# Optimal Control Analysis of Pneumonia and Meningitis Coinfection

Authors: Getachew Teshome Tilahun
Journal: Computational and Mathematical Methods in Medicine
 (2019)

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## Abstract

In this paper, we proposed a deterministic model of pneumonia-meningitis coinfection. We used a system of seven ordinary differential equations. Firstly, the qualitative behaviours of the model such as positivity of the solution, existence of the solution, the equilibrium points, basic reproduction number, analysis of equilibrium points, and sensitivity analysis are studied. The disease-free equilibrium is locally asymptotically stable if the basic reproduction number is kept less than unity, and conditions for global stability are established. Then, the basic model is extended to optimal control by incorporating four control interventions, such as prevention of pneumonia as well as meningitis and also treatment of pneumonia and meningitis diseases. The optimality system is obtained by using Pontryagin’s maximum principle. For simulation of the optimality system, we proposed five strategies to check the effect of the controls. First, we consider prevention only for both diseases, and the result shows that applying prevention control has a great impact in bringing down the expansion of pneumonia, meningitis, and their coinfection in the specified period of time. The other strategies are prevention effort for pneumonia and treatment effort for meningitis, prevention effort for meningitis and treatment effort for pneumonia, treatment effort for both diseases, and using all interventions. We obtained that each of the listed strategies is effective in minimizing the expansion of pneumonia-only, meningitis-only, and coinfectious population in the specified period of time.


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## Body

## 1. Introduction

Pneumonia, which can be categorized as one of the airborne diseases, claims for the death of millions of human beings through inhaling pathogenic organism, mainlyStreptococcus pneumoniae [1]. These bacteria are also responsible for the cause of other diseases, such as meningitis, ear infections, and sinus infections. Pneumonia can affect human beings of all ages, from children to the elderly, and it becomes dangerous when the immunity level is lowered, as well as when it is coinfected with other diseases like meningitis [2]. Meningitis, an infection which covers the brain and spinal cord, is caused by both bacteria and virus. Bacterial infection of meningitis is the most common one, particularly, Streptococcus pneumoniae, Haemophilus influenzae, and Neisseria meningitidis are responsible for 80% cause of meningitis [3]. To control these diseases, a lot of scholars proposed different methods. In this aspect, mathematical models played a great role in proposing controlling strategies. Several scholars proposed different models to describe the dynamics of infectious diseases in the community. Some of them [4–8] proposed a mathematical model of pneumonia only, and the others [9–11] proposed a mathematical model of meningitis only. Few scholars like Tilahun et al. [12] proposed a mathematical model of pneumonia and typhoid fever coinfection using optimal control strategies. Moreover, Onyinge et al. and Akinyi et al. [13, 14] developed a mathematical model for coinfection of pneumonia with malaria and HIV. More recently, Tilahun [15] proposed a mathematical model of pneumonia and meningitis and investigated their coinfection using an SIR approach. However, to the best of our knowledge, no one has proposed a mathematical model by incorporating optimal control strategies for coinfection of pneumonia and meningitis. Therefore, this work is devoted in fulfilling this gap.This paper is organized as follows. Section2 presents the description of the model. Qualitative behaviour of the model is discussed in Section 3. In Section 4, the basic model is extended to optimal control analysis. In Section 5, numerical simulation of the optimality system is presented. A brief discussion and conclusion are presented in Section 6.


## 2. Description of the Model

In this section, a deterministic mathematical model of pneumonia-meningitis coinfection is presented. The model is proposed using seven compartments with total population size denoted byNt. The compartment that has individuals who are healthy but able to be infected is denoted by St. Individuals that are affected by pneumonia and can transmit the disease to others are denoted by Ipt. Similarly, meningitis-infected individuals’ compartment is denoted by Imt, and coinfectious individuals’ compartment is represented by Ipmt. Additionally, recovered /removed compartments from pneumonia, meningitis, and coinfection of both diseases are denoted by Rpt,Rmt, and Rpmt, respectively. Then, the total population is Nt=St+Ipt+Imt+Ipmt+Rpt+Rmt+Rpmt. Susceptible compartment increase by recruitment rate of π and also from Pneumonia recovered compartment with rate of δ1, meningitis recovered compartment with rate of δ2 and from co-infectious recovered compartment with rate of δ3. Force of infection of pneumonia and meningitis is f1=aIpt+Ipmt/N and f2=bImt+Ipmt/N, respectively, where a is the contact rate of pneumonia and b is the contact rate of meningitis. Pneumonia-only recovered compartment is increased due to the recovery rate of pneumonia denoted by σ1, and meningitis-only and coinfectious recovered compartments increase their number with a rate of recovery of σ2 and σ, respectively. In the coinfectious recovered/removed compartment, individuals either recovered only from pneumonia, meningitis, or from both diseases with a probability of σ1−e,σg1−e or σ1−g1−e, respectively, where sigma, e, and g are any number between zero and one. The natural death rate is denoted by μ and pneumonia-causing death rate and meningitis-causing death rate are represented by α1 and α2, respectively. All parameters described in this model are assumed as nonnegative. The above description of the model is plotted in Figure 1.Figure 1
Flow diagram of the model.From the flow diagram (Figure1) of the model, the following system of differential equations is obtained:(1)dStdt=π+δ1Rpt+δ2Rmt+δ3Rpmt−f1+f2+μSt,dIptdt=f1St−f2+σ1+α1+μIpt,dImtdt=f2St−f1+σ2+α2+μImt,dIpmtdt=f2Ipt+f1Imt−σ+α1+α2+μIpmt,dRptdt=σ1Ipt+σeIpmt−δ1+μRpt,dRmtdt=σ2Imt+σg1−eIpmt−δ2+μRmt,dRpmtdt=σ1−g1−eIpmt−δ3+μRpmt.Initial conditions of system (1) are(2)S0≥0,Ip0≥0,Im0≥0,Ipm0≥0,Rp0≥0,Rm0≥0,Rpm0≥0.


## 3. Qualitative Analysis

In this section, the qualitative behaviours of the model such as the invariant region, positivity of future solution, equilibrium points and their stability analysis, basic reproduction number, and sensitivity analysis are investigated.


### 3.1. Invariant Region

To get the invariant region in which the solution of the model is bounded, we first considerNt=St+Ipt+Imt+Ipmt+Rpt+Rmt+Rpmt. Then,(3)dNtdt=π−μNt−α1Ipt+Ipmt−α2Imt+Ipmt.Ifα1=α2=0, then equation (3) becomes(4)dNtdt≤π−μNt.After solving equation (4), we get(5)Ω=S,Ip,Im,Ipm,Rp,Rm,Rpm∈ℜ7:0≤N≤πμ.Therefore, the invariant region of the model becomesΩ.


### 3.2. Positivity of the Solution

Theorem 1.
IfS0>0,Ip0>0,Im0>0,Ipm0>0,Rp0>0,Rm0>0,Rpm0>0, then all the solution sets St,Ipt,Imt,Ipmt,Rpt,Rmt,Rpmt are positive for future time.Proof.
First, let us taket1 as(6)t1=supt>0:Sτ>0,Ipτ>0,Imτ>0,Ipmτ>0,Rpτ>0,Rmτ>0,Rpmτ>0,for allτ∈0,t.
ConsiderS0≥0,Ip0≥0,Im0≥0,Ipm0≥0,Rp0≥0,Rm0≥0,Rpm0≥0; thus, t1>0. If t1<∞, then necessarily S or Ip or Im or Ipm or Rp or Rm or Rpm is equal to zero at t1. From equation (1),(7)dStdt=π+δ1Rpt+δ2Rmt+δ3Rpmt−f1+f2+μSt.
Using variation formula, equation (7) can be solved at t1:(8)St1=S0exp−∫0t1f1+f2+μsds+∫0t1π+δ1Rp+δ2Rm+δ3Rpmexp−∫st1f1+f2+μτdτds.
Accordingly, all the variables are nonnegative in0,t1; then, St1>0.
In a similar fashion, we can showIpt1>0,Imt1>0,Ipmt1>0,Rpt1>0,Rmt1>0, and Rpmt1>0 which is a contradiction. Hence, t1=∞.


### 3.3. Disease-Free Equilibrium (DFE)

EliminatingIpt,Imt, and Ipmt from equation (1) and solving for St give DFE:(9)E0=πμ,0,0,0,0,0,0.


### 3.4. Basic Reproduction Number (ℜ0)

Considering only the infected compartment and applying the next generation matrix give the following eigenvalues:(10)λ1∗=aπμσ1+α1+μ=ℜ0p,λ2∗=bπμσ2+α2+μ=ℜ0m,λ3∗=0.Since the basic reproduction number is the dominant eigenvalue of the next generation matrix,(11)ℜ0=maxℜ0p,ℜ0m.


### 3.5. Local Stability of Disease-Free Equilibrium

After obtaining the Jacobian matrix of the system at the disease-free equilibrium point, we obtained the following theorem.Theorem 2.
The disease-free equilibrium point is locally asymptotically stable ifℜ0<1, otherwise unstable.


### 3.6. Global Asymptotic Stability of Disease-Free Equilibrium

To investigate the global stability of disease-free equilibrium, we used the technique implemented in [9]. First, full pneumonia-meningitis model (1) can be re-written as(12)dXdt=FX,Z,dZdt=GX,Z,GX,0=0,where X stands for the uninfected population, that is X=S,Rp,Rm,Rpm, and Z stands for the infected population, that is Z=Ip,Im,Ipm. The disease-free equilibrium point of the model is denoted by U=X∗,0.For the pointU=X∗,0 to be globally asymptotically stable equilibrium for the model provided that ℜ0<1 (which is locally asymptotically stable) and the following conditions must be met:H1: for dX/dt=FX,0,X∗ is globally asymptotically stable.H2: GX,Z=AZ−G˜X,Z,G˜X,Z≥0 for X,Z∈ΩIf model (1) met the aforementioned two criteria, then the following theorem holds.Theorem 3.
The pointU=X∗,0 is globally asymptotically stable equilibrium provided that ℜ0<1 and the conditions H1 and H2 are satisfied.Proof.
From system (1), we can get FX,Z and GX,Z:(13)FX,Z=π+δ1Rp+δ2Rm+δ3Rpm−f1+f2+μSσ1Ip+σeIpm−δ1+μRpσ2Im+σg1−eIpm−δ2+μRmσ1−g1−eIpm−δ3+μRpm,GX,Z=f1S−f2+σ1+α1+μIpf2S−f1+σ2+α2+μImf2Ip+f1Im−σ+α1+α2+μIpm.
Consider the reduced system:(14)dXdtZ=0=π−μS000.
From equation (14), it is obvious that X∗=π/μ,0 is the global asymptotic point. This can be verified from the solution, namely, S=π/μ+S0−π/μe−μt. As t⟶∞, the solution S⟶π/μ, implying the global convergence of (14) in Ω.
Let(15)A=a−sigma1+α1+μ0a0−σ2+α2+μ000−σ+α1+α2+μ.
Then,GX,Z can be written as GX,Z=AZ−G˜X,Z, where(16)G˜X,Z=G˜1X,ZG˜2X,ZG˜3X,Z=aIm+Ipm1−SN+f2Ipf1Im−f2Ip+f1Im,where G˜2X,Z<0 which leads to G˜X,Z<0, which means the second condition (H2) is not satisfied, so U=X∗,0 may not be globally asymptotically stable when ℜ0<1.


### 3.7. Sensitivity Analysis

Here we performed sensitivity analysis in order to check the effect of each parameter in the expansion as well in controlling pneumonia and meningitis infection as well as their coinfection. To perform sensitivity analysis, we used a method outlined in [7]. Sensitivity index of ℜ0 with respect to parameter, say y, is given by Λyℜ0=∂ℜ0/∂yy/ℜ0. Since ℜ0=maxℜ0p,ℜ0m, we obtained sensitivity analysis of ℜ0p and ℜ0m separately in the following way:(17)Λaℜ0p=∂ℜ0p∂aaℜ0p=πμσ1+α1+μaμσ1+α1+μaπ=1>0,Λα1ℜ0p=∂ℜ0p∂α1α1ℜ0p=−α1σ1+α1+μ<0,Λσ1ℜ0p=∂ℜ0p∂σ1σ1ℜ0p=−σ1σ1+α1+μ<0,Λμℜ0p=∂ℜ0p∂μμℜ0p=−σ1+α1+2μσ1+α1+μ<0,Λbℜ0m=∂ℜ0m∂bbℜ0m=πμσ2+α2+μbμσ2+α2+μbπ=1>0,Λα2ℜ0m=∂ℜ0m∂α2α2ℜ0m=−α2σ2+α2+μ<0,Λσ2ℜ02=∂ℜ0m∂σ2σ2ℜ0m=−σ2σ2+α2+μ<0,Λμℜ0m=∂ℜ0m∂μμℜ0m=−σ2+α2+2μσ2+α2+μ<0.The above computation shows that pneumonia, meningitis, and their coinfection will be expanded if these parameters have a positive index, which is increased by keeping the other parameters constant. However, those parameters whose indices are negative have a great role in decreasing the diseases if their values are increased by keeping other parameters constant. From this result, we took prevention and treatment for both diseases to be considered in optimal control analysis in the next section.


## 3.1. Invariant Region

To get the invariant region in which the solution of the model is bounded, we first considerNt=St+Ipt+Imt+Ipmt+Rpt+Rmt+Rpmt. Then,(3)dNtdt=π−μNt−α1Ipt+Ipmt−α2Imt+Ipmt.Ifα1=α2=0, then equation (3) becomes(4)dNtdt≤π−μNt.After solving equation (4), we get(5)Ω=S,Ip,Im,Ipm,Rp,Rm,Rpm∈ℜ7:0≤N≤πμ.Therefore, the invariant region of the model becomesΩ.


## 3.2. Positivity of the Solution

Theorem 1.
IfS0>0,Ip0>0,Im0>0,Ipm0>0,Rp0>0,Rm0>0,Rpm0>0, then all the solution sets St,Ipt,Imt,Ipmt,Rpt,Rmt,Rpmt are positive for future time.Proof.
First, let us taket1 as(6)t1=supt>0:Sτ>0,Ipτ>0,Imτ>0,Ipmτ>0,Rpτ>0,Rmτ>0,Rpmτ>0,for allτ∈0,t.
ConsiderS0≥0,Ip0≥0,Im0≥0,Ipm0≥0,Rp0≥0,Rm0≥0,Rpm0≥0; thus, t1>0. If t1<∞, then necessarily S or Ip or Im or Ipm or Rp or Rm or Rpm is equal to zero at t1. From equation (1),(7)dStdt=π+δ1Rpt+δ2Rmt+δ3Rpmt−f1+f2+μSt.
Using variation formula, equation (7) can be solved at t1:(8)St1=S0exp−∫0t1f1+f2+μsds+∫0t1π+δ1Rp+δ2Rm+δ3Rpmexp−∫st1f1+f2+μτdτds.
Accordingly, all the variables are nonnegative in0,t1; then, St1>0.
In a similar fashion, we can showIpt1>0,Imt1>0,Ipmt1>0,Rpt1>0,Rmt1>0, and Rpmt1>0 which is a contradiction. Hence, t1=∞.


## 3.3. Disease-Free Equilibrium (DFE)

EliminatingIpt,Imt, and Ipmt from equation (1) and solving for St give DFE:(9)E0=πμ,0,0,0,0,0,0.


## 3.4. Basic Reproduction Number (ℜ0)

Considering only the infected compartment and applying the next generation matrix give the following eigenvalues:(10)λ1∗=aπμσ1+α1+μ=ℜ0p,λ2∗=bπμσ2+α2+μ=ℜ0m,λ3∗=0.Since the basic reproduction number is the dominant eigenvalue of the next generation matrix,(11)ℜ0=maxℜ0p,ℜ0m.


## 3.5. Local Stability of Disease-Free Equilibrium

After obtaining the Jacobian matrix of the system at the disease-free equilibrium point, we obtained the following theorem.Theorem 2.
The disease-free equilibrium point is locally asymptotically stable ifℜ0<1, otherwise unstable.


## 3.6. Global Asymptotic Stability of Disease-Free Equilibrium

To investigate the global stability of disease-free equilibrium, we used the technique implemented in [9]. First, full pneumonia-meningitis model (1) can be re-written as(12)dXdt=FX,Z,dZdt=GX,Z,GX,0=0,where X stands for the uninfected population, that is X=S,Rp,Rm,Rpm, and Z stands for the infected population, that is Z=Ip,Im,Ipm. The disease-free equilibrium point of the model is denoted by U=X∗,0.For the pointU=X∗,0 to be globally asymptotically stable equilibrium for the model provided that ℜ0<1 (which is locally asymptotically stable) and the following conditions must be met:H1: for dX/dt=FX,0,X∗ is globally asymptotically stable.H2: GX,Z=AZ−G˜X,Z,G˜X,Z≥0 for X,Z∈ΩIf model (1) met the aforementioned two criteria, then the following theorem holds.Theorem 3.
The pointU=X∗,0 is globally asymptotically stable equilibrium provided that ℜ0<1 and the conditions H1 and H2 are satisfied.Proof.
From system (1), we can get FX,Z and GX,Z:(13)FX,Z=π+δ1Rp+δ2Rm+δ3Rpm−f1+f2+μSσ1Ip+σeIpm−δ1+μRpσ2Im+σg1−eIpm−δ2+μRmσ1−g1−eIpm−δ3+μRpm,GX,Z=f1S−f2+σ1+α1+μIpf2S−f1+σ2+α2+μImf2Ip+f1Im−σ+α1+α2+μIpm.
Consider the reduced system:(14)dXdtZ=0=π−μS000.
From equation (14), it is obvious that X∗=π/μ,0 is the global asymptotic point. This can be verified from the solution, namely, S=π/μ+S0−π/μe−μt. As t⟶∞, the solution S⟶π/μ, implying the global convergence of (14) in Ω.
Let(15)A=a−sigma1+α1+μ0a0−σ2+α2+μ000−σ+α1+α2+μ.
Then,GX,Z can be written as GX,Z=AZ−G˜X,Z, where(16)G˜X,Z=G˜1X,ZG˜2X,ZG˜3X,Z=aIm+Ipm1−SN+f2Ipf1Im−f2Ip+f1Im,where G˜2X,Z<0 which leads to G˜X,Z<0, which means the second condition (H2) is not satisfied, so U=X∗,0 may not be globally asymptotically stable when ℜ0<1.


## 3.7. Sensitivity Analysis

Here we performed sensitivity analysis in order to check the effect of each parameter in the expansion as well in controlling pneumonia and meningitis infection as well as their coinfection. To perform sensitivity analysis, we used a method outlined in [7]. Sensitivity index of ℜ0 with respect to parameter, say y, is given by Λyℜ0=∂ℜ0/∂yy/ℜ0. Since ℜ0=maxℜ0p,ℜ0m, we obtained sensitivity analysis of ℜ0p and ℜ0m separately in the following way:(17)Λaℜ0p=∂ℜ0p∂aaℜ0p=πμσ1+α1+μaμσ1+α1+μaπ=1>0,Λα1ℜ0p=∂ℜ0p∂α1α1ℜ0p=−α1σ1+α1+μ<0,Λσ1ℜ0p=∂ℜ0p∂σ1σ1ℜ0p=−σ1σ1+α1+μ<0,Λμℜ0p=∂ℜ0p∂μμℜ0p=−σ1+α1+2μσ1+α1+μ<0,Λbℜ0m=∂ℜ0m∂bbℜ0m=πμσ2+α2+μbμσ2+α2+μbπ=1>0,Λα2ℜ0m=∂ℜ0m∂α2α2ℜ0m=−α2σ2+α2+μ<0,Λσ2ℜ02=∂ℜ0m∂σ2σ2ℜ0m=−σ2σ2+α2+μ<0,Λμℜ0m=∂ℜ0m∂μμℜ0m=−σ2+α2+2μσ2+α2+μ<0.The above computation shows that pneumonia, meningitis, and their coinfection will be expanded if these parameters have a positive index, which is increased by keeping the other parameters constant. However, those parameters whose indices are negative have a great role in decreasing the diseases if their values are increased by keeping other parameters constant. From this result, we took prevention and treatment for both diseases to be considered in optimal control analysis in the next section.


## 4. Optimal Control Analysis

In this section, we extended the basic model in equation (1) to optimal control by incorporating five controls which have a significant effect in controlling the expansion of coepidemics of pneumonia and meningitis. The interventions are as follows:(1)
u1: pneumonia prevention effort(2)
u2: meningitis prevention effort(3)
u3: pneumonia treating effort(4)
u4: meningitis treating effortAfter incorporating the above controls, the extended model becomes(18)dStdt=π+δ1Rpt+δ2Rmt+δ3Rpmt−1−u1f1+1−u2f2+μSt,dIptdt=1−u1f1St−1−u2f2Ipt−σ1+u3Ipt−α1+μIpt,dImtdt=1−u2f2St−1−u1f1Imt−σ2+u4Imt−α2+μImt,dIpmtdt=1−u2f2Ipt+1−u1f1Imt−σ+u3+u4Ipmt−α1+α2+μIpmt,dRptdt=σ1+u3Ipt+σe+u3Ipmt−δ1+μRpt,dRmtdt=σ2+u4Imt+σg1−e+u4Ipmt−δ2+μRmt,dRpmtdt=σ1−g1−e+u3+u4Ipmt−δ3+μRpmt.Being Lebesgue measurable ofU is crucial for studying the optimal levels of U=u1t,u2t,u3t,u4t:0≤u1<1,0≤u2<1,0≤u3<1,0≤u4<1,0≤t≤T. The main target is to get U and Ipt,Imt, and Ipmt, which minimize the objective function J, given by(19)J=minu1,u2,u3,u4∫0tfc1Ipt+c2Imt+c3Ipmt+12∑i=14wiui2dt,where c1,c2,c3, and wi are positive. The expression 1/2wiui2 represents costs. Our aim is to minimize infectious compartments and costs. Therefore, we want to get optimal controls u1∗,u2∗,u3∗,u4∗ in which(20)Ju1∗,u2∗,u3∗,u4∗=minJu1,u2,u3,u4/ui∈U.


### 4.1. The Hamiltonian and Optimality System

Here, Hamiltonian (H) is derived by applying Pontryagin’s maximum principle in the same way as described in [11], which is defined as(21)HSt,Ipt,Imt,Ipmt,Rpt,Rmt,Rpmt=LIpt,Imt,Ipmt,u1,u2,u3,u4,t+λ1dstdt+λ2tdIptdt+λ3tdImtdt+λ4tdIpmtdt+λ5tdRptdt+λ6tdRmtdt+λ7tdRpmtdt,where LIpt,Imt,Ipmt,u1,u2,u3,u4,t=c1Ipt+c2Imt+c3Ipmt+1/2∑i=14wiui2, λi,i=1,2,3,4,5,6,7 are the adjoint variable functions.Theorem 4.
For an optimal control setu1,u2,u3,u4 that minimizes J over U, there are adjoint variables, λ1,…,λ7 such that(22)dλ1dt=λ11−u1f1+1−u2f2+μ−λ21−u1f1−λ31−u2f2,dλ2dt=−c1−λ11−u1aSN−λ21−u1aSN+1−u2f2+σ1+u3+α1+μ−λ41−u2f2+1−u1aImN−λ5σ1+u3,dλ3dt=−c2+λ11−u2bSN+λ21−u2bIpN−λ3σ2+α2+u4+1−u1f1+1−u1f1−1−u2bSN−λ41−u2bIpN+1−u1f1−λ6σ2+u4,dλ4dt=−c3+λ11−u1aSN+1−u2bSN−λ21−u2bIpN−1−u1aSN−λ31−u1aImN−1−u2bSN−λ4σ+α1+α2+μ+u3+u4−1−u2bIpN−1−u1aImN−λ5σe+u3−λ6σg1−e+u4−λ7σ1−g1−e+u3+u4,dλ5dt=λ5δ1+μ−λ1δ1,dλ6dt=λ6δ2+μ−λ1δ2,dλ7dt=λ7δ3+μ−λ1δ3,with the condition λitf=0,i=1,…,7.
The characterized control sets are:(23)u1∗t=max0,min1,Φ1,u2∗t=max0,min1,Φ2,u3∗t=max0,min1,Φ3,u4∗t=max0,min1,Φ4,where(24)Φ1=f1Sλ2−λ1+f1Ipλ4−λ3w1,Φ2=f2Sλ3−λ1+f2Ipλ4−λ2w2,Φ3=λ2Ip+Ipmλ4−λ7−λ5Ip+Ipmw3,Φ4=λ3Im+Ipmλ4−λ7−λ6Im+Ipmw4.Proof.
Applying Pontryagin’s maximum principle gives the adjoint systems:(25)dλ1dt=−dHdS=λ11−u1f1+1−u2f2+μ−λ21−u1f1−λ31−u2f2,dλ2dt=−dHdIp=−c1−λ11−u1aSN−λ21−u1aSN+1−u2f2+σ1+u3+α1+μ−λ41−u2f2+1−u1aImN−λ5σ1+u3,dλ3dt=−dHdIm=−c2+λ11−u2bSN+λ21−u2bIpN−λ3σ2+α2+u4+1−u1f1−1−u2bSN−λ41−u2bIpN+1−u1f1−λ6σ2+u4,dλ4dt=−dHdIpm=−c3+λ11−u1aSN+1−u2bSN−λ21−u2bIpN−1−u1aSN−λ31−u1aImN−1−u2bSN−λ4σ+α1+α2+μ+u3+u4−1−u2bIpN−1−u1aImN−λ5σe+u3−λ6σg1−e+u4−λ7σ1−g1−e+u3+u4,dλ5dt=−dHdRp=λ5δ1+μ−λ1δ1,dλ6dt=−dHdRm=λ6δ2+μ−λ1δ2,dλ7dt=−dHdRpm=λ7δ3+μ−λ1δ3.
Now to obtain the time varying controls, we used the equation,∂H/∂ui=0atui∗, for i=1,…,4 and obtained the following:(26)u1∗=f1Sλ2−λ1+f1Ipλ4−λ3w1,u2∗=f2Sλ3−λ1+f2Ipλ4−λ2w2,u3∗=λ2Ip+Ipmλ4−λ7−λ5Ip+Ipmw3,u4∗=λ3Im+Ipmλ4−λ7−λ6Im+Ipmw4.
The controls can be written as(27)u1∗=Φ1,if0<Φ1<1,0,ifΦ1≤0,1,ifΦ1≥1,u2∗=Φ2,if0<Φ2<1,0,ifΦ2≤0,1,ifΦ2≥1,u3∗=Φ3,if0<Φ3<1,0,ifΦ3≤0,1,ifΦ3≥1,u4∗=Φ4,if0<Φ4<1,0,ifΦ4≤0,1,ifΦ4≥1,u5∗=Φ5,if0<Φ5<1,0,ifΦ5≤0,1,ifΦ5≥1.
The compact representation of the controls:(28)u1∗t=max0,min1,Φ1,u2∗t=max0,min1,Φ2,u3∗t=max0,min1,Φ3,u4∗t=max0,min1,Φ4,u5∗t=max0,min1,Φ5,Φ1=f1Sλ2−λ1+f1Ipλ4−λ3w1,Φ2=f2Sλ3−λ1+f2Ipλ4−λ2w2,Φ3=λ2Ip+Ipmλ4−λ7−λ5Ip+Ipmw3,Φ4=λ3Im+Ipmλ4−λ7−λ6Im+Ipmw4.
Then, the obtained optimality system is(29)dSdt=π+δ1Rp+δ2Rm+δ3Rpm−1−u1f1+1−u2f2+μS,dIpdt=1−u1f1S−1−u2f2Ip−σ1+u3Ip−α1+μIp,dImdt=1−u2f2S−1−u1f1Im−σ2+u4Im−α2+μIm,dIpmdt=1−u2f2Ip+1−u1f1Im−σ+u3+u4Ipm−α1+α2+μIpm,dRpdt=σ1+u3Ip+σe+u3Ipm−δ1+μRp,dRmdt=σ2+u4Im+σg1−e+u4Ipm−δ2+μRm,dRpmdt=σ1−g1−e+u3+u4Ipm−δ3+μRpm,dλ1dt=λ11−u1f1+1−u2f2+μ−λ21−u1f1−λ31−u2f2,dλ2dt=−c1−λ11−u1aSN−λ21−u1aSN+1−u2f2+σ1+u3+α1+μ−λ41−u2f2+1−u1aImN−λ5σ1+u3,dλ3dt=−c2+λ11−u2bSN+λ21−u2bIpN−λ3σ2+α2+u4+1−u1f1−1−u2bSN−λ41−u2bIpN+1−u1f1−λ6σ2+u4,dλ4dt=−c3+λ11−u1aSN+1−u2bSN−λ21−u2bIpN−1−u1aSN−λ31−u1aImN−1−u2bSN−λ4σ+α1+α2+μ+u3+u4−1−u2bIpN−1−u1aImN−λ5σe+u3−λ6σg1−e+u4−λ7σ1−g1−e+u3+u4,dλ5dt=λ5δ1+μ−λ1δ1,dλ6dt=λ6δ2+μ−λ1δ2,dλ7dt=λ7δ3+μ−λ1δ3.(30)λitf=0,i=1,…,7,S0=S0,Ip0=Ip0,Im0=Im0,Ipm0=Ipm0,Rp0=Rp0,Rm0=Rm0,Rpm0=Rpm0.


## 4.1. The Hamiltonian and Optimality System

Here, Hamiltonian (H) is derived by applying Pontryagin’s maximum principle in the same way as described in [11], which is defined as(21)HSt,Ipt,Imt,Ipmt,Rpt,Rmt,Rpmt=LIpt,Imt,Ipmt,u1,u2,u3,u4,t+λ1dstdt+λ2tdIptdt+λ3tdImtdt+λ4tdIpmtdt+λ5tdRptdt+λ6tdRmtdt+λ7tdRpmtdt,where LIpt,Imt,Ipmt,u1,u2,u3,u4,t=c1Ipt+c2Imt+c3Ipmt+1/2∑i=14wiui2, λi,i=1,2,3,4,5,6,7 are the adjoint variable functions.Theorem 4.
For an optimal control setu1,u2,u3,u4 that minimizes J over U, there are adjoint variables, λ1,…,λ7 such that(22)dλ1dt=λ11−u1f1+1−u2f2+μ−λ21−u1f1−λ31−u2f2,dλ2dt=−c1−λ11−u1aSN−λ21−u1aSN+1−u2f2+σ1+u3+α1+μ−λ41−u2f2+1−u1aImN−λ5σ1+u3,dλ3dt=−c2+λ11−u2bSN+λ21−u2bIpN−λ3σ2+α2+u4+1−u1f1+1−u1f1−1−u2bSN−λ41−u2bIpN+1−u1f1−λ6σ2+u4,dλ4dt=−c3+λ11−u1aSN+1−u2bSN−λ21−u2bIpN−1−u1aSN−λ31−u1aImN−1−u2bSN−λ4σ+α1+α2+μ+u3+u4−1−u2bIpN−1−u1aImN−λ5σe+u3−λ6σg1−e+u4−λ7σ1−g1−e+u3+u4,dλ5dt=λ5δ1+μ−λ1δ1,dλ6dt=λ6δ2+μ−λ1δ2,dλ7dt=λ7δ3+μ−λ1δ3,with the condition λitf=0,i=1,…,7.
The characterized control sets are:(23)u1∗t=max0,min1,Φ1,u2∗t=max0,min1,Φ2,u3∗t=max0,min1,Φ3,u4∗t=max0,min1,Φ4,where(24)Φ1=f1Sλ2−λ1+f1Ipλ4−λ3w1,Φ2=f2Sλ3−λ1+f2Ipλ4−λ2w2,Φ3=λ2Ip+Ipmλ4−λ7−λ5Ip+Ipmw3,Φ4=λ3Im+Ipmλ4−λ7−λ6Im+Ipmw4.Proof.
Applying Pontryagin’s maximum principle gives the adjoint systems:(25)dλ1dt=−dHdS=λ11−u1f1+1−u2f2+μ−λ21−u1f1−λ31−u2f2,dλ2dt=−dHdIp=−c1−λ11−u1aSN−λ21−u1aSN+1−u2f2+σ1+u3+α1+μ−λ41−u2f2+1−u1aImN−λ5σ1+u3,dλ3dt=−dHdIm=−c2+λ11−u2bSN+λ21−u2bIpN−λ3σ2+α2+u4+1−u1f1−1−u2bSN−λ41−u2bIpN+1−u1f1−λ6σ2+u4,dλ4dt=−dHdIpm=−c3+λ11−u1aSN+1−u2bSN−λ21−u2bIpN−1−u1aSN−λ31−u1aImN−1−u2bSN−λ4σ+α1+α2+μ+u3+u4−1−u2bIpN−1−u1aImN−λ5σe+u3−λ6σg1−e+u4−λ7σ1−g1−e+u3+u4,dλ5dt=−dHdRp=λ5δ1+μ−λ1δ1,dλ6dt=−dHdRm=λ6δ2+μ−λ1δ2,dλ7dt=−dHdRpm=λ7δ3+μ−λ1δ3.
Now to obtain the time varying controls, we used the equation,∂H/∂ui=0atui∗, for i=1,…,4 and obtained the following:(26)u1∗=f1Sλ2−λ1+f1Ipλ4−λ3w1,u2∗=f2Sλ3−λ1+f2Ipλ4−λ2w2,u3∗=λ2Ip+Ipmλ4−λ7−λ5Ip+Ipmw3,u4∗=λ3Im+Ipmλ4−λ7−λ6Im+Ipmw4.
The controls can be written as(27)u1∗=Φ1,if0<Φ1<1,0,ifΦ1≤0,1,ifΦ1≥1,u2∗=Φ2,if0<Φ2<1,0,ifΦ2≤0,1,ifΦ2≥1,u3∗=Φ3,if0<Φ3<1,0,ifΦ3≤0,1,ifΦ3≥1,u4∗=Φ4,if0<Φ4<1,0,ifΦ4≤0,1,ifΦ4≥1,u5∗=Φ5,if0<Φ5<1,0,ifΦ5≤0,1,ifΦ5≥1.
The compact representation of the controls:(28)u1∗t=max0,min1,Φ1,u2∗t=max0,min1,Φ2,u3∗t=max0,min1,Φ3,u4∗t=max0,min1,Φ4,u5∗t=max0,min1,Φ5,Φ1=f1Sλ2−λ1+f1Ipλ4−λ3w1,Φ2=f2Sλ3−λ1+f2Ipλ4−λ2w2,Φ3=λ2Ip+Ipmλ4−λ7−λ5Ip+Ipmw3,Φ4=λ3Im+Ipmλ4−λ7−λ6Im+Ipmw4.
Then, the obtained optimality system is(29)dSdt=π+δ1Rp+δ2Rm+δ3Rpm−1−u1f1+1−u2f2+μS,dIpdt=1−u1f1S−1−u2f2Ip−σ1+u3Ip−α1+μIp,dImdt=1−u2f2S−1−u1f1Im−σ2+u4Im−α2+μIm,dIpmdt=1−u2f2Ip+1−u1f1Im−σ+u3+u4Ipm−α1+α2+μIpm,dRpdt=σ1+u3Ip+σe+u3Ipm−δ1+μRp,dRmdt=σ2+u4Im+σg1−e+u4Ipm−δ2+μRm,dRpmdt=σ1−g1−e+u3+u4Ipm−δ3+μRpm,dλ1dt=λ11−u1f1+1−u2f2+μ−λ21−u1f1−λ31−u2f2,dλ2dt=−c1−λ11−u1aSN−λ21−u1aSN+1−u2f2+σ1+u3+α1+μ−λ41−u2f2+1−u1aImN−λ5σ1+u3,dλ3dt=−c2+λ11−u2bSN+λ21−u2bIpN−λ3σ2+α2+u4+1−u1f1−1−u2bSN−λ41−u2bIpN+1−u1f1−λ6σ2+u4,dλ4dt=−c3+λ11−u1aSN+1−u2bSN−λ21−u2bIpN−1−u1aSN−λ31−u1aImN−1−u2bSN−λ4σ+α1+α2+μ+u3+u4−1−u2bIpN−1−u1aImN−λ5σe+u3−λ6σg1−e+u4−λ7σ1−g1−e+u3+u4,dλ5dt=λ5δ1+μ−λ1δ1,dλ6dt=λ6δ2+μ−λ1δ2,dλ7dt=λ7δ3+μ−λ1δ3.(30)λitf=0,i=1,…,7,S0=S0,Ip0=Ip0,Im0=Im0,Ipm0=Ipm0,Rp0=Rp0,Rm0=Rm0,Rpm0=Rpm0.


## 5. Numerical Simulation

In this section, we performed numerical simulation of the optimality system. To simulate the system, we used the forward fourth-order Runge–Kutta method to solve the state system and the backward fourth-order Runge–Kutta method for solving the costate system. We have used Maple 18, for simulation.We proposed the following five strategies for numerical simulation of the optimality system:(i)
Using prevention effort for both diseasesu1≠0,u2≠0,u3=0,u4=0(ii)
Prevention effort for pneumonia disease and treatment effort for meningitis disease (u1≠0,u4≠0,u2=u3=0)(iii)
Using prevention effort for meningitis disease and treatment effort for pneumonia diseaseu2≠0,u3≠0,u1=0,u4=0(iv)
Using treatment effort for both diseases (u3≠0,u4≠0,u1=u2=0)(v)
Using all the intervention efforts (u1≠0,u2≠0,u3≠0,u4≠0)For simulation, we used parameter values listed in Table1, and we assumed c1=35,c2=45,c3=26,w1=4,w2=3,w3=5, and w4=6 for simulation. Initial conditions that are used are S0=1500,Ip0=456,Im0=564,Ipm0=250,Rp0=123,Rm0=248, and Rpm0=346.Table 1
Parameter values of the model.
Parameter symbol
Value
Source
δ
1
0.003–0.1
[7]
δ
2
0.00904–0.99
[3]
δ
3
0.01
Assumed
a
0.007–0.6
[9]
b
0.9
[3]
α
1
0.006–0.5
Estimated
α
2
0.002–0.2
Estimated
σ
0.1
[7]
g
0.5–1
[9]
e
0.5–1
[9]
μ
0.01
Assumed
σ
1
0.9
Assumed
σ
2
0.8
Assumed


### 5.1. Control with Prevention for Both Diseases

Here we applied prevention of both pneumonia and meningitis diseases as the intervention strategy. From the simulation results of Figures2 and 3, we see that prevention has a great impact in controlling pneumonia-only and meningitis-only infectious population and also in eradicating coinfection of pneumonia and meningitis diseases in the specified time.Figure 2
Simulations of optimality system with prevention only.
(a)
(b)Figure 3
Effect of prevention on coinfectious populations.


### 5.2. Control with Prevention Effort for Pneumonia and Treatment Effort for Meningitis

Here we investigated the effect of combination of prevention and treatment as the intervention strategy. Figures4 and 5 show that prevention for pneumonia only and treatment for meningitis only contribute in controlling pneumonia-only infectious population, meningitis-only infectious population, and coinfectious population.Figure 4
Simulations of optimality system with prevention of pneumonia and treatment of meningitis.
(a)
(b)Figure 5
Effect of prevention of pneumonia and treatment of meningitis on coinfectious populations.


### 5.3. Control with Prevention Effort for Meningitis and Treatment Effort for Pneumonia

In this section, we used prevention of meningitis and treatment of pneumonia as the controlling mechanism. Figures6 and 7 show that pneumonia and meningitis infectious populations and coinfectious populations are found to increase due to lack of intervention, but when meningitis prevention and pneumonia treatment are used as the intervention mechanism, the infectious population is found to decreaseat the specified time.Figure 6
Simulations of optimality system with prevention of meningitis and treatment of pneumonia.
(a)
(b)Figure 7
Effect of prevention of meningitis and treatment of pneumonia on coinfectious populations.


### 5.4. Control with Treatment Effort for Both Diseases

In this section, treatment of pneumonia and meningitis diseases is used as the controlling strategy. The results of the applications of the strategies are shown in Figures8 and 9. From the figures, we see that the infectious population due to pneumonia and meningitis diseases as well as the coinfectious population is found to decrease due to treatment strategy for both diseases.Figure 8
Simulations of optimality system with treatment effort.
(a)
(b)Figure 9
Effect of treating both diseases on coinfectious populations.


### 5.5. Control with All Intervention Strategies

In this section, we used all the four controlling strategies to tackle pneumonia, meningitis, and their coinfection. The results from Figures10 and 11 show that the proposed intervention strategies are effective in bringing down the infectious population in the specified period of time.Figure 10
Simulations of optimality system with all interventions.
(a)
(b)Figure 11
Effect of all control interventions on coinfectious populations.


## 5.1. Control with Prevention for Both Diseases

Here we applied prevention of both pneumonia and meningitis diseases as the intervention strategy. From the simulation results of Figures2 and 3, we see that prevention has a great impact in controlling pneumonia-only and meningitis-only infectious population and also in eradicating coinfection of pneumonia and meningitis diseases in the specified time.Figure 2
Simulations of optimality system with prevention only.
(a)
(b)Figure 3
Effect of prevention on coinfectious populations.


## 5.2. Control with Prevention Effort for Pneumonia and Treatment Effort for Meningitis

Here we investigated the effect of combination of prevention and treatment as the intervention strategy. Figures4 and 5 show that prevention for pneumonia only and treatment for meningitis only contribute in controlling pneumonia-only infectious population, meningitis-only infectious population, and coinfectious population.Figure 4
Simulations of optimality system with prevention of pneumonia and treatment of meningitis.
(a)
(b)Figure 5
Effect of prevention of pneumonia and treatment of meningitis on coinfectious populations.


## 5.3. Control with Prevention Effort for Meningitis and Treatment Effort for Pneumonia

In this section, we used prevention of meningitis and treatment of pneumonia as the controlling mechanism. Figures6 and 7 show that pneumonia and meningitis infectious populations and coinfectious populations are found to increase due to lack of intervention, but when meningitis prevention and pneumonia treatment are used as the intervention mechanism, the infectious population is found to decreaseat the specified time.Figure 6
Simulations of optimality system with prevention of meningitis and treatment of pneumonia.
(a)
(b)Figure 7
Effect of prevention of meningitis and treatment of pneumonia on coinfectious populations.


## 5.4. Control with Treatment Effort for Both Diseases

In this section, treatment of pneumonia and meningitis diseases is used as the controlling strategy. The results of the applications of the strategies are shown in Figures8 and 9. From the figures, we see that the infectious population due to pneumonia and meningitis diseases as well as the coinfectious population is found to decrease due to treatment strategy for both diseases.Figure 8
Simulations of optimality system with treatment effort.
(a)
(b)Figure 9
Effect of treating both diseases on coinfectious populations.


## 5.5. Control with All Intervention Strategies

In this section, we used all the four controlling strategies to tackle pneumonia, meningitis, and their coinfection. The results from Figures10 and 11 show that the proposed intervention strategies are effective in bringing down the infectious population in the specified period of time.Figure 10
Simulations of optimality system with all interventions.
(a)
(b)Figure 11
Effect of all control interventions on coinfectious populations.


## 6. Discussion and Conclusion

In Section2, the basic model is described. The total system is subdivided into seven compartments by using ordinary differential equations. The qualitative behaviours including the invariant region, the positivity of solution, the disease-free equilibrium, basic reproduction number, analysis of disease-free equilibrium points, and checking the sensitivity of each parameter are presented in Section 3. In Section 4, the basic model is extended to optimal control by incorporating four controls such as prevention of pneumonia, prevention of meningitis, treatment of pneumonia, and treatment of meningitis. In this section, we characterized the optimal controls in terms of optimality system solutions. In Section 5, the optimality system is simulated by applying the Runge–Kutta forward-backward sweep method. For simulation of the optimality system, we proposed five strategies to check the effect of the controls. First, we considered prevention only for both diseases, and the result shows that applying prevention control has a great impact in bringing down the expansion of pneumonia, meningitis, and their coinfection in the specified period of time. The other strategies are prevention effort for pneumonia and treatment effort for meningitis, prevention effort for meningitis and treatment effort for pneumonia, treatment effort for both diseases, and using all interventions. We obtained that each of the listed strategies is effective in minimizing the expansion of pneumonia-only infectious population, meningitis-only infectious population, and coinfectious population in the specified period of time.



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