneumonia vaccinated individual takes meningitis vaccinationε1The rate at which a meningitis vaccinated individual takes pneumonia vaccinationϕPneumonia vaccine wanes rateφMeningitis vaccine wanes rateρThe portion in which pneumonia vaccine is givenUsing the above assumptions and parameters, we have constructed the following schematic diagram that is given in Figure1.Figure 1
Schematic diagram of full pneumonia and meningitis codynamics.From the diagram given in Figure1, the corresponding dynamical systems will be as follows from the assumptions of the model and using the above basic model assumption we have the following flow chart.
(1)dSdt=1−πλ+ϕPV+φMV+YR–α1+α2+μS,dMVdt=1−ρπλ−μ+φ+ε1MV,dPVdt=ρπλ−μ+ϕ+ε2PV,dMPVdt=ε1MV+ε2PV−μMPV,dMIdt=α1S–ω2α2+τ1+δ1+μMI,dPIdt=α2S−ω1α1+τ2+δ2+μPI,dMPIdt=ω1α1PI+ω2α2MI−τ3+δ3+μMPI,dTdt=τ1MI+τ2PI+τ3MPI−β+μT,dRdt=βT−Y+μR.This system of differential equation is the mathematical representation of full meningitis and pneumonia model which is the combination of the two diseases. In the next section, we have studied the qualitative behavior of the constructed model. For simplification of our work, we split the full meningitis-pneumonia coinfection model into submodels, which are meningitis-only and pneumonia-only models. First, we will study the qualitative behavior of the submodel and then qualitative behavior of the full model is followed.


## 2.2. Positivity of Solutions and Invariant Region of the Only Pneumonia-Infected Model

In this subsection, we have considered the model of pneumonia only in the absence of meningitis disease. This procedure will help us to summarize and conclude some properties of the full coinfected model depending on the properties of sub models. To gate this submodel from the full model, we set,MI=MPI=0, and we have the following dynamical system.The corresponding dynamical systems are as follow.(2)dSdt=1−πλ+ϕPV+YR–α2+μS,dPIdt=α2S−τ2+δ2+μPI,dPVdt=πλ−μ+ϕPV,dTdt=τ2PI−β+μT,dRdt=βT−Y+μR.For the dynamical systems to be epidemiologically meaningful as well as well-posed, we need to prove that all the state variables of dynamical systems are nonnegative.Theorem 1.
All the populations of the system with positive initial conditions are positive.
Proof: assumeS0>0,PI0>0,PV0>0,T0>0andR0>0 are positive for time t>0 and for all nonnegative parameters.
First, let us takeT=supt>0 such that S t′>0,PIt′>0,PVt′>0,Tt′>0 and Rt′>0,t′∈0,t}.
From the first equation of system (2), we do have dS/dt=1−πλ+ϕPV+YR–α2+μS⇒St=S0e−∫0Tα2+μdt′+e−∫0Tα2+μdt′∫0Te∫0Tα2+μdt′1−πλ+YR+ϕPVdt>0⇒St>0.
There,St is positive. Following the same procedure, all the remaining state variables are nonnegative. Therefore, from proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.Theorem 2.
The total human population of the dynamical system (2) is positively closed in the closed invariant setΩ1=S,PI,PV,T,Rϵℝ+4:N1≤λ/μ.
Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [14].
Proof: assume the total population of the model isN1. To get an invariant region, which shows boundedness of solution, it can be obtained as follows.
(3)N1=S+PI+PV+T+R⇒dN1dt=λ−μS−δ2PI−μPI−μPV−μT−μR,(4)dN1dt=λ−μN1−δ2PI⇒dN1dt+μN1≤λ,(5)N1t≤N0e−μt+λμ⇒limt⟶∞N1t≤λμ,
⇒0≤N1≤λ/μ .
Therefore, the dynamical system that we do have is bounded.


## 2.3. Existence and Stability of Disease-Free Equilibrium Point

The disease-free equilibrium point is obtained by making all the equations equal to zero, provided thatPI=0 and the obtained disease-free equilibrium point is given by
(6)Ep0=SO,PV,OPIO,T0,RO=λμ1−πμ+ϕ+πϕμ+ϕ,πλμ+ϕ,0,0,0.


### 2.3.1. Effective Reproduction Number

The reproduction number is the number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [15–17].Using the next generation matrix method, we have obtained the effective reproduction number of pneumonia-infected-only submodel, which isRefp=λb/μ1−πμ+ϕ+πϕ/μ+ϕτ2+δ2+μ.Theorem 3.
The disease-free equilibrium pointEp0 of the model in system (2) is locally asymptotically stable if the effective reproduction number Refp<1 and is unstable ifRefp>1.
Proof: from the Jacobean matrixJEp0 of the model (2), with respect to S,PV,PI,T,R at the disease-free equilibrium point, we have the following characteristics equation.
(7)–μ−λ1r1−r2−λ2−μ+ϕ−λ3−β+μ−λ4−Y+μ−λ5=0
Wherer1=Refpr2 and r2=τ2+δ2+μ,(8)⇒λ1=–μ,λ3=−μ+ϕ,λ4=−β+μ,λ5=−Y+μ.
Hence, all the parameters are nonnegative, and all the eigenvalues of the corresponding Jacobean matrix are negative. Butλ2=r1−r2=Refpr2−r2=r2Refp−1,(9)⇒λ2=r2Refp−1,⇒λ2<0iffRefp<1.Therefore, the disease-free equilibrium point is locally asymptotically stable if and only if Refp<1, otherwise it is unstable, that is, ifRefp>1.


### 2.3.2. Global Stability of Disease-Free Equilibrium Point of the Model

To verify the global stability of the disease-free equilibrium point of the pneumonia monoinfection model, we have used an adopted method of Castillo-Chavez et al. used by others scholar such as [18, 19].Lemma 4.
If the pneumonia monoinfection model can be written as(10)dYdt=GY,W,dZdt=HY,W,HY,0=0,
whereY∈ℝm be the components of noninfected individuals and W∈ℝn be the components of infected individuals including treated class and Ep0=Y∗,0 denotes the disease-free equilibrium point of dynamical system (2).
Assume(i)
FordY/dt=GY,0, Y∗ is globally asymptotically stable (GAS)(ii)
HY,W=BW−HˇY,W, HˇY,W≥0 for Y,W∈Ω1 where B=DWHY∗,0 is an M-matrix, i.e., the off diagonal elements of B are nonnegative and Ω1 is the region in which the system makes biological sense. Then the fixed point Ep0=Y∗,0 is globally asymptotically stable equilibrium point of the system (2) wheneverRefp<1.Lemma 5.
The disease-free equilibrium pointEp0 of the pneumonia monoinfection model (2) is globally asymptotically stable if Refp<1 and the two sufficient conditions given in Lemma 4 are satisfied.
Proof: here we are applying Lemma5 on the pneumonia monoinfection model (2) and we have gotten the following matrices.
(11)dYdt=GY,W=1−πλ+ϕPV+YR–α2+μSπλ−μ+ϕPV,dWdt=HY,W=α2S−τ2+δ2+μPIτ2PI−β+μTβT−Y+μR,GY,0=1−πλ+ϕPV−μSπλ−μ+ϕPV,and
(12)HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=bλμ1−πμ+ϕ+πϕμ+ϕPI−α2S00,HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=S0−SbPI00.
SinceS≤S0, we have Hˇ1Y,W≥0, thus, the disease-free equilibrium point Ep0 is globally asymptotically stable if Refp<1. Biologically, whenever Refp<1, the only pneumonia infection disease dies out while the total population increases [18].


## 2.3.1. Effective Reproduction Number

The reproduction number is the number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [15–17].Using the next generation matrix method, we have obtained the effective reproduction number of pneumonia-infected-only submodel, which isRefp=λb/μ1−πμ+ϕ+πϕ/μ+ϕτ2+δ2+μ.Theorem 3.
The disease-free equilibrium pointEp0 of the model in system (2) is locally asymptotically stable if the effective reproduction number Refp<1 and is unstable ifRefp>1.
Proof: from the Jacobean matrixJEp0 of the model (2), with respect to S,PV,PI,T,R at the disease-free equilibrium point, we have the following characteristics equation.
(7)–μ−λ1r1−r2−λ2−μ+ϕ−λ3−β+μ−λ4−Y+μ−λ5=0
Wherer1=Refpr2 and r2=τ2+δ2+μ,(8)⇒λ1=–μ,λ3=−μ+ϕ,λ4=−β+μ,λ5=−Y+μ.
Hence, all the parameters are nonnegative, and all the eigenvalues of the corresponding Jacobean matrix are negative. Butλ2=r1−r2=Refpr2−r2=r2Refp−1,(9)⇒λ2=r2Refp−1,⇒λ2<0iffRefp<1.Therefore, the disease-free equilibrium point is locally asymptotically stable if and only if Refp<1, otherwise it is unstable, that is, ifRefp>1.


## 2.3.2. Global Stability of Disease-Free Equilibrium Point of the Model

To verify the global stability of the disease-free equilibrium point of the pneumonia monoinfection model, we have used an adopted method of Castillo-Chavez et al. used by others scholar such as [18, 19].Lemma 4.
If the pneumonia monoinfection model can be written as(10)dYdt=GY,W,dZdt=HY,W,HY,0=0,
whereY∈ℝm be the components of noninfected individuals and W∈ℝn be the components of infected individuals including treated class and Ep0=Y∗,0 denotes the disease-free equilibrium point of dynamical system (2).
Assume(i)
FordY/dt=GY,0, Y∗ is globally asymptotically stable (GAS)(ii)
HY,W=BW−HˇY,W, HˇY,W≥0 for Y,W∈Ω1 where B=DWHY∗,0 is an M-matrix, i.e., the off diagonal elements of B are nonnegative and Ω1 is the region in which the system makes biological sense. Then the fixed point Ep0=Y∗,0 is globally asymptotically stable equilibrium point of the system (2) wheneverRefp<1.Lemma 5.
The disease-free equilibrium pointEp0 of the pneumonia monoinfection model (2) is globally asymptotically stable if Refp<1 and the two sufficient conditions given in Lemma 4 are satisfied.
Proof: here we are applying Lemma5 on the pneumonia monoinfection model (2) and we have gotten the following matrices.
(11)dYdt=GY,W=1−πλ+ϕPV+YR–α2+μSπλ−μ+ϕPV,dWdt=HY,W=α2S−τ2+δ2+μPIτ2PI−β+μTβT−Y+μR,GY,0=1−πλ+ϕPV−μSπλ−μ+ϕPV,and
(12)HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=bλμ1−πμ+ϕ+πϕμ+ϕPI−α2S00,HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=S0−SbPI00.
SinceS≤S0, we have Hˇ1Y,W≥0, thus, the disease-free equilibrium point Ep0 is globally asymptotically stable if Refp<1. Biologically, whenever Refp<1, the only pneumonia infection disease dies out while the total population increases [18].


## 2.4. The Existence and Stability of Endemic Equilibrium Point

The endemic equilibrium point of the dynamical system of (2) is obtained by making the right side of the system equal to zero, providing that PI≠0. We have supposed that the endemic equilibrium point of the model is denoted by Ep∗=S∗,PI∗,Pv∗,T∗,R∗ and the corresponding force of infection isα2t=bPI∗t.For simplification of algebraic manipulation, we have assumed the parameters in the model by another variable as follows,k1=1−πλ,k2=τ2+μ,k3=τ2+δ2+μ,k4=πλ,k5=ϕ+μ,k6=β+μ,k7=Y+μ,k8=γβτ2,k9=k3k6k7, k10=k1+ϕk4/k5, k11=k9k10, k12=k2k9k13=k3k12 and k14=k8k3. Now the equation of force of infection can be rearranged as
(13)α2∗k14α2∗+bk11−k13=0⇒α2∗=0ork14α2∗+bk11−k13=0,⇒α2∗=k13−bk11/k14 but α2∗=k13−bk11/k14=β+μY+μτ2+δ2+μτ2+μϕ+μ−bλϕ+μ1−π−ϕπ/γβτ2ϕ+μ=β+μY+μτ2+δ2+μτ2+μϕ+μ/γβτ2ϕ+μ−λbϕ+μ1−π+ϕπ/γβτ2ϕ+μ,α2∗=β+μY+μτ2+δ2+μτ2+μϕ+μ/γβτ2λb/μ1−πμ+ϕ+πϕ/μ+ϕτ2+δ2+μ−1⇒α2∗=β+μY+μτ2+δ2+μτ2+μϕ+μ/γβτ2Rp−1⇒α2∗>0 if Refp>1.Therefore, there is a unique endemic equilibrium point for pneumonia monoinfected model as given byEp∗=S∗,PI∗,Pv∗,T∗,R∗where
(14)S∗=τ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗,PI∗=τ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗α2∗τ2+δ2+μ,Pv∗=πλϕ+μ,T∗=α2∗τ2τ2+δ2+μβ+μτ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗,R∗=τ2+δ2+μβ+μY+μϕ+μ−ϕ+μ1−πλ−ϕπλϕ+μτ2+μτ2+δ2+μβ+μY+μ−γβτ2α2∗βτ2α2∗τ2+δ2+μβ+μY+μ.Theorem 6.
The endemic equilibrium point of system (2) Ep∗=S∗,PI∗,Pv∗,T∗,R∗ is locally asymptotically stable for the reproduction number Reffp>1.
Proof: to show that the local stability of the endemic equilibrium point, we have used the method of the Jacobian matrix and the Routh Hurwitz stability criteria.
Then the corresponding characteristic equation is obtained from the determinant of(15)A−λBC0DEF−λ00000G−λ000H0I−λ0000JK−λ=0,
whereA=–α2Ip∗+μ, B=−α2S∗, C=ϕ, D=Y, E=α2Ip∗, F=α2S∗−τ2+δ2+μ,
(16)G=−μ+ϕ,H=τ2,I=−β+μ,J=βandk=−Y+μ,⇒a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5=0,
wherea0=1,−I+A+F+G+K=a1, IA+IF+AF+IG+AG+FG+IK+AK+FK+GK=a2, −−B+IAF+IAG+IFG+AFG+IAK+IFK+AFK+IGK+AGK+FGK=a3, −IB−IAFG+DEHJ+BK−IAFK−IAGK−IFGK−AFGK=a4, a5=DEGHJ+IBK−IAFGK.To apply the Routh-Hurwitz stability criteria, it is obligatory to check if the necessary condition of all the coefficients have the same sign or not. Sincea0=1 is positive in sign, all a1, a2,a3, a4, and a5 should be positives in sign. All the coefficients of the characteristic’s polynomial are positives whenever Reffp>1. We have observed that the first column of the Routh Hurwitz array has no sign change, thus the root of the characteristics equation of the dynamical system are negative. Hence, the endemic equilibrium point of the dynamical system is locally asymptotically stable.


## 2.5. Positivity of Solutions and Invariant Region of the Only Meningitis-Infected Model

We have madePI=MPI=0 from the full pneumonia and meningitis coinfection model to obtain this submeningitis-only model, and got the following dynamical system.
(17)dSdt=1−πλ+φMV+YR–α1+μS,dMIdt=α1S–τ1+δ1+μMI,dMVdt=πλ−μ+φMV,dTdt=τ1MI−β+μT,dRdt=βT−Y+μR.The above dynamical systems are needed to be epidemiologically meaningful as well as well-posed. To prove that, we have intimated that all the state variables of dynamical systems are nonnegative.Theorem 7.
All the populations of the system with positive initial conditions are positive.
Proof: assumeS0>0,MI0>0,MV0>0,T0>0 and R0>0 are positive for time t>0 and for all nonnegative. First let us take T=supt>0suchthatSt′>0,MIt′>0,MVt′>0,Tt′>0andRt′>0,t′∈0,t.
From the first equation of system (17), we do have
(18)dSdt=1−πλ+φMV+YR–α1+μS,St=S0e−∫0Tα1+μdt′+e−∫0Tα1+μdt′∫0Te∫0Tα1+μdt′1−πλ+YR+φMVdt>0,⇒St>0.
Therefore,St is positive. Subsequent to the same procedure, the remaining state variables are nonnegative. Therefore, from the stated proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.Theorem 8.
All the populations of the system with positive initial conditions are nonnegative
The total human population of the dynamical system (17) is positively closed in the closed invariant set Ω2=S,MI,MV,T,Rϵℝ+5:N2≤λ/μ. Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [14].
Proof: to get an invariant region, boundedness of solution is obtained as follow.(19)N2=S+MI+MV+T+R,dN2dt=λ−μN2−δ1MI⇒0≤N2≤λμ.
Therefore, the dynamical system that we do have is bounded.


## 2.6. Existence and Stability of Disease-Free Equilibrium Point

The disease-free equilibrium point is obtained by making all the equations in the system equal to zero, provided that providing thatMI=0. Therefore, the disease-free equilibrium point is
(20)Ep0=SO,MVO,MIO,T0,RO=λμ1−πμ+φ+πφμ+φ,πλμ+φ,0,0,0.


### 2.6.1. Effective Reproduction Number

The reproduction number can be defined as a number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [16, 17, 20].To compute the reproduction number, first distinguishing the new infected from all other changes in the host population is mandatory as follows.LetFix:be the rate of appearance of new infected in compartmenti,V+ix:be the rate of transfer of individuals in to compartment i,V−ix:be the rate of transfer of individuals out of compartmenti.And thenVix=V−ix−V+i but F=∂Fi/∂XjXo and V=∂Vi/∂XjXo, where F and V are mxm matrix with m is number of infected compartment. Fv−1 is the next generation matrix, and the spectral radius of next generation matrix is needed for the reproduction number we are seeking for.Thus∂FiX/∂Xj=aS and ∂ViX/∂Xj and FV−1=λa/μ1−πμ+φ+πφ/μ+φτ1+δ1+μTherefore, the effective reproduction number of meningitis monoinfected submodel is(21)Refm=λaμ1−πμ+φ+πφμ+φτ1+δ1+μ.Theorem 9.
The disease-free equilibrium pointEm0 of the model in system (17) is locally asymptotically stable if the effective reproduction number Refm<1, and it is unstable if Refm>1.
Proof:
Using the Jacobean matrixJEp0 of the model (17) with respect to S,MV,MI,T,R at the disease-free equilibrium point, we have the characteristic equation
(22)⇒–μ−λ1t1−t2−λ2−μ+φ−λ3−β+μ−λ4−Y+μ−λ5=0wheret1=Refmt2andt2=τ1+δ1+μ,⇒λ1=–μ,λ3=−μ+φ,λ4=−β+μ,λ5=−Y+μ.
Hence, all the parameters are nonnegative, all the eigenvalues of the corresponding Jacobean matrix are negative other thanλ2.
Forλ2,λ2=t2Reffm−1⇒λ2=t2Reffm−1,(23)⇒λ2<0iffRefm<1.
Therefore, the disease-free equilibrium point of the meningitis monoinfected model is locally asymptotically stable if the effective reproduction numberRefm<1 and is unstable if Refm>1.


### 2.6.2. Global Stability of Disease-Free Equilibrium Point

We utilized the approach developed by Castillo-Chavez et al. and used it to confirm the overall stability of the disease-free equilibrium point of the meningitis monoinfection model [18, 19].Lemma 10.
If the pneumonia monoinfection model can be written as(24)dYdt=GY,W,dZdt=HY,W,HY,0=0,
whereY∈ℝm be the components of noninfected individuals and W∈ℝn be the components of infected individuals including the treated class, and Em0=Y∗,0 denotes the disease-free equilibrium point of the dynamical system (3).
Assume(i)
FordY/dt=GY,0,Y∗ is globally asymptotically stable (GAS)(ii)
HY,W=BW−HˇY,W, HˇY,W≥0 for Y,W∈Ω1 where B=DWHY∗,0 is an M-matrix, i.e., the off diagonal elements of B are nonnegative and Ω2 is the region in which the system makes biological sense
Then the fixed pointEm0=Y∗,0 is globally asymptotically stable equilibrium point of the system (17) whenever Refm<1.Lemma 11.
The disease-free equilibrium pointEm0 of the pneumonia monoinfection model (17) is globally asymptotically stable if Refm<1 and the two sufficient conditions given in Lemma 10 are satisfied.
Proof: here we are applying Lemma11 on the meningitis monoinfection model (17) and we have gotten the following matrices dY/dt=GY,W=1−πλ+φMV+YR–α1+μSπλ−μ+φMV,(25)dWdt=HY,W=aλμ1−πμ+ϕ+πϕμ+ϕ−τ1+δ1+μ00τ1−β+μ00β−Y+μ,GY,0=1−πλ+φMV−μSπλ−μ+φMV.
Here after some steps of calculations, we have determined that(26)HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=S0−SaMI00.
SinceS≤S0, we have Hˇ1Y,W≥0, thus, the disease-free equilibrium point Em0 of model (17) is globally asymptotically stable if Refm<1. Biologically, whenever Refm<1, the meningitis monoinfection disease dies out while the total population increases [18].


## 2.6.1. Effective Reproduction Number

The reproduction number can be defined as a number of secondary cases produced by one infectious individual joining in a completely susceptible population during its infectious period [16, 17, 20].To compute the reproduction number, first distinguishing the new infected from all other changes in the host population is mandatory as follows.LetFix:be the rate of appearance of new infected in compartmenti,V+ix:be the rate of transfer of individuals in to compartment i,V−ix:be the rate of transfer of individuals out of compartmenti.And thenVix=V−ix−V+i but F=∂Fi/∂XjXo and V=∂Vi/∂XjXo, where F and V are mxm matrix with m is number of infected compartment. Fv−1 is the next generation matrix, and the spectral radius of next generation matrix is needed for the reproduction number we are seeking for.Thus∂FiX/∂Xj=aS and ∂ViX/∂Xj and FV−1=λa/μ1−πμ+φ+πφ/μ+φτ1+δ1+μTherefore, the effective reproduction number of meningitis monoinfected submodel is(21)Refm=λaμ1−πμ+φ+πφμ+φτ1+δ1+μ.Theorem 9.
The disease-free equilibrium pointEm0 of the model in system (17) is locally asymptotically stable if the effective reproduction number Refm<1, and it is unstable if Refm>1.
Proof:
Using the Jacobean matrixJEp0 of the model (17) with respect to S,MV,MI,T,R at the disease-free equilibrium point, we have the characteristic equation
(22)⇒–μ−λ1t1−t2−λ2−μ+φ−λ3−β+μ−λ4−Y+μ−λ5=0wheret1=Refmt2andt2=τ1+δ1+μ,⇒λ1=–μ,λ3=−μ+φ,λ4=−β+μ,λ5=−Y+μ.
Hence, all the parameters are nonnegative, all the eigenvalues of the corresponding Jacobean matrix are negative other thanλ2.
Forλ2,λ2=t2Reffm−1⇒λ2=t2Reffm−1,(23)⇒λ2<0iffRefm<1.
Therefore, the disease-free equilibrium point of the meningitis monoinfected model is locally asymptotically stable if the effective reproduction numberRefm<1 and is unstable if Refm>1.


## 2.6.2. Global Stability of Disease-Free Equilibrium Point

We utilized the approach developed by Castillo-Chavez et al. and used it to confirm the overall stability of the disease-free equilibrium point of the meningitis monoinfection model [18, 19].Lemma 10.
If the pneumonia monoinfection model can be written as(24)dYdt=GY,W,dZdt=HY,W,HY,0=0,
whereY∈ℝm be the components of noninfected individuals and W∈ℝn be the components of infected individuals including the treated class, and Em0=Y∗,0 denotes the disease-free equilibrium point of the dynamical system (3).
Assume(i)
FordY/dt=GY,0,Y∗ is globally asymptotically stable (GAS)(ii)
HY,W=BW−HˇY,W, HˇY,W≥0 for Y,W∈Ω1 where B=DWHY∗,0 is an M-matrix, i.e., the off diagonal elements of B are nonnegative and Ω2 is the region in which the system makes biological sense
Then the fixed pointEm0=Y∗,0 is globally asymptotically stable equilibrium point of the system (17) whenever Refm<1.Lemma 11.
The disease-free equilibrium pointEm0 of the pneumonia monoinfection model (17) is globally asymptotically stable if Refm<1 and the two sufficient conditions given in Lemma 10 are satisfied.
Proof: here we are applying Lemma11 on the meningitis monoinfection model (17) and we have gotten the following matrices dY/dt=GY,W=1−πλ+φMV+YR–α1+μSπλ−μ+φMV,(25)dWdt=HY,W=aλμ1−πμ+ϕ+πϕμ+ϕ−τ1+δ1+μ00τ1−β+μ00β−Y+μ,GY,0=1−πλ+φMV−μSπλ−μ+φMV.
Here after some steps of calculations, we have determined that(26)HˇY,W=Hˇ1Y,WHˇ2Y,WHˇ3Y,W=S0−SaMI00.
SinceS≤S0, we have Hˇ1Y,W≥0, thus, the disease-free equilibrium point Em0 of model (17) is globally asymptotically stable if Refm<1. Biologically, whenever Refm<1, the meningitis monoinfection disease dies out while the total population increases [18].


## 2.7. Existence and Stability of Endemic Equilibrium Point

The endemic equilibrium point of the dynamical system of (3) is obtained by making the right side of the system equal to zero, providing that MI≠0. We have supposed that the endemic equilibrium point of the model is denoted by Em∗=S∗,MI∗,Mv∗,T∗,R∗ and the corresponding force of infection is α1t=bMI∗t. For simplification of algebraic manipulation, we have assumed the parameters in the model by another variable as follows.
(27)n1=1−πλ,n2=τ1+μ,n3=τ1+δ1+μ,n4=πλ,n5=μ+φ,n6=β+μ,n7=Y+μ,n8=γβτ1,n9=n3n6n7,n10=n1+φn4/n5,n11=n9n10,n12=n2n9n13=n3n12 and n14=n8n3 . Now the equation of force of infection can be rearranged as
(28)α1∗n14α1∗+an11−n13=0⇒α1∗=0orn14α1∗+an11−n13=0⇒α1∗=n13−an11n14,butα1∗=n13−an11n14=β+μY+μτ1+δ1+μτ1+μφ+μ−aλφ+μ1−π−φπγβτ1φ+μ=β+μY+μτ1+δ1+μτ1+μφ+μγβτ2ϕ+μ−λaφ+μ1−π+φπγβτ1φ+μ,α1∗=β+μY+μτ1+δ1+μτ1+μφ+μγβτ1λaμ1−πφ+μ+πφφ+μτ1+δ1+μ−1,⇒α1∗=β+μY+μτ1+δ1+μτ1+μφ+μγβτ1Reffm−1⇒α1∗>0ifReffm>1.Therefore, the unique endemic equilibrium point for the meningitis monoinfected model is given byEm∗=S∗,MI∗,Mv∗,T∗,R∗, where
(29)S∗=τ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ1+μτ1+δ1+μβ+μY+μ−γβτ1α1∗,MI∗=τ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ1+μτ1+δ1+μβ+μY+μ−γβτ1α1∗α1∗τ1+δ1+μ,Mv∗=πλφ+μ,T∗=α1∗τ1τ1+δ1+μβ+μτ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ1+μτ1+δ1+μβ+μY+μ−γβτ1α1∗,R∗=τ1+δ1+μβ+μY+μφ+μ−φ+μ1−πλ−φπλφ+μτ2+μτ1+δ1+μβ+μY+μ−γβτ2α2∗βτ2α2∗τ1+δ1+μβ+μY+μ.Theorem 12.
The endemic equilibrium point of system (17) Em∗=S∗,MI∗,Mv∗,T∗,R∗ is locally asymptotically stable for the reproduction number Reffm>1.
Proof: To show the local stability of the endemic equilibrium point we have used the method of Jacobian matrix and Routh Hurwitz stability criteria.
From Jacobian matrix, we have obtained the following characteristic equation(30)a−λf−λg−λi−λk−λ−bi−λk−λ−deg−λhj=0
Wherea=–α1Im∗+μ, b=−α1S∗, c=φ, d=Y, e=α1Im∗, f=α1S∗−τ1+δ1+μ,
(31)g=−μ+φ,h=τ1,i=−β+μ,j=βandk=−Y+μ.⇒a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5=0
Wherea0=1, −i+a+f+g+k=a1, ai+if+af+ig+ag+fg+ik+ak+fk+g=a2,−−b+aif+aig+ifg+afg+aik+ifk+afk+gik+agk+fgk=a3, −bi−aifg+dehj+bk−aifk−aigk−ifgk−afgk=a4
a5=aeghj+bik−aifgk.To apply Routh-Hurwitz stability criteria, it is the must to check the necessary condition of all the coefficients have the same sign or not. Sincea0=1 is positive in sign, all a1, a2,a3, a4 and a5 should be positives in sign. All the coefficients of the characteristic’s polynomial are positives wheneverReffp>1.We have observed that the first column of the Routh Hurwitz array has no sign change, thus the root of the characteristics equation of the dynamical systema0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5=0 are negative. Hence, the endemic equilibrium point of the dynamical system is locally asymptotically stable.


## 2.8. Positivity and Boundedness of Full Pneumonia and Meningitis Coinfected Model

The corresponding dynamical system of the full pneumonia and meningitis coinfection model is given in Equation (1).The constructed model is expected to be meaningful epidemiologically as well as well-posed. We need to prove that all the state variables of the dynamical system are positive.Theorem 13.
All the population of the system with positive initial conditions are nonnegative
Proof: AssumeS0>0,MV0>0,PV0>0,MPV0>0, MI0>0,PI0>0,MPI0>0T0>0, and R0>0 are positive for time t>0 and for all nonnegative parameters. Let us define τ=supt>0suchthatSt>0,MVt>0,PVt>0,MPVt>0,MIt>0,PIt>0,MPIt>0,Tt>0,Rt>0andt∈0,t.
Since allSt,MVt,PVt,MPVt,MIt,PIt,MPIt,TtandRt are continuous, we can say for τ>0. If τ=+∞, then positivity holds.
Nevertheless, if0<τ<+∞, then all the state variables are zeros.
From the first equation of system (1) we do have
(32)dSdt=1−πλ+ϕPV+φMV+YR–α1+α2+μSSt=S0e−∫0τα1+α2+μdt+e−∫0τα1+α2+μdt∫0τe∫0τα1+α2+μdt1−πλ+ϕPV+φMV+YRdt>0⇒St>0
Following same procedure, all the remaining state variables are nonnegative.
Therefore, from proof, we can conclude that whenever the initial values of the systems are all nonnegative, then all the solutions of our dynamical system are positive.Theorem 14.
The total human population is assumed to beN and the dynamical system (1) is positively invariant in the closed invariant set Ω=S,MV,PV,MPV,MI,PI,MPI,T,Rϵℝ+9:N≤λ/μ. Furthermore, the system’s nonnegative solutions are all constrained, and it may exhibit the persistence property under any nonnegative initial concentration conditions [24]. Proof: to get an invariant region, which shows boundedness of solution, is obtained as follow.
(33)dNdt=dSdt+dMVdt+dPVdt+dMPVdt+dMIdt+dPIdt+dMPIdt+dTdt+dRdt,(34)⇒dNdt=λ−μN−δ2PI−δ1MI−δ3MPI,(35)⇒Nt≤N0e−μt+λμ⇒0≤N≤λμ.
Therefore, the dynamical system that we have constructed is bounded.


## 2.9. Disease-Free Equilibrium Point and Its Stability

The disease-free equilibrium point of full pneumonia and meningitis coinfection modelEpmO is obtained by making all the right-hand-side of equation in system (1), providing that all the infectious classes are equal to zero.
(36)SO,MV,OPVO,MPV0,MI0,PI0,MPI0,T0,RO=1−πλα1+α2+μ+πλα1+α2+μϕρμ+ϕ+ε2+1−ρφμ+φ+ε1,1−ρπλμ+φ+ε1,ρπλμ+ϕ+ε2,1−ρπλε1μμ+φ+ε1+ρπλε2μμ+ϕ+ε2,0,0,0,0,0


### 2.9.1. Effective Reproduction Number

The reproduction number is the average number of people that become infected because of the entry of one infectious person into a completely susceptible population in the absence of intervention. Moreover, reproduction number is utilized to determine the effect of the control measures and to understand the capability of the spread of the infection to disseminate in the entire community when the control strategies are applied [15, 17, 21].The reproduction number of pneumonia and meningitis confection model denoted byReff, which is manipulated by the Van den Driesch, Pauline, and James Warmouth next generation matrix approach [20], is the largest eigenvalue of the next generation matrix FV−1=∂FiErO/∂xj∂νiErO/∂xj−1, where Fi is the rate of appearance of new infection in compartment i, νi is the transfer of infections from one compartment i to another, and Epm0 is the disease-free equilibrium point.The(37)Fix=α1SOα2SO0000000,νi=π−1λ−ϕPVO−φMVO−YR0+μSOμ+φ+ε1MVO−1−ρπλμ+ϕ+ε2PVO−ρπλμMPV0−ε1MVO−ε2PVOω2α2+τ1+δ1+μMI0ω1α1+τ2+δ2+μPI0τ3+μMPI0−ω1α1PI−ω2α2MI0β+μT−τ1MI0−τ2PI−τ3MPI0Y+μR0−βT0.Then(38)F=1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1a0a0bb000,and
(39)V=τ1+δ1+μ000τ2+δ2+μ000τ3+μ⇒FV−1=aτ1+δ1+μ0aτ3+μ0bτ2+δ2+μbτ3+μ000.The eigenvalues of the next generation matrixF.V−1are
(40)0,aλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ.Therefore, the effective reproduction number of full meningitis and pneumonia model is(41)Reff=maxaλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ,Reff=maxR1,R2 where R1=aλ1−π/τ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2/μ+ϕ+ε2μπλa/τ1+δ1+μ, and R2=bλ1−π/τ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2/μ+ϕ+ε2μπλb/τ2+δ2+μ.Theorem 15.
The disease-free equilibrium pointEpmO of the model in system (17) is locally asymptotically stable if the effective reproduction number Reff<1 and is unstable ifReff>1.
Proof:
The Jacobean matrixJEpmO of the model (1) with respect to S,MV,PV,MPV,MI,PI,MPI,T,R at the disease-free equilibrium point is the following:
(42)JEpmO=–μφϕ0r1r2r30Y0r4000000000r50000000ε1ε2−μ000000000r6000000000r7000000000r8000000τ1τ2τ3r900000000βr10,
where(43)r1=−a1−πλ/μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r2=−b1−πλ/α1+α2+μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r3=−aℯ1+bℯ21−πλ/μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r4=−μ+φ+ε1,r5=−μ+ϕ+ε2,r6=a1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1–τ1+δ1+μ,r7=b1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1−τ2+δ2+μ,r8=−τ3+μ,r9=−β+μandr10=−Y+μ⇒λ1+μλ4+μ−λ2+r4−λ3+r5−λ5+r6−λ6+r7−λ7+r8−λ8+r9−λ9+r10⇒λ1=−μ,λ4=−μ,λ2=r4=−μ+φ+ε1,λ3=r5=−μ+ϕ+ε2,λ7=r8=−τ3+μ,λ8=r9=−β+μ,λ5=τ1+δ1+μaλμ+φμ1−πμ+φ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1aλ1−πμ+φ+πφμμ+φτ1+δ1+μ−1⇒λ5=τ1+δ1+μaλμ+φμ1−πμ+φ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1Reffm−1.λ6=τ2+δ2+μbλμ+ϕμ1−πμ+ϕ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1λbμ1−πμ+ϕ+πϕμ+ϕτ2+δ2+μ−1⇒λ6=τ2+δ2+μbλμ+ϕμ1−πμ+ϕ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1Reffp−1andλ9=r10=−Y+μ.Hence, all the parameters are nonnegative as well as all the eigenvalues of the corresponding Jacobean matrix are negative exceptλ5 andλ6.The sign of eigenvaluesλ5 and λ6 depends on the values of Reffm and Reffp, respectively. Moreover, λ5<0 whenever Reffm<1 and λ6<0 if and only if Reffp<1.Therefore, the disease-free equilibrium point of the full meningitis and pneumonia coinfection model is local asymptotically stable only ifReffm<1 and Reffp<1, otherwise it is unstable.


### 2.9.2. Global Stability of Disease-Free Equilibrium Point

The disease-free equilibriums of the meningitis monoinfection model and the pneumonia monoinfection model are both globally asymptotically stable whenever their corresponding effective reproduction numbers values are smaller than unity, as demonstrated in Section3 Figure 2 and 3, respectively. In light of this conclusion, the disease-free equilibrium point of the pneumonia and meningitis coinfection model (1) is globally asymptotically stable if Reff=maxR1,R2<1.Figure 2
Sensitivity indices forR2.Figure 3
Stability of endemic equilibrium whenReff=maxR1,R2>1.


## 2.9.1. Effective Reproduction Number

The reproduction number is the average number of people that become infected because of the entry of one infectious person into a completely susceptible population in the absence of intervention. Moreover, reproduction number is utilized to determine the effect of the control measures and to understand the capability of the spread of the infection to disseminate in the entire community when the control strategies are applied [15, 17, 21].The reproduction number of pneumonia and meningitis confection model denoted byReff, which is manipulated by the Van den Driesch, Pauline, and James Warmouth next generation matrix approach [20], is the largest eigenvalue of the next generation matrix FV−1=∂FiErO/∂xj∂νiErO/∂xj−1, where Fi is the rate of appearance of new infection in compartment i, νi is the transfer of infections from one compartment i to another, and Epm0 is the disease-free equilibrium point.The(37)Fix=α1SOα2SO0000000,νi=π−1λ−ϕPVO−φMVO−YR0+μSOμ+φ+ε1MVO−1−ρπλμ+ϕ+ε2PVO−ρπλμMPV0−ε1MVO−ε2PVOω2α2+τ1+δ1+μMI0ω1α1+τ2+δ2+μPI0τ3+μMPI0−ω1α1PI−ω2α2MI0β+μT−τ1MI0−τ2PI−τ3MPI0Y+μR0−βT0.Then(38)F=1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1a0a0bb000,and
(39)V=τ1+δ1+μ000τ2+δ2+μ000τ3+μ⇒FV−1=aτ1+δ1+μ0aτ3+μ0bτ2+δ2+μbτ3+μ000.The eigenvalues of the next generation matrixF.V−1are
(40)0,aλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ.Therefore, the effective reproduction number of full meningitis and pneumonia model is(41)Reff=maxaλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ,Reff=maxR1,R2 where R1=aλ1−π/τ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2/μ+ϕ+ε2μπλa/τ1+δ1+μ, and R2=bλ1−π/τ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2/μ+ϕ+ε2μπλb/τ2+δ2+μ.Theorem 15.
The disease-free equilibrium pointEpmO of the model in system (17) is locally asymptotically stable if the effective reproduction number Reff<1 and is unstable ifReff>1.
Proof:
The Jacobean matrixJEpmO of the model (1) with respect to S,MV,PV,MPV,MI,PI,MPI,T,R at the disease-free equilibrium point is the following:
(42)JEpmO=–μφϕ0r1r2r30Y0r4000000000r50000000ε1ε2−μ000000000r6000000000r7000000000r8000000τ1τ2τ3r900000000βr10,
where(43)r1=−a1−πλ/μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r2=−b1−πλ/α1+α2+μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r3=−aℯ1+bℯ21−πλ/μ+πλ/μϕρ/μ+ϕ+ε2+1−ρφ/μ+φ+ε1,r4=−μ+φ+ε1,r5=−μ+ϕ+ε2,r6=a1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1–τ1+δ1+μ,r7=b1−πλμ+πλμϕρμ+ϕ+ε2+1−ρφμ+φ+ε1−τ2+δ2+μ,r8=−τ3+μ,r9=−β+μandr10=−Y+μ⇒λ1+μλ4+μ−λ2+r4−λ3+r5−λ5+r6−λ6+r7−λ7+r8−λ8+r9−λ9+r10⇒λ1=−μ,λ4=−μ,λ2=r4=−μ+φ+ε1,λ3=r5=−μ+ϕ+ε2,λ7=r8=−τ3+μ,λ8=r9=−β+μ,λ5=τ1+δ1+μaλμ+φμ1−πμ+φ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1aλ1−πμ+φ+πφμμ+φτ1+δ1+μ−1⇒λ5=τ1+δ1+μaλμ+φμ1−πμ+φ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1Reffm−1.λ6=τ2+δ2+μbλμ+ϕμ1−πμ+ϕ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1λbμ1−πμ+ϕ+πϕμ+ϕτ2+δ2+μ−1⇒λ6=τ2+δ2+μbλμ+ϕμ1−πμ+ϕ+πφμ+ϕ+ε2+πϕρμ+ϕ+ε2+1−ρπφμ+φ+ε1Reffp−1andλ9=r10=−Y+μ.Hence, all the parameters are nonnegative as well as all the eigenvalues of the corresponding Jacobean matrix are negative exceptλ5 andλ6.The sign of eigenvaluesλ5 and λ6 depends on the values of Reffm and Reffp, respectively. Moreover, λ5<0 whenever Reffm<1 and λ6<0 if and only if Reffp<1.Therefore, the disease-free equilibrium point of the full meningitis and pneumonia coinfection model is local asymptotically stable only ifReffm<1 and Reffp<1, otherwise it is unstable.


## 2.9.2. Global Stability of Disease-Free Equilibrium Point

The disease-free equilibriums of the meningitis monoinfection model and the pneumonia monoinfection model are both globally asymptotically stable whenever their corresponding effective reproduction numbers values are smaller than unity, as demonstrated in Section3 Figure 2 and 3, respectively. In light of this conclusion, the disease-free equilibrium point of the pneumonia and meningitis coinfection model (1) is globally asymptotically stable if Reff=maxR1,R2<1.Figure 2
Sensitivity indices forR2.Figure 3
Stability of endemic equilibrium whenReff=maxR1,R2>1.


## 2.10. Endemic Equilibrium Point and Its Stability

The endemic equilibrium point of the dynamical system (1) is obtained by making the right side of the system equal to zero providing that MI≠0, PI≠0 and MPI≠0. We have supposed the endemic equilibrium point of the model is denoted by Epm∗=S∗,Mv∗,Pv∗,MPv∗,MI∗,PI∗,MPI∗,T∗,R∗ and the corresponding forces of infection are
(44)α1∗=aMI∗+ℯ1MPI∗andα2∗=bPI∗+ℯ2MPI∗,S∗=1α1∗+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,Mv∗=1−ρπλμ+φ+ε1,Pv∗=ρπλμ+ϕ+ε2,MPv∗=ε1μ1−ρπλμ+φ+ε1+ε2μρπλμ+ϕ+ε2,MI∗=α1ω2α2∗+τ1+δ1+μ1α1∗+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,PI∗=α2ω1α1∗+τ2+δ2+μ1α1+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,MPI∗=1τ3+μω1α1∗α2ω1α1∗+τ2+δ2+μ1α1∗+α2∗+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ+α1ω2α2∗ω2α1∗+τ1+δ1+μα1∗+α2∗+μτ3+μ1−πλ+ϕρπλμ+ϕ+ε2+1−ρφπλμ+φ+ε1+YβT∗Y+μ,T∗=τ1MI∗+τ2PI∗+τ3MPI∗β+μ,R∗=βT∗Y+μ.Theorem 16.
The endemic equilibrium pointEpm∗ is locally asymptotically stable if the Reff>1, otherwise it is unstable.
Proof: the local stability of the endemic equilibrium point of the full model is verified by using a numerical simulation in Section4 Figure 3.


## 3. Sensitivity Analysis and Numerical Simulations

In this section, we have carried out the sensitivity analysis to find the possible sensitive parameters having important implications to prevent and control the meningitis and pneumonia coinfection spread and the numerical simulations of model parameters and model solutions to approve the analytical results that we have done in Section3.In the numerical simulation of the meningitis and pneumonia coinfection model, we assessed the possible impact of controlling strategies on the dynamics of the disease.


### 3.1. Sensitivity Analysis

Definition. the normalized forward sensitivity index of a variable meningitis and pneumonia reproduction number denoted by the symbol Reff that depends differentially on a parameter ξ is defined as SIp=∂Reff/∂ξ∗ξ/Reff [2, 17].Conducting sensitivity analysis provides a number of benefits for decision-makers. First, it acts as an in-depth study of all the variables. Secondly, it allows decision-makers to identify where they can make improvements in the future. In our case, it helps us to determine the relative and importance of different parameters in meningitis and pneumonia incidence and prevalence. The most sensitive parameter has the magnitude of the sensitivity index larger than that of all other parameters. We have manipulated the sensitivity index in terms of(45)Reff=maxaλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ.We could have manipulated the sensitivity index in terms ofR1 and R2 since Reff=maxR1,R2.Sensitivity analysis results and the numerical simulation are given in this section with parameter values given in Table2 with the initial population of the full meningitis and pneumonia coinfection model.Table 2
Parameter values used for the coinfection model simulation.
ParameterValuesUnitSourcea0.9size−1∗Time−1[8]b0.007-0.6size−1∗Time−1[8]τ10.02Time−1[8]τ20.012Time−1[8]τ30.31Time−1[8]β0.06 -0.13Time−1[16, 17]μ0.01Time−1[8]δ10.002-0.2Time−1[8]δ20.006-05Time−1[8]δ30.008-0.7Time−1[8]ω11Time−1Assumedω21Time−1AssumedY0.007-0.4Time−1[8]π0.105Time−1[8]λ0.0413∗N0Size∗Time−1[8]ε20.007Time−1Assumedε10.025Time−1Assumedϕ0.0115Time−1[22]φ0.5Time−1[23]ρ0.05Time−1[21]


#### 3.1.1. The Sensitivity Indices forR1

In this section, we have stated the sensitivity indices forR1. Using the values of the parameters in Table 2, the sensitivity indices for R1 is calculated in the following:
(1)
SIa=∂R1/∂a∗a/R1∂R1/∂a=λ1−π/μμ+δ1+τ1+πλϕρ1+μ+φ+1+μ+ϕφ1−ρ/μ1+μ+ϕμ+δ1+τ1⇒SIa=1(2)
SIδ1=∂R1/∂δ1∗δ1/R1 ⇒∂R1/∂δ1=−aλ1−π/μμ+δ1+τ12−aπλϕρ1+μ+φ+1+μ+ϕφ1−ρ/μ1+μ+ϕμ+δ1+τ12⇒SIδ1=−δ1/μ+δ1+τ1Following the procedures we have generalized and stated the sensitivity parameters as follows:In this section, the obtained figure was explored with parameter values given in Table2, and we have gotten R1=12.33 at a=0.9, which implies meningitis has been expanded throughout the considered community, and additionally, we have manipulated the sensitivity indices for other parameters as show in Figure 4. The obtained sensitivity analysis shows that the recruitment rate λ and meningitis effective contact rate a have the highest impact onR1.Figure 4
Sensitivity indices forR1.


#### 3.1.2. The Sensitivity Indices forR2

In this section we have presented the sensitivity indices forR2. Using the values of parameters in Table 2, the sensitivity indices for R2 is calculated in following chart.
(1)
SIb=∂R2/∂b∗b/R2∂R2/∂b=πλϕρ1+μ+φ+1+μ+ϕλ1−π+πλφ1−ρ/μ1+μ+ϕμ+δ2+τ2⇒SIb=1(2)
SIδ2=∂R2/∂δ2∗δ2/R2⇒∂R2/∂δ2=bπλ1+μρϕ−φ+1+μ+ϕφ+1+μ+ϕλ1−π/μ1+μ+ϕμ+δ2+τ2Following the procedures we have generalized and stated the sensitivity parameters as followIn this section, the obtained figure was explored with parameters values given in Table2 and we have got R2=15.1 at b=0.85 which implies pneumonia has been expanded throughout the considered community and additionally we have manipulated the sensitivity indices for another parameters as show in Figure 2 above. The obtained sensitivity analysis shows that the recruitment rate λ and pneumonia effective contact rate b have the highest impact onR2.


### 3.2. Numerical Simulations

MATAB software is used to ensure the accuracy of the mathematical terminology descriptions and the validity of the analytical solution. To show the verification of the analytical solution we acquired in the previous part, we utilized the MATLAB code ODE 45. Additionally, we have shown and investigated the effects of a number of traits (parameters) that are related to meningitis and pneumonia coinfection illness.


#### 3.2.1. Local Stability of the Endemic Equilibrium Point

The parameters from Table2 were used to run a numerical simulation using the ODE 45, which produced the results shown in Figure 3. From Figure 3, we can observe that after a year, the solutions of the meningitis and pneumonia coinfection dynamical system (1) will be approaching to the endemic equilibrium point of the meningitis and pneumonia coinfection depends on the value of Reff=maxR1,R2. More specifically, Figure 3 shows that after a time, the solutions of the meningitis and pneumonia infection transmission dynamics will be converging to its endemic equilibrium point, i.e., the endemic equilibrium point is locally asymptotically stable whenever a=0.9 and Reff=maxR1,R2=15.1>1. This mathematical conclusion illustrates the physical phenomena known as the proliferation and spread of meningitis and pneumonia coinfection regularly happening throughout the population which is confined to a certain location.Figure5 demonstrates that while the Reff=maxR1,R2<1, the solution of the system is converged to the disease-free equilibrium point.Figure 5
model representation whenReff=maxR1,R2 <1


#### 3.2.2. The Impact of Meningitis Treatment Rate on Meningitis and Pneumonia Coinfected Populations

We looked at how the coinfected class was affected byτ3 in this part. We can see from Figure 6 that the prevalence of meningitis and pneumonia coinfection decreases as values of τ3 rise. To manage the disease in the community, public authorities must focus on increasing the values of the treatment rateτ3.Figure 6
The effect of treatment rate on the coinfected population.


#### 3.2.3. The Impact of Meningitis Treatment Rate on Coinfected Class

The endeavor we conducted about the impact ofτ1 on the coinfected class is illustrated in Figure 7. The figure shows that the population with coinfections of meningitis and pneumonia is decreasing as the values of the meningitis treatment rate, indicated by τ1, rise. The most crucial thing to remember is that treating just meningitis-infected people can significantly reduce the cooccurrence of meningitis and pneumonia infections in communities, which is one of the study’s main findings.Figure 7
Plot effect of meningitis treatment rate on the coinfected groups.As a result, we advise the stakeholders to maximize the meningitis treatment rate in order to confine and stop the spread of the meningitis and pneumonia coinfection illness in the community.


#### 3.2.4. The Impact of Pneumonia Treatment Rate on Coinfected Class

The influence of pneumonia treatment rate (τ2) on the coinfected class is shown in Figure 8.Figure 8
Plot effect of pneumonia treatment rate on the coinfected class.The graph shows that when the incidence of pneumonia treatment rises, the population with coinfections of meningitis and pneumonia declines. This study’s other key observation is that treating pneumonia-infected people only has a substantial impact on reducing the cooccurrence of meningitis and pneumonia infections in communities, which is perhaps most relevant. In order to prevent and slow the spread of the meningitis and pneumonia coinfection illness in the community, we advise the stakeholders to maximize the values of the pneumonia treatment rate ofτ2.


#### 3.2.5. The Impact of Parameters onReff=maxR1,R2

In this subsection, as we see in Figure9, we have investigated the effect of vaccination ε2 and the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ε2 increases, both R1 and R2 decrease, which implies that the maximum of them also declines.Figure 9
Effect of vaccinationε2 on reproduction numbers.Moreover, the valueReff=maxR1,R2 becomes smaller than one when the value of ε2>0.39.In this subsection, as we can see in Figure10, we have investigated the effect of vaccination ε1 and the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ε1 increases, both R1 and R2 decrease, which implies that the maximum of them also declines.Figure 10
Effect of vaccinationε1 on reproduction numbers.Moreover, the valueReff=maxR1,R2 becomes smaller than one when the value of ε1>0.258.Figure11 elaborates the investigation of effect contact rates a and b on Reff=maxR1,R2.The figure represents that as the values of aandb increase, both R1 and R2 increase. To have the minimum values of Reff=maxR1,R2, the value of a and b should be less than 0.19. As a result, we urge all relevant parties to pay close attention to the effectiveness of the pneumonia vaccine in preventing the coinfection of meningitis and pneumonia in society.Figure 11
Effect of effective contact rate on reproduction numbers.In this subsection, as we see in Figure12, we have investigated the effect of a portion of vaccination of pneumonia protection portion ρ on the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ρ increases, both R1 and R2 decrease, which implies that the maximum of them also declines. Moreover, when the value of ρ>0.629, the value of R2 is less than one; when the value of ρ>0.831, the value of R1 is less than one.Figure 12
Effect of a portion of pneumonia vaccination on reproduction numbers.This suggests that if and only if the percentage of pneumonia vaccination rate is higher than 0.831, thenReff=maxR1,R2<1. Therefore, to avoid and regulate the dynamic transmission of meningitis and pneumonia coinfection, public authorities must look to increase the gains of the pneumonia vaccination portion.Figure13 elaborates the investigation of the effect of pneumonia vaccination wanes (ϕ) on Reff=maxR1,R2.The figure represents as the value of ϕ increases, both R1 and R2 increase. To have the minimum values of Reff=maxR1,R2, the value ϕ should be less than 0.4. As a result, we urge all relevant parties to pay close attention to the effectiveness of the pneumonia vaccine in preventing the coinfection of meningitis and pneumonia in society.Figure 13
Effect of pneumonia vaccination wanes rate on reproduction numbers.Figure14 explains the investigation of the effect of meningitis vaccination wanes φ on Reff=maxR1,R2. The graph illustrates how both R1 and R2 grow when the value of φ rises. Our conclusion from this finding is that when the value of φ is smaller than 0.396, the minimal value of Reff=maxR1,R2 is reached. The effectiveness of the meningitis vaccination in avoiding the coinfection of meningitis and pneumonia in society should thus be closely monitored, thus we encourage all pertinent parties to pay attention for it.Figure 14
Effect of meningitis vaccination wanes rate on reproduction numbers.Analytic solution of some differential equations is complicated and tough. In such case, we need a numerical simulation of the system. In our case we have used a numerical simulation for stability analysis of steady state and obtained the result; after a year, the solutions of the meningitis and pneumonia coinfection dynamical system will tend to the endemic equilibrium point of the model. This expression of the plot illustrates the biological meaning of the proliferation and spread of meningitis and pneumonia coinfection which is regularly happening throughout the population of which is confined to a certain location. Additionally, we have used a numerical simulation to investigate the effects of a number of traits (parameters) that are related to the meningitis and pneumonia coinfection illness, such as, treatment rate, meningitis treatment rate, pneumonia treatment rate, effect of vaccination, and effective contact rate and vaccination wanes rate. As a result, we have shown that increasing availability to meningitis and pneumonia prevention, treatment, and meningitis and pneumonia coinfection vaccine has a significant influence on the frequency of meningitis and pneumonia in a particular population.


## 3.1. Sensitivity Analysis

Definition. the normalized forward sensitivity index of a variable meningitis and pneumonia reproduction number denoted by the symbol Reff that depends differentially on a parameter ξ is defined as SIp=∂Reff/∂ξ∗ξ/Reff [2, 17].Conducting sensitivity analysis provides a number of benefits for decision-makers. First, it acts as an in-depth study of all the variables. Secondly, it allows decision-makers to identify where they can make improvements in the future. In our case, it helps us to determine the relative and importance of different parameters in meningitis and pneumonia incidence and prevalence. The most sensitive parameter has the magnitude of the sensitivity index larger than that of all other parameters. We have manipulated the sensitivity index in terms of(45)Reff=maxaλ1−πτ1+δ1+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλaτ1+δ1+μ,bλ1−πτ2+δ2+μμ+ϕρμ+φ+ε1+φ1−ρμ+ϕ+ε2μ+ϕ+ε2μπλbτ2+δ2+μ.We could have manipulated the sensitivity index in terms ofR1 and R2 since Reff=maxR1,R2.Sensitivity analysis results and the numerical simulation are given in this section with parameter values given in Table2 with the initial population of the full meningitis and pneumonia coinfection model.Table 2
Parameter values used for the coinfection model simulation.
ParameterValuesUnitSourcea0.9size−1∗Time−1[8]b0.007-0.6size−1∗Time−1[8]τ10.02Time−1[8]τ20.012Time−1[8]τ30.31Time−1[8]β0.06 -0.13Time−1[16, 17]μ0.01Time−1[8]δ10.002-0.2Time−1[8]δ20.006-05Time−1[8]δ30.008-0.7Time−1[8]ω11Time−1Assumedω21Time−1AssumedY0.007-0.4Time−1[8]π0.105Time−1[8]λ0.0413∗N0Size∗Time−1[8]ε20.007Time−1Assumedε10.025Time−1Assumedϕ0.0115Time−1[22]φ0.5Time−1[23]ρ0.05Time−1[21]


### 3.1.1. The Sensitivity Indices forR1

In this section, we have stated the sensitivity indices forR1. Using the values of the parameters in Table 2, the sensitivity indices for R1 is calculated in the following:
(1)
SIa=∂R1/∂a∗a/R1∂R1/∂a=λ1−π/μμ+δ1+τ1+πλϕρ1+μ+φ+1+μ+ϕφ1−ρ/μ1+μ+ϕμ+δ1+τ1⇒SIa=1(2)
SIδ1=∂R1/∂δ1∗δ1/R1 ⇒∂R1/∂δ1=−aλ1−π/μμ+δ1+τ12−aπλϕρ1+μ+φ+1+μ+ϕφ1−ρ/μ1+μ+ϕμ+δ1+τ12⇒SIδ1=−δ1/μ+δ1+τ1Following the procedures we have generalized and stated the sensitivity parameters as follows:In this section, the obtained figure was explored with parameter values given in Table2, and we have gotten R1=12.33 at a=0.9, which implies meningitis has been expanded throughout the considered community, and additionally, we have manipulated the sensitivity indices for other parameters as show in Figure 4. The obtained sensitivity analysis shows that the recruitment rate λ and meningitis effective contact rate a have the highest impact onR1.Figure 4
Sensitivity indices forR1.


### 3.1.2. The Sensitivity Indices forR2

In this section we have presented the sensitivity indices forR2. Using the values of parameters in Table 2, the sensitivity indices for R2 is calculated in following chart.
(1)
SIb=∂R2/∂b∗b/R2∂R2/∂b=πλϕρ1+μ+φ+1+μ+ϕλ1−π+πλφ1−ρ/μ1+μ+ϕμ+δ2+τ2⇒SIb=1(2)
SIδ2=∂R2/∂δ2∗δ2/R2⇒∂R2/∂δ2=bπλ1+μρϕ−φ+1+μ+ϕφ+1+μ+ϕλ1−π/μ1+μ+ϕμ+δ2+τ2Following the procedures we have generalized and stated the sensitivity parameters as followIn this section, the obtained figure was explored with parameters values given in Table2 and we have got R2=15.1 at b=0.85 which implies pneumonia has been expanded throughout the considered community and additionally we have manipulated the sensitivity indices for another parameters as show in Figure 2 above. The obtained sensitivity analysis shows that the recruitment rate λ and pneumonia effective contact rate b have the highest impact onR2.


## 3.1.1. The Sensitivity Indices forR1

In this section, we have stated the sensitivity indices forR1. Using the values of the parameters in Table 2, the sensitivity indices for R1 is calculated in the following:
(1)
SIa=∂R1/∂a∗a/R1∂R1/∂a=λ1−π/μμ+δ1+τ1+πλϕρ1+μ+φ+1+μ+ϕφ1−ρ/μ1+μ+ϕμ+δ1+τ1⇒SIa=1(2)
SIδ1=∂R1/∂δ1∗δ1/R1 ⇒∂R1/∂δ1=−aλ1−π/μμ+δ1+τ12−aπλϕρ1+μ+φ+1+μ+ϕφ1−ρ/μ1+μ+ϕμ+δ1+τ12⇒SIδ1=−δ1/μ+δ1+τ1Following the procedures we have generalized and stated the sensitivity parameters as follows:In this section, the obtained figure was explored with parameter values given in Table2, and we have gotten R1=12.33 at a=0.9, which implies meningitis has been expanded throughout the considered community, and additionally, we have manipulated the sensitivity indices for other parameters as show in Figure 4. The obtained sensitivity analysis shows that the recruitment rate λ and meningitis effective contact rate a have the highest impact onR1.Figure 4
Sensitivity indices forR1.


## 3.1.2. The Sensitivity Indices forR2

In this section we have presented the sensitivity indices forR2. Using the values of parameters in Table 2, the sensitivity indices for R2 is calculated in following chart.
(1)
SIb=∂R2/∂b∗b/R2∂R2/∂b=πλϕρ1+μ+φ+1+μ+ϕλ1−π+πλφ1−ρ/μ1+μ+ϕμ+δ2+τ2⇒SIb=1(2)
SIδ2=∂R2/∂δ2∗δ2/R2⇒∂R2/∂δ2=bπλ1+μρϕ−φ+1+μ+ϕφ+1+μ+ϕλ1−π/μ1+μ+ϕμ+δ2+τ2Following the procedures we have generalized and stated the sensitivity parameters as followIn this section, the obtained figure was explored with parameters values given in Table2 and we have got R2=15.1 at b=0.85 which implies pneumonia has been expanded throughout the considered community and additionally we have manipulated the sensitivity indices for another parameters as show in Figure 2 above. The obtained sensitivity analysis shows that the recruitment rate λ and pneumonia effective contact rate b have the highest impact onR2.


## 3.2. Numerical Simulations

MATAB software is used to ensure the accuracy of the mathematical terminology descriptions and the validity of the analytical solution. To show the verification of the analytical solution we acquired in the previous part, we utilized the MATLAB code ODE 45. Additionally, we have shown and investigated the effects of a number of traits (parameters) that are related to meningitis and pneumonia coinfection illness.


### 3.2.1. Local Stability of the Endemic Equilibrium Point

The parameters from Table2 were used to run a numerical simulation using the ODE 45, which produced the results shown in Figure 3. From Figure 3, we can observe that after a year, the solutions of the meningitis and pneumonia coinfection dynamical system (1) will be approaching to the endemic equilibrium point of the meningitis and pneumonia coinfection depends on the value of Reff=maxR1,R2. More specifically, Figure 3 shows that after a time, the solutions of the meningitis and pneumonia infection transmission dynamics will be converging to its endemic equilibrium point, i.e., the endemic equilibrium point is locally asymptotically stable whenever a=0.9 and Reff=maxR1,R2=15.1>1. This mathematical conclusion illustrates the physical phenomena known as the proliferation and spread of meningitis and pneumonia coinfection regularly happening throughout the population which is confined to a certain location.Figure5 demonstrates that while the Reff=maxR1,R2<1, the solution of the system is converged to the disease-free equilibrium point.Figure 5
model representation whenReff=maxR1,R2 <1


### 3.2.2. The Impact of Meningitis Treatment Rate on Meningitis and Pneumonia Coinfected Populations

We looked at how the coinfected class was affected byτ3 in this part. We can see from Figure 6 that the prevalence of meningitis and pneumonia coinfection decreases as values of τ3 rise. To manage the disease in the community, public authorities must focus on increasing the values of the treatment rateτ3.Figure 6
The effect of treatment rate on the coinfected population.


### 3.2.3. The Impact of Meningitis Treatment Rate on Coinfected Class

The endeavor we conducted about the impact ofτ1 on the coinfected class is illustrated in Figure 7. The figure shows that the population with coinfections of meningitis and pneumonia is decreasing as the values of the meningitis treatment rate, indicated by τ1, rise. The most crucial thing to remember is that treating just meningitis-infected people can significantly reduce the cooccurrence of meningitis and pneumonia infections in communities, which is one of the study’s main findings.Figure 7
Plot effect of meningitis treatment rate on the coinfected groups.As a result, we advise the stakeholders to maximize the meningitis treatment rate in order to confine and stop the spread of the meningitis and pneumonia coinfection illness in the community.


### 3.2.4. The Impact of Pneumonia Treatment Rate on Coinfected Class

The influence of pneumonia treatment rate (τ2) on the coinfected class is shown in Figure 8.Figure 8
Plot effect of pneumonia treatment rate on the coinfected class.The graph shows that when the incidence of pneumonia treatment rises, the population with coinfections of meningitis and pneumonia declines. This study’s other key observation is that treating pneumonia-infected people only has a substantial impact on reducing the cooccurrence of meningitis and pneumonia infections in communities, which is perhaps most relevant. In order to prevent and slow the spread of the meningitis and pneumonia coinfection illness in the community, we advise the stakeholders to maximize the values of the pneumonia treatment rate ofτ2.


### 3.2.5. The Impact of Parameters onReff=maxR1,R2

In this subsection, as we see in Figure9, we have investigated the effect of vaccination ε2 and the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ε2 increases, both R1 and R2 decrease, which implies that the maximum of them also declines.Figure 9
Effect of vaccinationε2 on reproduction numbers.Moreover, the valueReff=maxR1,R2 becomes smaller than one when the value of ε2>0.39.In this subsection, as we can see in Figure10, we have investigated the effect of vaccination ε1 and the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ε1 increases, both R1 and R2 decrease, which implies that the maximum of them also declines.Figure 10
Effect of vaccinationε1 on reproduction numbers.Moreover, the valueReff=maxR1,R2 becomes smaller than one when the value of ε1>0.258.Figure11 elaborates the investigation of effect contact rates a and b on Reff=maxR1,R2.The figure represents that as the values of aandb increase, both R1 and R2 increase. To have the minimum values of Reff=maxR1,R2, the value of a and b should be less than 0.19. As a result, we urge all relevant parties to pay close attention to the effectiveness of the pneumonia vaccine in preventing the coinfection of meningitis and pneumonia in society.Figure 11
Effect of effective contact rate on reproduction numbers.In this subsection, as we see in Figure12, we have investigated the effect of a portion of vaccination of pneumonia protection portion ρ on the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ρ increases, both R1 and R2 decrease, which implies that the maximum of them also declines. Moreover, when the value of ρ>0.629, the value of R2 is less than one; when the value of ρ>0.831, the value of R1 is less than one.Figure 12
Effect of a portion of pneumonia vaccination on reproduction numbers.This suggests that if and only if the percentage of pneumonia vaccination rate is higher than 0.831, thenReff=maxR1,R2<1. Therefore, to avoid and regulate the dynamic transmission of meningitis and pneumonia coinfection, public authorities must look to increase the gains of the pneumonia vaccination portion.Figure13 elaborates the investigation of the effect of pneumonia vaccination wanes (ϕ) on Reff=maxR1,R2.The figure represents as the value of ϕ increases, both R1 and R2 increase. To have the minimum values of Reff=maxR1,R2, the value ϕ should be less than 0.4. As a result, we urge all relevant parties to pay close attention to the effectiveness of the pneumonia vaccine in preventing the coinfection of meningitis and pneumonia in society.Figure 13
Effect of pneumonia vaccination wanes rate on reproduction numbers.Figure14 explains the investigation of the effect of meningitis vaccination wanes φ on Reff=maxR1,R2. The graph illustrates how both R1 and R2 grow when the value of φ rises. Our conclusion from this finding is that when the value of φ is smaller than 0.396, the minimal value of Reff=maxR1,R2 is reached. The effectiveness of the meningitis vaccination in avoiding the coinfection of meningitis and pneumonia in society should thus be closely monitored, thus we encourage all pertinent parties to pay attention for it.Figure 14
Effect of meningitis vaccination wanes rate on reproduction numbers.Analytic solution of some differential equations is complicated and tough. In such case, we need a numerical simulation of the system. In our case we have used a numerical simulation for stability analysis of steady state and obtained the result; after a year, the solutions of the meningitis and pneumonia coinfection dynamical system will tend to the endemic equilibrium point of the model. This expression of the plot illustrates the biological meaning of the proliferation and spread of meningitis and pneumonia coinfection which is regularly happening throughout the population of which is confined to a certain location. Additionally, we have used a numerical simulation to investigate the effects of a number of traits (parameters) that are related to the meningitis and pneumonia coinfection illness, such as, treatment rate, meningitis treatment rate, pneumonia treatment rate, effect of vaccination, and effective contact rate and vaccination wanes rate. As a result, we have shown that increasing availability to meningitis and pneumonia prevention, treatment, and meningitis and pneumonia coinfection vaccine has a significant influence on the frequency of meningitis and pneumonia in a particular population.


## 3.2.1. Local Stability of the Endemic Equilibrium Point

The parameters from Table2 were used to run a numerical simulation using the ODE 45, which produced the results shown in Figure 3. From Figure 3, we can observe that after a year, the solutions of the meningitis and pneumonia coinfection dynamical system (1) will be approaching to the endemic equilibrium point of the meningitis and pneumonia coinfection depends on the value of Reff=maxR1,R2. More specifically, Figure 3 shows that after a time, the solutions of the meningitis and pneumonia infection transmission dynamics will be converging to its endemic equilibrium point, i.e., the endemic equilibrium point is locally asymptotically stable whenever a=0.9 and Reff=maxR1,R2=15.1>1. This mathematical conclusion illustrates the physical phenomena known as the proliferation and spread of meningitis and pneumonia coinfection regularly happening throughout the population which is confined to a certain location.Figure5 demonstrates that while the Reff=maxR1,R2<1, the solution of the system is converged to the disease-free equilibrium point.Figure 5
model representation whenReff=maxR1,R2 <1


## 3.2.2. The Impact of Meningitis Treatment Rate on Meningitis and Pneumonia Coinfected Populations

We looked at how the coinfected class was affected byτ3 in this part. We can see from Figure 6 that the prevalence of meningitis and pneumonia coinfection decreases as values of τ3 rise. To manage the disease in the community, public authorities must focus on increasing the values of the treatment rateτ3.Figure 6
The effect of treatment rate on the coinfected population.


## 3.2.3. The Impact of Meningitis Treatment Rate on Coinfected Class

The endeavor we conducted about the impact ofτ1 on the coinfected class is illustrated in Figure 7. The figure shows that the population with coinfections of meningitis and pneumonia is decreasing as the values of the meningitis treatment rate, indicated by τ1, rise. The most crucial thing to remember is that treating just meningitis-infected people can significantly reduce the cooccurrence of meningitis and pneumonia infections in communities, which is one of the study’s main findings.Figure 7
Plot effect of meningitis treatment rate on the coinfected groups.As a result, we advise the stakeholders to maximize the meningitis treatment rate in order to confine and stop the spread of the meningitis and pneumonia coinfection illness in the community.


## 3.2.4. The Impact of Pneumonia Treatment Rate on Coinfected Class

The influence of pneumonia treatment rate (τ2) on the coinfected class is shown in Figure 8.Figure 8
Plot effect of pneumonia treatment rate on the coinfected class.The graph shows that when the incidence of pneumonia treatment rises, the population with coinfections of meningitis and pneumonia declines. This study’s other key observation is that treating pneumonia-infected people only has a substantial impact on reducing the cooccurrence of meningitis and pneumonia infections in communities, which is perhaps most relevant. In order to prevent and slow the spread of the meningitis and pneumonia coinfection illness in the community, we advise the stakeholders to maximize the values of the pneumonia treatment rate ofτ2.


## 3.2.5. The Impact of Parameters onReff=maxR1,R2

In this subsection, as we see in Figure9, we have investigated the effect of vaccination ε2 and the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ε2 increases, both R1 and R2 decrease, which implies that the maximum of them also declines.Figure 9
Effect of vaccinationε2 on reproduction numbers.Moreover, the valueReff=maxR1,R2 becomes smaller than one when the value of ε2>0.39.In this subsection, as we can see in Figure10, we have investigated the effect of vaccination ε1 and the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ε1 increases, both R1 and R2 decrease, which implies that the maximum of them also declines.Figure 10
Effect of vaccinationε1 on reproduction numbers.Moreover, the valueReff=maxR1,R2 becomes smaller than one when the value of ε1>0.258.Figure11 elaborates the investigation of effect contact rates a and b on Reff=maxR1,R2.The figure represents that as the values of aandb increase, both R1 and R2 increase. To have the minimum values of Reff=maxR1,R2, the value of a and b should be less than 0.19. As a result, we urge all relevant parties to pay close attention to the effectiveness of the pneumonia vaccine in preventing the coinfection of meningitis and pneumonia in society.Figure 11
Effect of effective contact rate on reproduction numbers.In this subsection, as we see in Figure12, we have investigated the effect of a portion of vaccination of pneumonia protection portion ρ on the effective reproduction number Reff=maxR1,R2. The figure reflects that when the value of ρ increases, both R1 and R2 decrease, which implies that the maximum of them also declines. Moreover, when the value of ρ>0.629, the value of R2 is less than one; when the value of ρ>0.831, the value of R1 is less than one.Figure 12
Effect of a portion of pneumonia vaccination on reproduction numbers.This suggests that if and only if the percentage of pneumonia vaccination rate is higher than 0.831, thenReff=maxR1,R2<1. Therefore, to avoid and regulate the dynamic transmission of meningitis and pneumonia coinfection, public authorities must look to increase the gains of the pneumonia vaccination portion.Figure13 elaborates the investigation of the effect of pneumonia vaccination wanes (ϕ) on Reff=maxR1,R2.The figure represents as the value of ϕ increases, both R1 and R2 increase. To have the minimum values of Reff=maxR1,R2, the value ϕ should be less than 0.4. As a result, we urge all relevant parties to pay close attention to the effectiveness of the pneumonia vaccine in preventing the coinfection of meningitis and pneumonia in society.Figure 13
Effect of pneumonia vaccination wanes rate on reproduction numbers.Figure14 explains the investigation of the effect of meningitis vaccination wanes φ on Reff=maxR1,R2. The graph illustrates how both R1 and R2 grow when the value of φ rises. Our conclusion from this finding is that when the value of φ is smaller than 0.396, the minimal value of Reff=maxR1,R2 is reached. The effectiveness of the meningitis vaccination in avoiding the coinfection of meningitis and pneumonia in society should thus be closely monitored, thus we encourage all pertinent parties to pay attention for it.Figure 14
Effect of meningitis vaccination wanes rate on reproduction numbers.Analytic solution of some differential equations is complicated and tough. In such case, we need a numerical simulation of the system. In our case we have used a numerical simulation for stability analysis of steady state and obtained the result; after a year, the solutions of the meningitis and pneumonia coinfection dynamical system will tend to the endemic equilibrium point of the model. This expression of the plot illustrates the biological meaning of the proliferation and spread of meningitis and pneumonia coinfection which is regularly happening throughout the population of which is confined to a certain location. Additionally, we have used a numerical simulation to investigate the effects of a number of traits (parameters) that are related to the meningitis and pneumonia coinfection illness, such as, treatment rate, meningitis treatment rate, pneumonia treatment rate, effect of vaccination, and effective contact rate and vaccination wanes rate. As a result, we have shown that increasing availability to meningitis and pneumonia prevention, treatment, and meningitis and pneumonia coinfection vaccine has a significant influence on the frequency of meningitis and pneumonia in a particular population.


## 4. Discussion

We addressed the epidemiology and historical context of meningitis and pneumonia in Section1.In Section2, the deterministic meningitis and pneumonia coinfection dynamical system was built by dividing the entire human population into nine groups based on the severity of the infection. Moreover, we looked at the model’s descriptive phenomena, including the model’s prospective solutions being positive, the dynamical system being bounded, the existence of a disease-free equilibrium point, the existence of an effective reproduction number using the next generation matrix technique, the existence of endemic equilibriums, and the stability analysis of the disease-free equilibrium point using the Routh-Hurwitz criteria in this section. Additionally, the Routh-Hurwitz criteria were used to demonstrate the local stability of endemic equilibrium points of submodels, and we were able to conclude that these points are locally asymptotically stable if the reproduction number is less than one. Nevertheless, the endemic equilibrium point of the combined meningitis and pneumonia coinfection model is revealed by numerical simulation presented in Section 3 by Figure 3, i.e., the endemic equilibrium point is locally asymptotically stable just when a=0.9, b=0.85, andReff=maxR1=12.33,R2=15.1>1.Through the use of numerical simulation, we have examined the effects of several parameters on the effective reproduction rate as well as the prevention and control of the development of meningitis and pneumonia infections in Section3. The results show that increasing availability to meningitis and pneumonia prevention, treatment, and meningitis and pneumonia coinfection vaccine has a significant influence on the frequency of meningitis and pneumonia in a particular population. The optimal control analysis of pneumonia and meningitis coinfection [8] having solely evaluated prevention for both meningitis and pneumonia, we found that preventative regulation had a significant influence on reducing the spread of meningitis, pneumonia, and their coinfection within the given time frame. Efforts to avoid pneumonia and treat meningitis are two more techniques addressed in this study. They discovered that each of the stated measures is successful in reducing the growth of the infectious populations that only have meningitis, pneumonia, or both throughout the allotted time frame.A mathematical model of seven nonlinear differential equations for the pneumonia and meningitis coinfection with PCV vaccination for a newly born population and treatment for coinfected class has also been developed and entitled as “mathematical model analysis and numerical simulation for codynamics of meningitis and pneumonia infection with intervention” [5]. They have taken into account the PCV13 (pneumococcal conjugate vaccine), which is a vaccination given to infants and protects against a variety of pneumococcal bacteria that can cause the most severe forms of pneumococcal illness, such as pneumonia and meningitis. Additionally, researchers [7] have demonstrated that the availability of hospital beds, medicines, and other treatment options all contribute to a decline in the number of cases of bacterial meningitis. However, they observe that an increased recruitment rate causes a backward split in a town with few hospital beds.Finally, they advise using vaccination when there is a large influx of new people. Numerous studies have demonstrated that meningitis and pneumonia infections may be wiped out in the community when treatment and immunization efforts are combined. Despite treatment interventions, these diseases and their combined forms continue to exist in the entire population. Three different vaccine combinations with similar treatments were not taken into account in any model across all studies. In contrast to the previous studies described above, the model developed in this study took into consideration all three vaccines, namely the meningitis vaccine, the pneumonia vaccine, and people who had received both the vaccine and therapy at the same time.


## 5. Conclusion

The study is aimed at building and examining a compartmental deterministic mathematical model of the dynamics of the spread of the coinfection of meningitis and pneumonia. In the population under the study, this model considered how vaccination and therapy of single infection of meningitis or pneumonia may reduce the risk of meningitis and pneumonia coinfection. When the effective reproduction number is smaller than unity, the model possesses a disease-free equilibrium point that is locally asymptotically stable. The outcome of the numerical simulation demonstrates that whenReff>1, the meningitis and pneumonia coinfection model’s endemic equilibrium point is locally asymptotically stable.Besides obtaining the model’s matching effective reproduction numberReff, we were able to identify the effects of changing specific parameter values and provide future guidance for the public’s partners. Our findings have significant public health implications because they affect whether a disease is eradicated or persists in the community under investigation.As a result of our numerical findings which show thatReff=15.1 at b=0.85 and a=0.9, we notify public authorities to focus on raising or maximizing the values of the involved individuals’ meningitis vaccination, pneumonia vaccination, and meningitis and pneumonia coinfection vaccination as well as treatment rates in order to reduce and eradicate the disease from the study’s target community. Finally, other key results of this study are meningitis contact rates, pneumonia contact rates, and vaccination deficits of corresponding diseases that have contributed immensely to the spread of meningitis and pneumonia coinfection in the community.



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*Source: 9945047-2022-11-28.xml*