# HIV/AIDS-Pneumonia Coinfection Model with Treatment at Each Infection Stage: Mathematical Analysis and Numerical Simulation

Authors: Shewafera Wondimagegnhu Teklu; Temesgen Tibebu Mekonnen
Journal: Journal of Applied Mathematics
 (2021)

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## Abstract

In the paper, we have considered a nonlinear compartmental mathematical modelthat assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia coinfection in a human population at different infection stages. Our model revealed that the disease-free equilibrium points of the HIV/AIDS and pneumonia submodels are both locally and globally asymptotically stable whenever the associated basic reproduction numbers (RH and RP) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers (RP and RH) are greater than unity. The full HIV/AIDS-pneumonia coinfection model has both locally and globally asymptotically stable disease-free equilibrium points whenever the basic reproduction number of the coinfection modelRHPis less than unity. Using standard values of parameters collected from different kinds of literature, we found that the numerical values of the basic reproduction numbers of the HIV/AIDS-only submodel and pneumonia-only submodel are 17 and 7, respectively, and the basic reproduction number of the HIV/AIDS-pneumonia coinfection model is max7,17=17. Applying sensitive analysis, we identified the most influential parameters to change the behavior of the solution of the considered coinfection dynamical system are the HIV/AIDS and pneumonia transmission rates β1 and β2, respectively. The coinfection model was numerically simulated to investigate the stability of the coinfection endemic equilibrium point, the impacts of transmission rates, and treatment strategies for HIV/AIDS-only, pneumonia-only, and HIV/AIDS-pneumonia coinfected individuals. Finally, we observed that numerical simulations indicate that treatment against infection at every stage lowers the rate of infection or disease prevalence.


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## Body

## 1. Introduction

Infectious diseases are a clinically evident illness, and commonly, they have a great influence on the human population. They are induced by a pathogenic microbial agent such as bacterial, viral, fungal, parasitic, or it can be toxic proteins, called prions. The most common ones are tuberculosis and pneumonia caused by bacteria, HIV, and influenza caused by the virus [1, 2].HIV/AIDS is a global health issue affecting approximately 70 million people worldwide causing significant morbidity and mortality [3]. Over two-thirds of people living with HIV live in the Sub-Saharan African region [4]. Human immunodeficiency virus (HIV) is a retrovirus virus which attacks and weakens the human body immunity and the central nervous system and if untreated it continues to multiply into the host until it reaches the peak leading into a very serious disease called AIDS, the stage where the symptoms of the disease occur frequently [5–7]. HIV is transmitted through sexual intercourse, needle sharing, and direct contact of blood or other body fluids containing the virus and mother to child during childbirth [1, 8]. According to the center for disease control and prevention (CDC), when individuals get HIV and do not receive treatment known as antiretroviral therapy (ART), they will typically progress through three stages of disease: acute HIV infection, clinical latency (HIV inactivity or dormancy), and acquired immunodeficiency syndrome (AIDS).Pneumonia is one of airborne infectious disease caused by caused by bacteria, viruses, fungi, or parasites which attacks the human lungs or alveoli [9–11]. Most of the time pneumonia affects older adults, babies, and people with other diseases or impaired immune systems worldwide. Its most common cause is the Streptococcus pneumoniae, also known as pneumococcus [10, 12, 13]. The basic controlling strategies of pneumonia infection are treatment and vaccination interventions [11].A coinfection is the infection of a host with two or more different pathogens or different strains of the same pathogens, leading to coexistence of strains (pathogens) at population level [14]. Mathematical modeling of infectious diseases such as coinfection of HIV/AIDS and opportunistic infection is regarded as a fundamental tool in understanding the dynamics and helpful in the decision-making processes regarding intervention strategies and measures required for disease elimination and/or control [15, 16]. Some mathematical models have been used to investigate the transmission dynamics of coinfection of two or more diseases where HIV/AIDS and pneumonia coinfection is among the diseases that infect a large number of individuals worldwide. HIV-infected persons are particularly susceptible to the development of severe pneumococcal disease, even in the setting of combination antiretroviral therapy (ART) [4, 7, 12].Over the past, mathematical model shave been developed to analyze the population dynamics of HIV/AIDS and pneumonia single infections. Authors in [3] developed and analyzed only a sex-structured population model and studied the HIV infection trends in males and females. Their model assumed that the main mode of HIV transmission is heterosexual. They showed that prevention of HIV infection still remains the most important way of controlling further spread in the community. HIV/AIDS patients under ART treatment are possibly capable of helping the eradication of HIV by convincing their sexual partners of the need to adhere to protection via use of preexposure prophylaxis (PrEP) or any other protection means and ART treatment. Huo and Chen [5] developed and analyzed a mathematical analysis to study the spread of HIV/AIDS with treatment at different stages. Their results show that early treatment for individuals in asymptomatic stage of HIV infection or the pre-AIDS stage is very important. Rahman [2] have analyzed a seven dimension in the living organism HIV model with optimal control of in-host HIV dynamics using different control strategies such as three drug combinations, that is, FIs, RTIs, and PIs to determine the optimal treatment regime. From their results, they recommend that RTIs be used as initial therapy for HIV and FI should be introduced to the patient after the RTIs but should never be used alone.Mbabazi et al. [13] formulated a mathematical model to study the global stability of pneumococcal pneumonia with awareness, and saturated treatment is presented. Their results showed that the family of decaying curves could help in providing mechanisms to design awareness strategies for containing pneumococcal pneumoniae threshold parameter could be reduced to less than unity if antibiotic resistance awareness and treatment are implemented simultaneously to ensure eradication of pneumococcus bacteria; thus, spread of pneumococcus pneumonia in the population will die out. Tilahun et al. [17] proposed and analyzed a nonlinear mathematical model for the transmission dynamics of pneumonia disease in a population of varying size with optimal control of pneumonia disease and cost-effective strategies. Their cost-effectiveness analysis of the adopted control strategies showed that the combination of prevention and treatment is the most cost-effective intervention strategies to fight the pneumonia pandemic. Ndelwa et al. [18] developed a mathematical model and analyzed treatment and screening strategies on pneumonia infection, and from their numerical results, they concluded screening and treating at the same time can eradicate the pneumonia epidemic from the community.Nwankwo and Okuonghae [19] formulated and analyzed a mathematical model for the transmission dynamics of syphilis and HIV coinfection in a community to assess the impact of treatment of syphilis on the coendemicity of both diseases in a population where treatment for HIV is not readily available (or easily accessible) but with syphilis treatment sufficiently available. Their syphilis-only model and the full coinfection models undergo the phenomenon of backward bifurcation due to syphilis reinfection after recovery from a previous syphilis infection. They have got the treatment of primary and secondary syphilis in both singly and dually infected individuals; especially with high treatment rates for primary syphilis, this will result in a reduction in the incidence of HIV and its coinfection with syphilis in the community. Kaur et al. [6] formulated and analyzed a mathematical model for HIV/AIDS-TB coinfection with screening and treatment of both HIV and TB infective. Their numerical results suggested that the rates of transmission of both TB and HIV should be decreased, as an increase causes a rise in the number of infective at the equilibrium level.However, most microbiology, epidemiology, and medical sources like [7, 12] shows the coexistence of HIV/AIDS and pneumonia infection but coinfection mathematical models of HIV/AIDS and pneumonia are rare in literature yet the coexistence between the two infections exist. In our review of literatures, we have got one mathematical model of HIV/AIDS and pneumonia infection in a population and we used it as initial literature reviewed as follows. According to Nthiiri et al. [4] maximum protection against the HIV/AIDS-pneumonia coinfection was analyzed where the maximum protection against HIV/AIDS and the maximum protection against pneumonia was the main concern of their project. In their model, they did not considered maximum protection against the coinfection rather considered maximum protection against single infections. Also, they did not considered treatment on either the submodels or the coinfection. Their analysis found that when protection is high, the number of HIV/AIDS and pneumonia cases is low.Our paper, therefore, presents a mathematical model describing the transmission dynamics of HIV/AIDS and pneumonia coinfection in a population where treatment for both HIV/AIDS and pneumonia are available in the community. Basically, the model will be used to evaluate the effect of treatment at every infection stage of either the single infected individuals with HIV/AIDS or pneumonia or HIV/AIDS and pneumonia coinfection as a control strategy for minimizing incidences of coinfections in the target population. In this work, we applied the center for disease control and prevention (CDC) human immunodeficiency virus (HIV) infection stages and the control measure treatment at each stage of the single infections and coinfection model. We have checked this case has never been done before. We discussed the effects of treatment for single infected individuals with either HIV/AIDS or pneumonia and the coinfected patient with HIV/AIDS and pneumonia at each infection stage. The paper is organized as follows. The model is formulated in Section2 and is analyzed in Section 3. Sensitivity analysis, numerical results, and discussion are carried out in Section 4. Finally, conclusion and limitations of the study are carried out in Sections 5 and 6, respectively.


## 2. The Mathematical Model

According to the three center for disease control and prevention (CDC) HIV infection stages, we have divide the total populationNt in to eleven mutually exclusive compartments stated in Table 1, so that Nt=St+IPt+H1t+H2t+H3t+C1t+C2t+C3t+TPt+THt+Tt.Table 1
Descriptions of state variables in model (3).
State variablesBiological meaningSSusceptible individuals for both HIV and pneumoniaIPIndividuals infected with pneumoniaH1Acute HIV-infected individualsH2Chronic HIV-infected individualsH2Number of AIDS patientsC1Individuals coinfected with acute HIV and pneumoniaC2Individuals coinfected with chronic HIV and pneumoniaC3Individuals coinfected with AIDS and pneumoniaTHIndividuals on treatment of HIV/AIDS at different stagesTPIndividuals on treatment of pneumonia infectionTIndividuals on treatment of coinfections at different stagesWe assumed that coinfected individuals can only transmit either pneumonia or HIV but not both infections simultaneously. Individuals acquire HIV infection following effective contacts with people infected with HIV only, (H1 and H2 classes), at the rate given by
(1)λHt=β1H1t+ρH2t,whereρ>1 is the modification parameter accounting for the assumed increased infectivity due to chronic HIV infected than acute HIV-infected one and β1 is the HIV transmission rate.Also, individuals acquire pneumonia infection from those in theIP,C1, C2, and C3 infectious at the rate
(2)λPt=β2Ipt+ω1C1t+ω2C2t+ω3C3t,whereω3>ω2>ω1 are modification parameters accounting for the assumed increased infectivity due to coinfections and β2 is the pneumonia transmission rate. The derivation of the model differential equations is given in “Appendix A.”


### 2.1. Flow Chart of the Dynamical System

Here, parameter descriptions in Table2, state variable descriptions in Table 1 above, and based on the model assumptions that led to the formulation of the model (as stated in”Appendix A“), the flow diagram for the transmission dynamics of HIV/AIDS and pneumonia coinfection is given by Figure 1.Table 2
Description of parameters in model (3).
ParameterBiological meaningUnitμNatural death rateTime−1ΛRecruitment rate of susceptible individualssize∗Time−1α1The progression rate from acute HIV infection to chronic HIVTime−1α2The progression rate from chronic HIV infection to AIDSTime−1φ1The modification parameter accounting that acute HIV-infected individual is more susceptible to pneumoniaTime−1φ2The modification parameter accounting that chronic HIV-infected individual is more susceptible to pneumoniaTime−1φ3The modification parameter accounting that AIDS patient individual is more susceptible to pneumonia infectionTime−1λHHIV/AIDS force of infectionsize−1∗Time−1λPPneumonia force of infectionsize−1∗Time−1α3The progression rate from acute HIV and pneumonia coinfection to chronic HIV and pneumonia coinfectionTime−1α4The progression rate from coinfection of chronic HIV-pneumonia to AIDS-pneumonia coinfectionTime−1δPPneumonia disease-induced death rateTime−1δ1AIDS disease-induced death rateTime−1δ2AIDS and pneumonia diseases-induced death rateTime−1γTreatment rate for pneumonia-infected individualsTime−1pPortion of pneumonia infected who become susceptible againDimensionlessγ1Treatment rate for acute HIV-infected individualsTime−1γ2Treatment rate for chronic HIV-infected individualsTime−1γ3Treatment rate for AIDS stage-infected individualsTime−1ε1Treatment rate for acute HIV and pneumonia coinfectedTime−1ε2Treatment rate for chronic HIV and pneumonia coinfectionTime−1ε3Treatment rate for AIDS and pneumonia coinfectionTime−1β1HIV/AIDS transmission ratesize−1∗Time−1β2Pneumonia transmission ratesize−1∗Time−1Figure 1
Flow chart of the HIV/AIDS-pneumonia coinfection model (3) where λH and λP are given in (1) and (2), respectively.


### 2.2. Dynamical System of HIV/AIDS-Pneumonia Coinfection

Based on Figure1 above, the dynamical system of HIV/AIDS-pneumonia coinfection becomes
(3)dSdt=Λ+pγTP−μ+λH+λPS,dIPdt=λPS−μ+γ+δPIP,dH1dt=λHS−μ+γ1+α1+φ1λP,dH2dt=α1H1−μ+γ2+α2+φ2λPH2,dH3dt=α2H2−μ+γ3+δ1+φ3λPH3,dC1dt=φ1λPH1−μ+δP+ε1+α3C1,dC2dt=φ2λPH2+α3C1−μ+δP+ε2+α4C2,dC3dt=φ3λPH3+α4C2−μ+δ2+ε3C3,dTPdt=γIP−μ+pγTP,dTHdt=γ1H1+γ2H2+γ3H3−μTH,dTdt=ε1C1+ε2C2+ε3C3−μT.With initial conditions,(4)S0>0,IP0≥0,H10≥0,H20≥0,H30≥0,C10≥0,C20≥0,C30≥0,TP0≥0,TH0≥0,T0≥0.The sum of all the differential equations in (3) is given by
(5)dNdt=Λ−μN−δPIP+δ1H3+δPC1+δPC2+δ2C3.Since model (3) monitors human population, it is assumed that all variables and parameters are nonnegative. The dynamics of model (3) will be analyzed in the following invariant region:
(6)Ω=S,H1,H2,H3,IP,C1,C2,C3,TP,TH,T∈ℝ11+,N≤Λμ.Then, we have proved the positivity and boundedness of solutions of (3) in Ω in “Appendix B.”


## 2.1. Flow Chart of the Dynamical System

Here, parameter descriptions in Table2, state variable descriptions in Table 1 above, and based on the model assumptions that led to the formulation of the model (as stated in”Appendix A“), the flow diagram for the transmission dynamics of HIV/AIDS and pneumonia coinfection is given by Figure 1.Table 2
Description of parameters in model (3).
ParameterBiological meaningUnitμNatural death rateTime−1ΛRecruitment rate of susceptible individualssize∗Time−1α1The progression rate from acute HIV infection to chronic HIVTime−1α2The progression rate from chronic HIV infection to AIDSTime−1φ1The modification parameter accounting that acute HIV-infected individual is more susceptible to pneumoniaTime−1φ2The modification parameter accounting that chronic HIV-infected individual is more susceptible to pneumoniaTime−1φ3The modification parameter accounting that AIDS patient individual is more susceptible to pneumonia infectionTime−1λHHIV/AIDS force of infectionsize−1∗Time−1λPPneumonia force of infectionsize−1∗Time−1α3The progression rate from acute HIV and pneumonia coinfection to chronic HIV and pneumonia coinfectionTime−1α4The progression rate from coinfection of chronic HIV-pneumonia to AIDS-pneumonia coinfectionTime−1δPPneumonia disease-induced death rateTime−1δ1AIDS disease-induced death rateTime−1δ2AIDS and pneumonia diseases-induced death rateTime−1γTreatment rate for pneumonia-infected individualsTime−1pPortion of pneumonia infected who become susceptible againDimensionlessγ1Treatment rate for acute HIV-infected individualsTime−1γ2Treatment rate for chronic HIV-infected individualsTime−1γ3Treatment rate for AIDS stage-infected individualsTime−1ε1Treatment rate for acute HIV and pneumonia coinfectedTime−1ε2Treatment rate for chronic HIV and pneumonia coinfectionTime−1ε3Treatment rate for AIDS and pneumonia coinfectionTime−1β1HIV/AIDS transmission ratesize−1∗Time−1β2Pneumonia transmission ratesize−1∗Time−1Figure 1
Flow chart of the HIV/AIDS-pneumonia coinfection model (3) where λH and λP are given in (1) and (2), respectively.


## 2.2. Dynamical System of HIV/AIDS-Pneumonia Coinfection

Based on Figure1 above, the dynamical system of HIV/AIDS-pneumonia coinfection becomes
(3)dSdt=Λ+pγTP−μ+λH+λPS,dIPdt=λPS−μ+γ+δPIP,dH1dt=λHS−μ+γ1+α1+φ1λP,dH2dt=α1H1−μ+γ2+α2+φ2λPH2,dH3dt=α2H2−μ+γ3+δ1+φ3λPH3,dC1dt=φ1λPH1−μ+δP+ε1+α3C1,dC2dt=φ2λPH2+α3C1−μ+δP+ε2+α4C2,dC3dt=φ3λPH3+α4C2−μ+δ2+ε3C3,dTPdt=γIP−μ+pγTP,dTHdt=γ1H1+γ2H2+γ3H3−μTH,dTdt=ε1C1+ε2C2+ε3C3−μT.With initial conditions,(4)S0>0,IP0≥0,H10≥0,H20≥0,H30≥0,C10≥0,C20≥0,C30≥0,TP0≥0,TH0≥0,T0≥0.The sum of all the differential equations in (3) is given by
(5)dNdt=Λ−μN−δPIP+δ1H3+δPC1+δPC2+δ2C3.Since model (3) monitors human population, it is assumed that all variables and parameters are nonnegative. The dynamics of model (3) will be analyzed in the following invariant region:
(6)Ω=S,H1,H2,H3,IP,C1,C2,C3,TP,TH,T∈ℝ11+,N≤Λμ.Then, we have proved the positivity and boundedness of solutions of (3) in Ω in “Appendix B.”


## 3. Mathematical Model Analysis

Before we analyzed the HIV/AIDS-pneumonia coinfection model (3), it is useful to gain some background about the HIV-only submodel and pneumonia-only submodel transmission dynamics.


### 3.1. HIV/AIDS-Only Submodel Analysis

The HIV submodel of (3) (obtained by settingIP=C1=C2=C3=TP=T=0 is given by
(7)dSdt=Λ−μ+λHS,dH1dt=λHS−μ+γ1+α1H1,dH2dt=α1H1−μ+γ2+α2H2,dH3dt=α2H2−μ+γ3+δ1H3,dTHdt=γ1H1+γ2H2+γ3H3−μTH,where the total population isN1t=St+H1t+H2t+H3t+THtand the HIV/AIDS force of infection is given byλH=β1H1+ρH2 with initial conditions:
(8)S0>0,H10≥0,H20≥0,H30≥0,TH0≥0.The sum of all the differential equations in (7) is obtained as
(9)dN1dt=Λ−μN1−δ1H3.Consider the regionΩ1=S,H1,H2,H3,TH∈ℝ5+,N1≤Λ/μ. It is easy to show that the set Ω1 is positively invariant and a global attractor of all positive solution of submodel. Hence, it is sufficient to consider the dynamics of model (7) in Ω1. In this region, the model is considered epidemiologically and mathematically well posed.


#### 3.1.1. Disease-Free Equilibrium Point of the HIV/AIDS-Only Submodel

The disease-free equilibrium point (DFE) of the HIV/AIDS-only submodel (7) is denoted by EH0=S0,H10,H20,H30,TH0 and obtained by making the right hand side of the system as zero and setting all the infectious classes and treatment class to zero as H1=H2=H3=TH=0 we have got S0=Λ/μ such that EH0=Λ/μ,0,0,0,0.


#### 3.1.2. The Basic Reproduction Number of the Submodel

The basic reproduction number of HIV/AIDS-infected individuals denoted byRH is defined as the average number of secondary infections produced by a single HIV/AIDS infectious individual introduced in a wholly susceptible population during his or her entire infectious period [14, 20]. This definition is given for the dynamical system that represents the spread of infection in a population. We calculate the basic reproduction number by using the next-generation operator method on the dynamical system (7). The basic reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEH0/∂xj∂νiEH0/∂xj−1 [21, 22] 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 EH0 is the disease-free equilibrium point.After some calculations we have gotF=β1Λ/μβ1ρΛ/μ00,V=μ+γ1+α10−α1μ+γ2+α2, V−1=1/μ+α1+γ10α1/μ+α1+γ1μ+α2+γ21/μ+α2+γ2, and FV−1=β1Λ/μμ+γ1+α1μμ+γ1+α1+β1ρΛα1/μμ+γ1+α1μ+γ2+α2μμ+γ1+α1μ+γ2+α2β1ρΛ/μμ+γ2+α200.Then, the spectral radius (reproduction numberRH) of FV−1 of the HIV/AIDS subdynamical system (7) is ρFV−1=RH=β1Λ/μμ+γ1+α1+β1ρΛα1/μμ+γ1+α1μ+γ2+α2=RH1+RH2.


#### 3.1.3. Local Stability of the Submodel Disease-Free Equilibrium Point

Theorem 1.
The disease-free equilibrium pointEH0 of the HIV/AIDSonly submodel is locally asymptotically stable (LAS) if RH<1, and unstable if RH<1.Proof.
“AppendixC”Biologically speaking, Theorem1 implies that HIV can be eliminated from the population when RH<1if the initial sizes of the subpopulation of the submodel are in the region of attraction of EH0.


#### 3.1.4. Existence and Stability of Endemic Equilibrium Point of the Submodel

Before investigating the global asymptotic stability of the DFE, it is instructive to determine the number of endemic equilibrium solutions of the model (7). Let an arbitrary equilibrium point of a HIV/AIDS-only dynamical system (7) is denoted by EH∗=S∗,H1∗,H2∗,H3∗,TH∗. Moreover, let λH∗=β1H1∗+ρH2∗ be the associated infection rate (“force of infection”) at endemic equilibrium point. After some calculations, we have got λH∗=μRH−1.⟹λH∗>0 if RH>1 and hence an endemic equilibrium point EH∗=S∗,H1∗,H2∗,H3∗,TH∗ of the HIV/AIDS submodel (7) exist whenever RH>1whereS∗=Λ/μR0H,H1∗=ΛRH−1/d1R0H with d1=μ+γ1+α1,H2∗=α1ΛRH−1/d1d2R0H with d2=μ+γ2+α2,H3∗=α2μRH−1/d3 with d3=μ+γ3+δ1 and TH∗=ΛRH−1γ1/d1R0HΛRH−1γ1/d1R0H+α1ΛRH−1γ2/d1d2R0Hα1ΛRH−1γ2/d1d2R0H+α2μRH−1γ3/d3/μ all are positives if RH>1.Theorem 2.
The submodel (7) has a unique endemic equilibrium point if and only if RH>1.


#### 3.1.5. Global Asymptotic Stability (GAS) of the Disease-Free Equilibrium Point

Theorem 3.
The disease-free equilibrium pointEH0=Λ/μ,0,0,0,0 of the dynamical system (7) is globally asymptotically stable if RH≤1 otherwise unstable.Proof.
To show global stability of the DFE applied Lyapunov function method as [13, 23].Let the Lyapunov functionL:R+5⟶R+ is defined by: LS,H1,H2,H3,TH=a2H1+a3H2whereEH0=Λ/μ,0,0,0,0 disease-free equilibrium is point where a2=1/μ+γ1+α1μ+γ2+α2 and a3=1−RH1/μ+γ2+α2 are positive constants. Then, dL/dt=a2dH1/dt+a3dH2/dt:(10)⟹dLdt=a2λHS−μ+γ1+α1H1+a3α1H1−μ+γ2+α2H2≤a2β1Λμ−a2μ+γ1+α1+a3α1H1+a2β1ρΛμ−a3μ+γ2+α2H2.SinceS≤S0=Λ/μ to find values of a2and a3 take a2β1Λ/μ−a2μ+γ1+α1+a3α1=0⟹a3=−a2β1Λ/μΛ/μ+a2μ+γ1+α1/α1; then, we obtained as dL/dt≤a2β1ρΛ/μ+a2β1Λμ+γ2+α2/μα1−a2μ+γ1+α1μ+γ2+α2/α1H2.Here, we can takea2=1/μ+γ1+α1μ+γ2+α2 and a3=1−RH1/μ+γ2+α2 with RH1=β1Λ/μμ+γ1+α1 which is the reproduction number for acute HIV infection (H1) and a3 is positive for RH1<1.Then, we obtained(11)dLdt≤β1ρΛμμ+γ1+α1μ+γ2+α2+β1Λμα1μ+γ1+α1−1α1=α1RH−1,⟹dL/dt≤α1RH−1⟹dL/dt≤0 if RH≤1and dL/dt=0 if and only if S=S0,H1=H10,H2=H20,H3=H30, and TH=TH0. Therefore, the largest compact invariant set in S,H1,H2,H3,TH∈Ω1:dL/dt=0 is the singleton EH0where EH0is the disease-free equilibrium point of the model (7).Thus, by LaSalle’s invariance principle [24], it implies that the disease-free equilibrium point EH0=Λ/μ,0,0,0,0 is globally asymptotically stable in Ω1 if RH≤1.


#### 3.1.6. Local and Global Stabilities of Endemic Equilibrium Point of the HIV/AIDS-Only Submodel

Theorem 4.
The endemic equilibrium pointEH∗=S∗,H1∗,H1∗,H1∗,H1∗ is locally asymptotically stable for the basic reproduction numberRH>1.Proof.
AppendixDTheorem 5.
The endemic equilibrium pointEH∗=S∗,H1∗,H2∗,H3∗,TH∗ is globally asymptotically stable for the basic reproduction number RH>1, otherwise unstable.Proof.
Let the Lyapunov functionV:R+5⟶R+ is defined by
(12)LS,H1,H2,H3,TH=a1S−S∗−S∗lnSS∗+a2H1−H1∗−H1∗lnH1H1∗+a3H2−H2∗−H2∗lnH2H2∗.Here, we havedL/dt=a1dS/dt−S∗/SdS/dt+a2dH1/dt−H1∗/H1H1∗/H1dH1/dt+a3dH2/dt−H2∗/H2H2∗/H2dH2/dt(13)⟹dLdt=a11−S∗SdSdt+a21−H1∗H1dH1dt+a31−H2∗H2dH2dt.At the endemic equilibrium pointEH∗=S∗,H1∗,H2∗,H3∗,TH∗, we obtain from the system (7): Λ=μ+λH∗S∗, λH∗S∗=μ+γ1+α1H1∗ and α1H1∗=μ+γ2+α2H2∗⟹Λ=μ+λH∗S∗,μ+γ1+α1H1=λH∗S∗H1/H1∗ and μ+γ2+α2H2=α1H1∗H2/H2∗.Then, we obtained(14)dLdt=a11−S∗Sμ+λH∗S∗−μ+λHS+a21−H1∗H1λHS−λH∗S∗H1H1∗+a31−H2∗H2α1H1−α1H1∗H2H2∗⟹dLdt=−a1μS−S∗2S+a1+a2β1H1∗S∗+a1+a2β1ρH2∗S∗+a2−a1β1H1S+a2−a1β1ρH2S−a1S∗2Sβ1H1∗−a1S∗2Sβ1ρH2∗+a3α1−a2β1ρS∗H2∗H1∗H1+a1β1ρS∗−a3α1H1∗H2∗H2+a1β1H1S∗−a2β1H1S∗−a2β1H1∗S−a2β1ρH2SH1∗H1−a3α1H1H2∗H2+a3α1H1∗.Choosea1,a2, and a3 such that the expressions in the brackets vanish and take a1=a2 and solve for a3, by making expressions in the closed bracket zero, we obtain as a3α1−a2β1ρS∗H2∗/H1∗=0(15)⟹a3=a1β1ρS∗H2∗α1H1∗fora1=a2,⟹dLdt=−a1μS−S∗2S+2a1β1H1∗S∗+2a1β1ρH2∗S∗−a1S∗2Sβ1H1∗−a1S∗2Sβ1ρH2∗+a1β1H1S∗−a1β1H1S∗−a1β1H1∗S−a1β1ρH2SH1∗H1−a1β1ρS∗H2∗2H1∗H1H2+a1β1ρS∗H2∗.Grouping some terms in the expression above yields(16)dLdt=−a1μS−S∗2S+a1β1H1∗S∗2−SS∗−S∗S+a1β1ρH2∗S∗3−S∗S−SH2H1∗S∗H2∗H1−H2∗H1H2H1∗.Using the arithmetic-geometric mean inequality property, we have2−S/S∗−S∗/S≤0 and 3−S∗/S−SH2H1∗/S∗H2∗H1−H2∗H1/H2H1∗≤0.Hence, we conclude thatdL/dt≤0, and hence, L is the representative Lyapunov function.Furthermore,dL/dt=0 if and only if S,H1,H2,H3,TH=S∗,H1∗,H2∗,H3∗,TH∗, and the largest invariant subset contained in the set E∞=S,H1,H2,H3,TH∈Ω1:dL/dt=0 is the set contained only the endemic equilibrium point:
(17)EH∗=S∗,H1∗,H2∗,H3∗,TH∗.Therefore, we conclude by LaSalle’s invariance principle [24] that EH∗=S∗,H1∗,H2∗,H3∗,TH∗ is globally asymptotically stable (GAS) if RH>1.


### 3.2. Pneumonia Submodel Analysis

We have the pneumonia submodel of (3) when H1=H2=H3=C1=C2=C3=TH=T=0, which is given by
(18)dSdt=Λ+pγTP−μ+λPS,dIPdt=λPS−μ+γ+δPIP,dTPdt=γIP−μ+pγTP,where the total population isN2t=St+Ipt+Tpt and the pneumonia force of infection is given by λP=β2IP with initial conditions
(19)S0>0,IP0≥0,TP0≥0.The sum of all the differential equations in (18) above is obtained as
(20)dN2dt=Λ−μN−δPIp.Consider the regionΩ2=S,IP,TP∈ℝ3+,N2≤Λ/μ. It is easy to show that the set Ω2 is positively invariant and a global attractor of all positive solution of submodel (18). Hence, it is sufficient to consider the dynamics of model (18) inΩ1. In this region, the model is epidemiologically and mathematically well posed.


#### 3.2.1. Disease-Free Equilibrium Point (DFE) of the Pneumonia-Only Submodel

The disease-free equilibrium point (DFE) of the system (18) is obtained by making the right hand side of the system as zero and setting all the infectious classes and treatment class to zero as IP=TP=0 we have got S0=Λ/μ such that EP0=Λ/μ,0,0.


#### 3.2.2. The Reproduction Number of the Pneumonia-Only Submodel

We calculate the basic reproduction number denoted byRP using the van den Driesch and Warmouth next-generation matrix approach from [22]. The basic reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEP0/∂xj∂νiEP0/∂xj−1, whereFiis the rate of appearance of new infection in compartmenti, νi is the transfer of infections from one compartment ito another, and EP0 is the disease-free equilibrium point. The reproduction numberRPof the pneumonia-only dynamical system (18) is obtained by rearrange the differential equation of the dynamical system (18) above in terms of dXi/dt=Fi−vi=Fi−νi−−νi+. Then, after some calculations, we have got FV−1=β2Λ/μμ+γ+δP000 and the spectral radius (reproduction numberRP) of FV−1 of the pneumonia subdynamical system (18) is RP=β2Λ/μμ+γ+δP .


#### 3.2.3. Local Stability of the Submodel Disease-Free Equilibrium Point

Theorem 6.
The disease-free equilibrium point of the pneumonia-only submodel is locally asymptotically stable (LAS) ifRP<1, otherwise unstable.Proof.
The local stability of the disease-free equilibrium of the system (18) can be studied from its Jacobian matrix at the disease-free equilibrium point EP0=S0,IP0,TP0=Λ/μ,0,0and Routh-Hurwitz stability criteria. Then, the Jacobian matrix of the dynamical system (18) at EP0=Λ/μ,0,0 is given by
(21)JEP0=−μβ2Λμpγ0−μ+γ+δP00γ−μ+pγ.Then, the characteristic equation of the above Jacobian matrix is given by(22)−μ−λβ2Λμpγ0β2Λμ−μ+γ+δP−λ00γ−μ+pγ−λ=0,⟹−μ−λβ2Λμ−μ+γ+δP−λ−μ+pγ−λ=0,⟹λ1=−μ,λ2=β2Λμ−μ+γ+δP,λ3=−μ+pγ,⟹λ2=β2Λμ−μ+γ+δP=μ+γ+δPβ2Λμμ+γ+δP−1,⟹λ2=μ+γ+δPRP−1<0ifRP<1.Therefore, since all the eigenvalues of the characteristics polynomial of the system (18) are negative for RP<1, the disease-free equilibrium point of the system (18) is locally asymptotically stable.


#### 3.2.4. Existence of Endemic Equilibrium Point of the Pneumonia Submodel

Before investigating the global asymptotic stability of the DFE, it is instructive to determine the number of endemic equilibrium solutions of the model (18). Let an arbitrary endemic equilibrium point of the pneumonia-only dynamical system (18) be denoted byEP∗=S∗,IP∗,TP∗. Now, after some calculations, we have got a unique endemic equilibrium point EP∗=S∗,IP∗,TP∗ where S∗=d2Λm2+d2Λm3D1RP−1+D2pγRP−1/d2m2+d2m3D1RP−1μ+D1RP−1>0,IP∗=d2ΛD1RP−1/m2+m3D1RP−1>0 and TP∗=D2RP−1/d2m2+d2m3D1RP−1>0 ifRP>1 where D1=μ+γ+δPμ+pγμ/μ+γ+δPμ+pγ−pγ2 and D2=d2γΛD1.Theorem 7.
The model (18) has a unique endemic equilibrium point if RP>1.


#### 3.2.5. Globally Asymptotically Stability (GAS) of the Disease-Free Equilibrium Point

Theorem 8.
The disease-free equilibrium pointEP0=S0,IP0,TP0=Λ/μ,0,0 of the dynamical system (18) is globally asymptotically stable if RP≤1, otherwise unstable.Proof.
Let the Lyapunov functionL:R+3⟶R+ is defined by: LS,IP,TP=a1IP
whereEP0=S0,IP0,TP0=Λ/μ,0,0 is disease-free equilibrium point where a1=1 is a positive constant.ThendL/dt=a1dIP/dt(23)⟹dLdt=a1dIPdt=a1λPS−μ+γ+δPIP=a1β2IPS−μ+γ+δPIP⟹dLdt=a1β2S−μ+γ+δPIP≤a1β2S0−μ+γ+δPIPsinceS≤S0=Λμ,(24)⟹dLdt≤a1β2Λμ−μ+γ+δPIP=a1μ+γ+δPβ2Λμμ+γ+δP−1,⟹dLdt≤a1μ+γ+δPRP−1=μ+γ+δPRP−1,⟹dL/dt≤0 if RP≤1. Hence, the disease-free equilibrium point EP0=Λ/μ,0,0 is globally asymptotically stable whenever RP≤1.


#### 3.2.6. Local and Global Stabilities of Endemic Equilibrium Point

The following theorem studies the local stability of the endemic equilibrium. The result is obtained by means of the Routh-Hurwitz stability criteria.Theorem 9.
The endemic equilibrium point of the system (18) is locally asymptotically stable ifRP>1.Proof.
To show the local stability of the endemic equilibrium point, we use the method of the Jacobian matrix and Routh Hurwitz stability criteria. Then, the Jacobian matrix of the dynamical system (18) at the endemic equilibrium point EP∗=S∗,IP∗,TP∗ whereS∗=d2Λm2+d2Λm3D1RP−1+D2pγRP−1/d2m2+d2m3D1RP−1μ+D1RP−1,IP∗=d2ΛD1RP−1/m2+m3D1RP−1 and TP∗=D2RP−1/d2m2+d2m3D1RP−1,D2=d2γΛD1 for the system (18) whenever RP>1 is given by
(25)JEP∗=−μ−β2IP∗−β2S∗pγβ2IP∗β2S∗−d100γ−d2,
⟹JEP∗=A1A4pγA5A200γA3 whereA1=−μ−β2IP∗,A2=β2S∗−d1,A3=−d2,A4=−β2S∗, and A5=β2IP∗.Then, the characteristic equation of the above Jacobian matrix is given by(26)A1−λA4pγA5A2−λ00γA3−λ=0,⟹a3λ3+a2λ2+a1λ+a0=0wherea3=1,a2=−A1+A2+A3,a1=A1A2+A1A3+A2A3−A4A5, and a0=−A1A2A3+A3A4A5−pγ2A5.Here, we apply the necessary condition of Routh-Hurwitz stability criteria sincea3=1 is positive in sign, indicating all a2,a1, and a0 should be positive.Hence,a2=−A1+A2+A3=μ+β2IP∗−β2S∗+d1+d2(27)⟹a2=μd2m2+d2m3D1RP−1μ+D1RP−1+β2D2RP−1μ+D1RP−1−β2d2Λm2−β2d2Λm3D1RP−1−β2D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1+d1+d2d2m2+d2m3D1RP−1μ+D1RP−1>0ifRP>1,a1=A1A2+A1A3+A2A3−A4A5=μ+β2d2ΛD1RP−1m2+m3D1RP−1β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1−d1+d2μ+β2d2ΛD1RP−1m2+m3D1RP−1+β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1β2d2ΛD1RP−1m2+m3D1RP−1>0ifRP>1,(28)a0=−A1A2A3+A3A4A5−pγ2A5=μ+β2IP∗β2S∗−d1d2+d2β2S∗β2IP∗−β2IP∗pγ2=μ+β2d2ΛD1RP−1m2+m3D1RP−1β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1−d1d2+d2β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1β2d2ΛD1RP−1m2+m3D1RP−1−β2d2ΛD1RP−1m2+m3D1RP−1pγ2>0ifRP>1.Thus, all the coefficients of the characteristic’s polynomiala0,a1,a2, anda3 have the same sign. Then, applying the Routh-Hurwitz criteria to determine the sign of the root without calculating their values of the root of the characteristics equationa3λ3+a2λ2+a1λ+a0=0 and we have got the Routh-Hurwitz array that has no sign change, indicating the roots of the characteristics equation of the dynamical system (18) are negatives. Hence, the endemic equilibrium point EP∗=S∗,IP∗,TP∗ of the dynamical system (18) is locally asymptotically stable.


#### 3.2.7. Global Stability of Endemic Equilibrium of Pneumonia Submodel

Theorem 10.
IfRP>1 the unique endemic equilibrium pointEP∗=S∗,IP∗,TP∗ is globally asymptotically stable in the interior of Ω2.Proof.
Let us take the Lyapunov functionL:ℝ3+⟶ℝ byLS,IP,TP=A1S−S∗+IP−IP∗+TP−TP∗2+A2IP−IP∗−IP∗lnIP/IP∗+A3TP−TP∗2 for positive constantsA1=1/2,A2=δP+2μ/β2 andA3=δP+2μ/2γ. Then, L is C1 on the interior of Ω2, EP∗=S∗,IP∗,TP∗ is the global minimum of L on Ω2, andLS∗,IP∗,TP∗=0.
Then, the time derivative ofL is given by
(29)dLdt=2A1S−S∗+IP−IP∗+TP−TP∗dS+IP+TPdt+A21−IP∗IPdIPdt+2A3TP−TP∗dTPdt,⟹dLdt=S−S∗+I−IP∗+TP−TP∗dS+IP+TPdt+δP+2μβ21−IP∗IPdIPdt+δP+2μγTP−TP∗dTPdt.Using expressions at the endemic equilibrium,(30)Λ=μS∗+IP∗+TP∗+δPIP∗,β2S∗=μ+γ+δP,0=μ+pγTP∗−γIP∗,we have got
(31)dLdt=S−S∗+I−IP∗+TP−TP∗μS∗+IP∗+TP∗+δPIP∗−μS+IP+TP−δPIP+δP+2μβ2IP−IP∗IPβ2IPS−β2k2S∗IP+δP+2μγTP−TP∗γIP−μ+pγTP+μ+pγTP∗−γIP∗,⟹dLdt=−μS−S∗+TP−TP∗2−δP+μIP−IP∗2−δP+2μμ+pγTP−TP∗2≤0.Hence,dL/dt is negative. Note that dL/dt=0 if and only if S=S∗,IP=IP∗ and TP=TP∗. Therefore, the largest compact invariant set inS,IP,TP∈Ω2:dL/dt=0 is the singleton {EP∗} set where EP∗ the endemic equilibrium point. Then, by LaSalle’s invariant principle, [24] then implies that EP∗is globally asymptotically stable in the interior of Ω2 if RP>1.


### 3.3. Analysis of the HIV-Pneumonia Coinfection Model (3)

In this section, we analyze the main dynamical system (3). Epidemiologically, the HIV/AIDS-pneumonia coinfection dynamical system (3) will have four equilibrium points, namely, HIV/AIDS-pneumonia coinfection model disease-free equilibrium pointEHP0, HIV/AIDS-only endemic equilibrium pointEH∗, pneumonia-only endemic equilibrium point EP∗, and the endemic equilibrium point of the coexistence of HIV-pneumonia denoted by EHP∗.


#### 3.3.1. Disease-Free Equilibrium Point of the HIV-Pneumonia Coinfection Model

The disease-free equilibrium pointEHP0=S0,H10,H20,H30,C10,C20,C30,TP0,TH0,T0 of the system (3) is obtained by setting all the infectious classes and treatment classes to zero and making the right hand side of the system as zero. Then, the disease-free equilibrium of the HIV/AIDS coinfection model is EHP0=Λ/μ,0,0,0,0,0,0,0,0,0.


#### 3.3.2. The Basic Reproduction NumberRHP of the HIV-Pneumonia Coinfection Model

The basic reproduction number of the dynamical system (3) by applying the next-generation operator method [21, 22] is the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEHP0/∂xj∂νiEHP0/∂xj−1, where Fi is the rate of appearance of new infection in compartment i, νi is the transfer of infections from one compartment ito another, and EHP0 is the disease-free equilibrium point. Here, we obtained the following matrices:
(32)F=β2Λμ000β2ω1Λμβ2ω2Λμβ2ω3Λμ0000β1Λμβ1ρΛμ000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,V=m10000000000m2000000000−α1m3000000000−α2m40000000000m5000000000−α3m6000000000−α4m7000−γ000000m8000−γ1−γ2−γ30000μ00000−ε1−ε2−ε300μ.Then, using Mathematica, we have got the spectral radius (dominant eigenvalue) of the matrixFV−1 is RHP=maxΛβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2,Λβ2/μμ+γ+δP where RP=Λβ2/μμ+γ+δP is the basic reproduction number for pneumonia infection andRH=Λβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2 is the basic reproduction number for HIV/AIDS infection, and hence, RHP=maxRP,RH is the basic reproduction number of the HIV and pneumonia co-infection.


#### 3.3.3. Local Stability of the Disease-Free Equilibrium Point

Theorem 11.
The disease-free equilibrium point of the model (3) above is locally asymptotically stable if RHP<1 and unstable ifRHP>1.Proof.
The Jacobian matrixJEHP0 of the model at EHP0 is given by
(33)JEHP0=−μβ2Λμ−β1Λμ−β1ρΛμ0−β2ω1Λμ−β2ω2Λμ−β2ω3Λμpγ000Z2000β2ω1Λμβ2ω2Λμβ2ω3Λμ00000Z3β1ρΛμ000000000α1Z40000000000α2Z500000000000Z60000000000α3Z70000000000α4Z8000γ0000000Z90000γ1γ2γ30000−μ000000ε1ε2ε300−μ.The eigenvalues of the above Jacobian matrix areλ1=λ2=λ3=−μ<0 or λ4=Z2=β2Λ/μ−μ+γ+δP or λ5=Z5=−μ+γ3+α3<0 or λ6=Z6=−μ+δP+ε1+α3<0 or λ7=Z7=−μ+δP+ε2+α4<0orλ8=Z8=−μ+δ2+ε3<0orλ9=Z9=−μ+pγ<0 or a2λ2+a1λ+a0=0 fora2=1,a1=−Z3−Z4anda0=Z3Z4−β1ρα1Λ/μ.Here,λ4=Z2=β2Λ/μ−μ+γ+δP=μ+γ+δPβ2Λ/μμ+γ+δP−1=μ+γ+δPRP−1⟹λ4=μ+γ+δPRP−1<0 ifRP<1.To check the remaining eigenvalues are negatives for the quadratic equationa2λ2+a1λ+a0=0, we can apply Routh-Hurwitz stability criteria since a2=1>0 both a1 and a0 should be positives.Now,a1=μ+γ2+α2+μ+γ1+α11−RH1⟹a1>0 if RH1<1 anda0=μ+γ2+α2μ+γ1+α11−β1ρα1Λ/μμ+γ2+α2μ+γ1+α1⟹a0=μ+γ2+α2μ+γ1+α11−RH2>0 if RH2<1.Hence, the last two eigenvalues are negatives ifRH<1 and thus all the eigenvalues are negatives if RHP=maxRH,RP<1. Thus, since all the eigenvalues are negatives, the disease-free equilibrium point of the HIV/AIDS-pneumonia coinfection dynamical system (3) above is locally asymptotically stable if
(34)RHP=maxRH,RP<1.


#### 3.3.4. Existence of Endemic Equilibrium Point for the Full HIV/AIDS-Pneumonia Coinfection Model

The endemic equilibrium point (EEP) of the model (3) above is denoted by EHP∗=S∗,IP∗,H1∗,H2∗,H3∗,C1∗,C2∗,C3∗,TP∗,TH∗,T∗ which occurs when the disease persist in the community. From the analysis of the HIV-only submodel (7) and the pneumonia-only submodel (18), we have shown that there is no endemic equilibrium point if RH<1and RP<1 implies that there is no endemic equilibrium point if RHP<1 for the coinfection model. The endemic equilibrium of the system (3) above can be obtained as
(35)S∗=Λ+pγTP∗μ+λH∗+λP∗,IP∗=λP∗S∗μ+γ+δP,H1∗=λH∗S∗μ+γ1+α1+φ1λP∗,H2∗=α1H1∗μ+γ2+α2+φ2λP∗,H3∗=α2H2∗μ+γ3+δ1+φ3λP∗,C1∗=φ1λP∗H1∗μ+δP+ε1+α3,C2∗=φ2λP∗H2∗+α3C1∗μ+δP+ε2+α4,C3∗=φ3λP∗H3∗+α4C2∗μ+δ2+ε3(36)TP∗=γIP∗μ+pγ,TH∗=γ1H1∗+γ2H2∗+γ3H3∗μ,T=ε1C1∗+ε2C2∗+ε3C3∗μ.Summary of endemic equilibrium point. The explicit computation of the endemic equilibrium of the full model (3) given in equations (35) and (36) in terms of model parameters is difficult analytically; however, the model (3) above endemic equilibrium EHP∗=S∗,IP∗,H1∗,H2∗,H3∗,C1∗,C2∗,C3∗,TP∗,TH∗,T∗ exists ifRH>1andRP>1, i.e., RHP>1. We will give an explanation of E3∗ in our numerical simulations.


## 3.1. HIV/AIDS-Only Submodel Analysis

The HIV submodel of (3) (obtained by settingIP=C1=C2=C3=TP=T=0 is given by
(7)dSdt=Λ−μ+λHS,dH1dt=λHS−μ+γ1+α1H1,dH2dt=α1H1−μ+γ2+α2H2,dH3dt=α2H2−μ+γ3+δ1H3,dTHdt=γ1H1+γ2H2+γ3H3−μTH,where the total population isN1t=St+H1t+H2t+H3t+THtand the HIV/AIDS force of infection is given byλH=β1H1+ρH2 with initial conditions:
(8)S0>0,H10≥0,H20≥0,H30≥0,TH0≥0.The sum of all the differential equations in (7) is obtained as
(9)dN1dt=Λ−μN1−δ1H3.Consider the regionΩ1=S,H1,H2,H3,TH∈ℝ5+,N1≤Λ/μ. It is easy to show that the set Ω1 is positively invariant and a global attractor of all positive solution of submodel. Hence, it is sufficient to consider the dynamics of model (7) in Ω1. In this region, the model is considered epidemiologically and mathematically well posed.


### 3.1.1. Disease-Free Equilibrium Point of the HIV/AIDS-Only Submodel

The disease-free equilibrium point (DFE) of the HIV/AIDS-only submodel (7) is denoted by EH0=S0,H10,H20,H30,TH0 and obtained by making the right hand side of the system as zero and setting all the infectious classes and treatment class to zero as H1=H2=H3=TH=0 we have got S0=Λ/μ such that EH0=Λ/μ,0,0,0,0.


### 3.1.2. The Basic Reproduction Number of the Submodel

The basic reproduction number of HIV/AIDS-infected individuals denoted byRH is defined as the average number of secondary infections produced by a single HIV/AIDS infectious individual introduced in a wholly susceptible population during his or her entire infectious period [14, 20]. This definition is given for the dynamical system that represents the spread of infection in a population. We calculate the basic reproduction number by using the next-generation operator method on the dynamical system (7). The basic reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEH0/∂xj∂νiEH0/∂xj−1 [21, 22] 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 EH0 is the disease-free equilibrium point.After some calculations we have gotF=β1Λ/μβ1ρΛ/μ00,V=μ+γ1+α10−α1μ+γ2+α2, V−1=1/μ+α1+γ10α1/μ+α1+γ1μ+α2+γ21/μ+α2+γ2, and FV−1=β1Λ/μμ+γ1+α1μμ+γ1+α1+β1ρΛα1/μμ+γ1+α1μ+γ2+α2μμ+γ1+α1μ+γ2+α2β1ρΛ/μμ+γ2+α200.Then, the spectral radius (reproduction numberRH) of FV−1 of the HIV/AIDS subdynamical system (7) is ρFV−1=RH=β1Λ/μμ+γ1+α1+β1ρΛα1/μμ+γ1+α1μ+γ2+α2=RH1+RH2.


### 3.1.3. Local Stability of the Submodel Disease-Free Equilibrium Point

Theorem 1.
The disease-free equilibrium pointEH0 of the HIV/AIDSonly submodel is locally asymptotically stable (LAS) if RH<1, and unstable if RH<1.Proof.
“AppendixC”Biologically speaking, Theorem1 implies that HIV can be eliminated from the population when RH<1if the initial sizes of the subpopulation of the submodel are in the region of attraction of EH0.


### 3.1.4. Existence and Stability of Endemic Equilibrium Point of the Submodel

Before investigating the global asymptotic stability of the DFE, it is instructive to determine the number of endemic equilibrium solutions of the model (7). Let an arbitrary equilibrium point of a HIV/AIDS-only dynamical system (7) is denoted by EH∗=S∗,H1∗,H2∗,H3∗,TH∗. Moreover, let λH∗=β1H1∗+ρH2∗ be the associated infection rate (“force of infection”) at endemic equilibrium point. After some calculations, we have got λH∗=μRH−1.⟹λH∗>0 if RH>1 and hence an endemic equilibrium point EH∗=S∗,H1∗,H2∗,H3∗,TH∗ of the HIV/AIDS submodel (7) exist whenever RH>1whereS∗=Λ/μR0H,H1∗=ΛRH−1/d1R0H with d1=μ+γ1+α1,H2∗=α1ΛRH−1/d1d2R0H with d2=μ+γ2+α2,H3∗=α2μRH−1/d3 with d3=μ+γ3+δ1 and TH∗=ΛRH−1γ1/d1R0HΛRH−1γ1/d1R0H+α1ΛRH−1γ2/d1d2R0Hα1ΛRH−1γ2/d1d2R0H+α2μRH−1γ3/d3/μ all are positives if RH>1.Theorem 2.
The submodel (7) has a unique endemic equilibrium point if and only if RH>1.


### 3.1.5. Global Asymptotic Stability (GAS) of the Disease-Free Equilibrium Point

Theorem 3.
The disease-free equilibrium pointEH0=Λ/μ,0,0,0,0 of the dynamical system (7) is globally asymptotically stable if RH≤1 otherwise unstable.Proof.
To show global stability of the DFE applied Lyapunov function method as [13, 23].Let the Lyapunov functionL:R+5⟶R+ is defined by: LS,H1,H2,H3,TH=a2H1+a3H2whereEH0=Λ/μ,0,0,0,0 disease-free equilibrium is point where a2=1/μ+γ1+α1μ+γ2+α2 and a3=1−RH1/μ+γ2+α2 are positive constants. Then, dL/dt=a2dH1/dt+a3dH2/dt:(10)⟹dLdt=a2λHS−μ+γ1+α1H1+a3α1H1−μ+γ2+α2H2≤a2β1Λμ−a2μ+γ1+α1+a3α1H1+a2β1ρΛμ−a3μ+γ2+α2H2.SinceS≤S0=Λ/μ to find values of a2and a3 take a2β1Λ/μ−a2μ+γ1+α1+a3α1=0⟹a3=−a2β1Λ/μΛ/μ+a2μ+γ1+α1/α1; then, we obtained as dL/dt≤a2β1ρΛ/μ+a2β1Λμ+γ2+α2/μα1−a2μ+γ1+α1μ+γ2+α2/α1H2.Here, we can takea2=1/μ+γ1+α1μ+γ2+α2 and a3=1−RH1/μ+γ2+α2 with RH1=β1Λ/μμ+γ1+α1 which is the reproduction number for acute HIV infection (H1) and a3 is positive for RH1<1.Then, we obtained(11)dLdt≤β1ρΛμμ+γ1+α1μ+γ2+α2+β1Λμα1μ+γ1+α1−1α1=α1RH−1,⟹dL/dt≤α1RH−1⟹dL/dt≤0 if RH≤1and dL/dt=0 if and only if S=S0,H1=H10,H2=H20,H3=H30, and TH=TH0. Therefore, the largest compact invariant set in S,H1,H2,H3,TH∈Ω1:dL/dt=0 is the singleton EH0where EH0is the disease-free equilibrium point of the model (7).Thus, by LaSalle’s invariance principle [24], it implies that the disease-free equilibrium point EH0=Λ/μ,0,0,0,0 is globally asymptotically stable in Ω1 if RH≤1.


### 3.1.6. Local and Global Stabilities of Endemic Equilibrium Point of the HIV/AIDS-Only Submodel

Theorem 4.
The endemic equilibrium pointEH∗=S∗,H1∗,H1∗,H1∗,H1∗ is locally asymptotically stable for the basic reproduction numberRH>1.Proof.
AppendixDTheorem 5.
The endemic equilibrium pointEH∗=S∗,H1∗,H2∗,H3∗,TH∗ is globally asymptotically stable for the basic reproduction number RH>1, otherwise unstable.Proof.
Let the Lyapunov functionV:R+5⟶R+ is defined by
(12)LS,H1,H2,H3,TH=a1S−S∗−S∗lnSS∗+a2H1−H1∗−H1∗lnH1H1∗+a3H2−H2∗−H2∗lnH2H2∗.Here, we havedL/dt=a1dS/dt−S∗/SdS/dt+a2dH1/dt−H1∗/H1H1∗/H1dH1/dt+a3dH2/dt−H2∗/H2H2∗/H2dH2/dt(13)⟹dLdt=a11−S∗SdSdt+a21−H1∗H1dH1dt+a31−H2∗H2dH2dt.At the endemic equilibrium pointEH∗=S∗,H1∗,H2∗,H3∗,TH∗, we obtain from the system (7): Λ=μ+λH∗S∗, λH∗S∗=μ+γ1+α1H1∗ and α1H1∗=μ+γ2+α2H2∗⟹Λ=μ+λH∗S∗,μ+γ1+α1H1=λH∗S∗H1/H1∗ and μ+γ2+α2H2=α1H1∗H2/H2∗.Then, we obtained(14)dLdt=a11−S∗Sμ+λH∗S∗−μ+λHS+a21−H1∗H1λHS−λH∗S∗H1H1∗+a31−H2∗H2α1H1−α1H1∗H2H2∗⟹dLdt=−a1μS−S∗2S+a1+a2β1H1∗S∗+a1+a2β1ρH2∗S∗+a2−a1β1H1S+a2−a1β1ρH2S−a1S∗2Sβ1H1∗−a1S∗2Sβ1ρH2∗+a3α1−a2β1ρS∗H2∗H1∗H1+a1β1ρS∗−a3α1H1∗H2∗H2+a1β1H1S∗−a2β1H1S∗−a2β1H1∗S−a2β1ρH2SH1∗H1−a3α1H1H2∗H2+a3α1H1∗.Choosea1,a2, and a3 such that the expressions in the brackets vanish and take a1=a2 and solve for a3, by making expressions in the closed bracket zero, we obtain as a3α1−a2β1ρS∗H2∗/H1∗=0(15)⟹a3=a1β1ρS∗H2∗α1H1∗fora1=a2,⟹dLdt=−a1μS−S∗2S+2a1β1H1∗S∗+2a1β1ρH2∗S∗−a1S∗2Sβ1H1∗−a1S∗2Sβ1ρH2∗+a1β1H1S∗−a1β1H1S∗−a1β1H1∗S−a1β1ρH2SH1∗H1−a1β1ρS∗H2∗2H1∗H1H2+a1β1ρS∗H2∗.Grouping some terms in the expression above yields(16)dLdt=−a1μS−S∗2S+a1β1H1∗S∗2−SS∗−S∗S+a1β1ρH2∗S∗3−S∗S−SH2H1∗S∗H2∗H1−H2∗H1H2H1∗.Using the arithmetic-geometric mean inequality property, we have2−S/S∗−S∗/S≤0 and 3−S∗/S−SH2H1∗/S∗H2∗H1−H2∗H1/H2H1∗≤0.Hence, we conclude thatdL/dt≤0, and hence, L is the representative Lyapunov function.Furthermore,dL/dt=0 if and only if S,H1,H2,H3,TH=S∗,H1∗,H2∗,H3∗,TH∗, and the largest invariant subset contained in the set E∞=S,H1,H2,H3,TH∈Ω1:dL/dt=0 is the set contained only the endemic equilibrium point:
(17)EH∗=S∗,H1∗,H2∗,H3∗,TH∗.Therefore, we conclude by LaSalle’s invariance principle [24] that EH∗=S∗,H1∗,H2∗,H3∗,TH∗ is globally asymptotically stable (GAS) if RH>1.


## 3.1.1. Disease-Free Equilibrium Point of the HIV/AIDS-Only Submodel

The disease-free equilibrium point (DFE) of the HIV/AIDS-only submodel (7) is denoted by EH0=S0,H10,H20,H30,TH0 and obtained by making the right hand side of the system as zero and setting all the infectious classes and treatment class to zero as H1=H2=H3=TH=0 we have got S0=Λ/μ such that EH0=Λ/μ,0,0,0,0.


## 3.1.2. The Basic Reproduction Number of the Submodel

The basic reproduction number of HIV/AIDS-infected individuals denoted byRH is defined as the average number of secondary infections produced by a single HIV/AIDS infectious individual introduced in a wholly susceptible population during his or her entire infectious period [14, 20]. This definition is given for the dynamical system that represents the spread of infection in a population. We calculate the basic reproduction number by using the next-generation operator method on the dynamical system (7). The basic reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEH0/∂xj∂νiEH0/∂xj−1 [21, 22] 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 EH0 is the disease-free equilibrium point.After some calculations we have gotF=β1Λ/μβ1ρΛ/μ00,V=μ+γ1+α10−α1μ+γ2+α2, V−1=1/μ+α1+γ10α1/μ+α1+γ1μ+α2+γ21/μ+α2+γ2, and FV−1=β1Λ/μμ+γ1+α1μμ+γ1+α1+β1ρΛα1/μμ+γ1+α1μ+γ2+α2μμ+γ1+α1μ+γ2+α2β1ρΛ/μμ+γ2+α200.Then, the spectral radius (reproduction numberRH) of FV−1 of the HIV/AIDS subdynamical system (7) is ρFV−1=RH=β1Λ/μμ+γ1+α1+β1ρΛα1/μμ+γ1+α1μ+γ2+α2=RH1+RH2.


## 3.1.3. Local Stability of the Submodel Disease-Free Equilibrium Point

Theorem 1.
The disease-free equilibrium pointEH0 of the HIV/AIDSonly submodel is locally asymptotically stable (LAS) if RH<1, and unstable if RH<1.Proof.
“AppendixC”Biologically speaking, Theorem1 implies that HIV can be eliminated from the population when RH<1if the initial sizes of the subpopulation of the submodel are in the region of attraction of EH0.


## 3.1.4. Existence and Stability of Endemic Equilibrium Point of the Submodel

Before investigating the global asymptotic stability of the DFE, it is instructive to determine the number of endemic equilibrium solutions of the model (7). Let an arbitrary equilibrium point of a HIV/AIDS-only dynamical system (7) is denoted by EH∗=S∗,H1∗,H2∗,H3∗,TH∗. Moreover, let λH∗=β1H1∗+ρH2∗ be the associated infection rate (“force of infection”) at endemic equilibrium point. After some calculations, we have got λH∗=μRH−1.⟹λH∗>0 if RH>1 and hence an endemic equilibrium point EH∗=S∗,H1∗,H2∗,H3∗,TH∗ of the HIV/AIDS submodel (7) exist whenever RH>1whereS∗=Λ/μR0H,H1∗=ΛRH−1/d1R0H with d1=μ+γ1+α1,H2∗=α1ΛRH−1/d1d2R0H with d2=μ+γ2+α2,H3∗=α2μRH−1/d3 with d3=μ+γ3+δ1 and TH∗=ΛRH−1γ1/d1R0HΛRH−1γ1/d1R0H+α1ΛRH−1γ2/d1d2R0Hα1ΛRH−1γ2/d1d2R0H+α2μRH−1γ3/d3/μ all are positives if RH>1.Theorem 2.
The submodel (7) has a unique endemic equilibrium point if and only if RH>1.


## 3.1.5. Global Asymptotic Stability (GAS) of the Disease-Free Equilibrium Point

Theorem 3.
The disease-free equilibrium pointEH0=Λ/μ,0,0,0,0 of the dynamical system (7) is globally asymptotically stable if RH≤1 otherwise unstable.Proof.
To show global stability of the DFE applied Lyapunov function method as [13, 23].Let the Lyapunov functionL:R+5⟶R+ is defined by: LS,H1,H2,H3,TH=a2H1+a3H2whereEH0=Λ/μ,0,0,0,0 disease-free equilibrium is point where a2=1/μ+γ1+α1μ+γ2+α2 and a3=1−RH1/μ+γ2+α2 are positive constants. Then, dL/dt=a2dH1/dt+a3dH2/dt:(10)⟹dLdt=a2λHS−μ+γ1+α1H1+a3α1H1−μ+γ2+α2H2≤a2β1Λμ−a2μ+γ1+α1+a3α1H1+a2β1ρΛμ−a3μ+γ2+α2H2.SinceS≤S0=Λ/μ to find values of a2and a3 take a2β1Λ/μ−a2μ+γ1+α1+a3α1=0⟹a3=−a2β1Λ/μΛ/μ+a2μ+γ1+α1/α1; then, we obtained as dL/dt≤a2β1ρΛ/μ+a2β1Λμ+γ2+α2/μα1−a2μ+γ1+α1μ+γ2+α2/α1H2.Here, we can takea2=1/μ+γ1+α1μ+γ2+α2 and a3=1−RH1/μ+γ2+α2 with RH1=β1Λ/μμ+γ1+α1 which is the reproduction number for acute HIV infection (H1) and a3 is positive for RH1<1.Then, we obtained(11)dLdt≤β1ρΛμμ+γ1+α1μ+γ2+α2+β1Λμα1μ+γ1+α1−1α1=α1RH−1,⟹dL/dt≤α1RH−1⟹dL/dt≤0 if RH≤1and dL/dt=0 if and only if S=S0,H1=H10,H2=H20,H3=H30, and TH=TH0. Therefore, the largest compact invariant set in S,H1,H2,H3,TH∈Ω1:dL/dt=0 is the singleton EH0where EH0is the disease-free equilibrium point of the model (7).Thus, by LaSalle’s invariance principle [24], it implies that the disease-free equilibrium point EH0=Λ/μ,0,0,0,0 is globally asymptotically stable in Ω1 if RH≤1.


## 3.1.6. Local and Global Stabilities of Endemic Equilibrium Point of the HIV/AIDS-Only Submodel

Theorem 4.
The endemic equilibrium pointEH∗=S∗,H1∗,H1∗,H1∗,H1∗ is locally asymptotically stable for the basic reproduction numberRH>1.Proof.
AppendixDTheorem 5.
The endemic equilibrium pointEH∗=S∗,H1∗,H2∗,H3∗,TH∗ is globally asymptotically stable for the basic reproduction number RH>1, otherwise unstable.Proof.
Let the Lyapunov functionV:R+5⟶R+ is defined by
(12)LS,H1,H2,H3,TH=a1S−S∗−S∗lnSS∗+a2H1−H1∗−H1∗lnH1H1∗+a3H2−H2∗−H2∗lnH2H2∗.Here, we havedL/dt=a1dS/dt−S∗/SdS/dt+a2dH1/dt−H1∗/H1H1∗/H1dH1/dt+a3dH2/dt−H2∗/H2H2∗/H2dH2/dt(13)⟹dLdt=a11−S∗SdSdt+a21−H1∗H1dH1dt+a31−H2∗H2dH2dt.At the endemic equilibrium pointEH∗=S∗,H1∗,H2∗,H3∗,TH∗, we obtain from the system (7): Λ=μ+λH∗S∗, λH∗S∗=μ+γ1+α1H1∗ and α1H1∗=μ+γ2+α2H2∗⟹Λ=μ+λH∗S∗,μ+γ1+α1H1=λH∗S∗H1/H1∗ and μ+γ2+α2H2=α1H1∗H2/H2∗.Then, we obtained(14)dLdt=a11−S∗Sμ+λH∗S∗−μ+λHS+a21−H1∗H1λHS−λH∗S∗H1H1∗+a31−H2∗H2α1H1−α1H1∗H2H2∗⟹dLdt=−a1μS−S∗2S+a1+a2β1H1∗S∗+a1+a2β1ρH2∗S∗+a2−a1β1H1S+a2−a1β1ρH2S−a1S∗2Sβ1H1∗−a1S∗2Sβ1ρH2∗+a3α1−a2β1ρS∗H2∗H1∗H1+a1β1ρS∗−a3α1H1∗H2∗H2+a1β1H1S∗−a2β1H1S∗−a2β1H1∗S−a2β1ρH2SH1∗H1−a3α1H1H2∗H2+a3α1H1∗.Choosea1,a2, and a3 such that the expressions in the brackets vanish and take a1=a2 and solve for a3, by making expressions in the closed bracket zero, we obtain as a3α1−a2β1ρS∗H2∗/H1∗=0(15)⟹a3=a1β1ρS∗H2∗α1H1∗fora1=a2,⟹dLdt=−a1μS−S∗2S+2a1β1H1∗S∗+2a1β1ρH2∗S∗−a1S∗2Sβ1H1∗−a1S∗2Sβ1ρH2∗+a1β1H1S∗−a1β1H1S∗−a1β1H1∗S−a1β1ρH2SH1∗H1−a1β1ρS∗H2∗2H1∗H1H2+a1β1ρS∗H2∗.Grouping some terms in the expression above yields(16)dLdt=−a1μS−S∗2S+a1β1H1∗S∗2−SS∗−S∗S+a1β1ρH2∗S∗3−S∗S−SH2H1∗S∗H2∗H1−H2∗H1H2H1∗.Using the arithmetic-geometric mean inequality property, we have2−S/S∗−S∗/S≤0 and 3−S∗/S−SH2H1∗/S∗H2∗H1−H2∗H1/H2H1∗≤0.Hence, we conclude thatdL/dt≤0, and hence, L is the representative Lyapunov function.Furthermore,dL/dt=0 if and only if S,H1,H2,H3,TH=S∗,H1∗,H2∗,H3∗,TH∗, and the largest invariant subset contained in the set E∞=S,H1,H2,H3,TH∈Ω1:dL/dt=0 is the set contained only the endemic equilibrium point:
(17)EH∗=S∗,H1∗,H2∗,H3∗,TH∗.Therefore, we conclude by LaSalle’s invariance principle [24] that EH∗=S∗,H1∗,H2∗,H3∗,TH∗ is globally asymptotically stable (GAS) if RH>1.


## 3.2. Pneumonia Submodel Analysis

We have the pneumonia submodel of (3) when H1=H2=H3=C1=C2=C3=TH=T=0, which is given by
(18)dSdt=Λ+pγTP−μ+λPS,dIPdt=λPS−μ+γ+δPIP,dTPdt=γIP−μ+pγTP,where the total population isN2t=St+Ipt+Tpt and the pneumonia force of infection is given by λP=β2IP with initial conditions
(19)S0>0,IP0≥0,TP0≥0.The sum of all the differential equations in (18) above is obtained as
(20)dN2dt=Λ−μN−δPIp.Consider the regionΩ2=S,IP,TP∈ℝ3+,N2≤Λ/μ. It is easy to show that the set Ω2 is positively invariant and a global attractor of all positive solution of submodel (18). Hence, it is sufficient to consider the dynamics of model (18) inΩ1. In this region, the model is epidemiologically and mathematically well posed.


### 3.2.1. Disease-Free Equilibrium Point (DFE) of the Pneumonia-Only Submodel

The disease-free equilibrium point (DFE) of the system (18) is obtained by making the right hand side of the system as zero and setting all the infectious classes and treatment class to zero as IP=TP=0 we have got S0=Λ/μ such that EP0=Λ/μ,0,0.


### 3.2.2. The Reproduction Number of the Pneumonia-Only Submodel

We calculate the basic reproduction number denoted byRP using the van den Driesch and Warmouth next-generation matrix approach from [22]. The basic reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEP0/∂xj∂νiEP0/∂xj−1, whereFiis the rate of appearance of new infection in compartmenti, νi is the transfer of infections from one compartment ito another, and EP0 is the disease-free equilibrium point. The reproduction numberRPof the pneumonia-only dynamical system (18) is obtained by rearrange the differential equation of the dynamical system (18) above in terms of dXi/dt=Fi−vi=Fi−νi−−νi+. Then, after some calculations, we have got FV−1=β2Λ/μμ+γ+δP000 and the spectral radius (reproduction numberRP) of FV−1 of the pneumonia subdynamical system (18) is RP=β2Λ/μμ+γ+δP .


### 3.2.3. Local Stability of the Submodel Disease-Free Equilibrium Point

Theorem 6.
The disease-free equilibrium point of the pneumonia-only submodel is locally asymptotically stable (LAS) ifRP<1, otherwise unstable.Proof.
The local stability of the disease-free equilibrium of the system (18) can be studied from its Jacobian matrix at the disease-free equilibrium point EP0=S0,IP0,TP0=Λ/μ,0,0and Routh-Hurwitz stability criteria. Then, the Jacobian matrix of the dynamical system (18) at EP0=Λ/μ,0,0 is given by
(21)JEP0=−μβ2Λμpγ0−μ+γ+δP00γ−μ+pγ.Then, the characteristic equation of the above Jacobian matrix is given by(22)−μ−λβ2Λμpγ0β2Λμ−μ+γ+δP−λ00γ−μ+pγ−λ=0,⟹−μ−λβ2Λμ−μ+γ+δP−λ−μ+pγ−λ=0,⟹λ1=−μ,λ2=β2Λμ−μ+γ+δP,λ3=−μ+pγ,⟹λ2=β2Λμ−μ+γ+δP=μ+γ+δPβ2Λμμ+γ+δP−1,⟹λ2=μ+γ+δPRP−1<0ifRP<1.Therefore, since all the eigenvalues of the characteristics polynomial of the system (18) are negative for RP<1, the disease-free equilibrium point of the system (18) is locally asymptotically stable.


### 3.2.4. Existence of Endemic Equilibrium Point of the Pneumonia Submodel

Before investigating the global asymptotic stability of the DFE, it is instructive to determine the number of endemic equilibrium solutions of the model (18). Let an arbitrary endemic equilibrium point of the pneumonia-only dynamical system (18) be denoted byEP∗=S∗,IP∗,TP∗. Now, after some calculations, we have got a unique endemic equilibrium point EP∗=S∗,IP∗,TP∗ where S∗=d2Λm2+d2Λm3D1RP−1+D2pγRP−1/d2m2+d2m3D1RP−1μ+D1RP−1>0,IP∗=d2ΛD1RP−1/m2+m3D1RP−1>0 and TP∗=D2RP−1/d2m2+d2m3D1RP−1>0 ifRP>1 where D1=μ+γ+δPμ+pγμ/μ+γ+δPμ+pγ−pγ2 and D2=d2γΛD1.Theorem 7.
The model (18) has a unique endemic equilibrium point if RP>1.


### 3.2.5. Globally Asymptotically Stability (GAS) of the Disease-Free Equilibrium Point

Theorem 8.
The disease-free equilibrium pointEP0=S0,IP0,TP0=Λ/μ,0,0 of the dynamical system (18) is globally asymptotically stable if RP≤1, otherwise unstable.Proof.
Let the Lyapunov functionL:R+3⟶R+ is defined by: LS,IP,TP=a1IP
whereEP0=S0,IP0,TP0=Λ/μ,0,0 is disease-free equilibrium point where a1=1 is a positive constant.ThendL/dt=a1dIP/dt(23)⟹dLdt=a1dIPdt=a1λPS−μ+γ+δPIP=a1β2IPS−μ+γ+δPIP⟹dLdt=a1β2S−μ+γ+δPIP≤a1β2S0−μ+γ+δPIPsinceS≤S0=Λμ,(24)⟹dLdt≤a1β2Λμ−μ+γ+δPIP=a1μ+γ+δPβ2Λμμ+γ+δP−1,⟹dLdt≤a1μ+γ+δPRP−1=μ+γ+δPRP−1,⟹dL/dt≤0 if RP≤1. Hence, the disease-free equilibrium point EP0=Λ/μ,0,0 is globally asymptotically stable whenever RP≤1.


### 3.2.6. Local and Global Stabilities of Endemic Equilibrium Point

The following theorem studies the local stability of the endemic equilibrium. The result is obtained by means of the Routh-Hurwitz stability criteria.Theorem 9.
The endemic equilibrium point of the system (18) is locally asymptotically stable ifRP>1.Proof.
To show the local stability of the endemic equilibrium point, we use the method of the Jacobian matrix and Routh Hurwitz stability criteria. Then, the Jacobian matrix of the dynamical system (18) at the endemic equilibrium point EP∗=S∗,IP∗,TP∗ whereS∗=d2Λm2+d2Λm3D1RP−1+D2pγRP−1/d2m2+d2m3D1RP−1μ+D1RP−1,IP∗=d2ΛD1RP−1/m2+m3D1RP−1 and TP∗=D2RP−1/d2m2+d2m3D1RP−1,D2=d2γΛD1 for the system (18) whenever RP>1 is given by
(25)JEP∗=−μ−β2IP∗−β2S∗pγβ2IP∗β2S∗−d100γ−d2,
⟹JEP∗=A1A4pγA5A200γA3 whereA1=−μ−β2IP∗,A2=β2S∗−d1,A3=−d2,A4=−β2S∗, and A5=β2IP∗.Then, the characteristic equation of the above Jacobian matrix is given by(26)A1−λA4pγA5A2−λ00γA3−λ=0,⟹a3λ3+a2λ2+a1λ+a0=0wherea3=1,a2=−A1+A2+A3,a1=A1A2+A1A3+A2A3−A4A5, and a0=−A1A2A3+A3A4A5−pγ2A5.Here, we apply the necessary condition of Routh-Hurwitz stability criteria sincea3=1 is positive in sign, indicating all a2,a1, and a0 should be positive.Hence,a2=−A1+A2+A3=μ+β2IP∗−β2S∗+d1+d2(27)⟹a2=μd2m2+d2m3D1RP−1μ+D1RP−1+β2D2RP−1μ+D1RP−1−β2d2Λm2−β2d2Λm3D1RP−1−β2D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1+d1+d2d2m2+d2m3D1RP−1μ+D1RP−1>0ifRP>1,a1=A1A2+A1A3+A2A3−A4A5=μ+β2d2ΛD1RP−1m2+m3D1RP−1β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1−d1+d2μ+β2d2ΛD1RP−1m2+m3D1RP−1+β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1β2d2ΛD1RP−1m2+m3D1RP−1>0ifRP>1,(28)a0=−A1A2A3+A3A4A5−pγ2A5=μ+β2IP∗β2S∗−d1d2+d2β2S∗β2IP∗−β2IP∗pγ2=μ+β2d2ΛD1RP−1m2+m3D1RP−1β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1−d1d2+d2β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1β2d2ΛD1RP−1m2+m3D1RP−1−β2d2ΛD1RP−1m2+m3D1RP−1pγ2>0ifRP>1.Thus, all the coefficients of the characteristic’s polynomiala0,a1,a2, anda3 have the same sign. Then, applying the Routh-Hurwitz criteria to determine the sign of the root without calculating their values of the root of the characteristics equationa3λ3+a2λ2+a1λ+a0=0 and we have got the Routh-Hurwitz array that has no sign change, indicating the roots of the characteristics equation of the dynamical system (18) are negatives. Hence, the endemic equilibrium point EP∗=S∗,IP∗,TP∗ of the dynamical system (18) is locally asymptotically stable.


### 3.2.7. Global Stability of Endemic Equilibrium of Pneumonia Submodel

Theorem 10.
IfRP>1 the unique endemic equilibrium pointEP∗=S∗,IP∗,TP∗ is globally asymptotically stable in the interior of Ω2.Proof.
Let us take the Lyapunov functionL:ℝ3+⟶ℝ byLS,IP,TP=A1S−S∗+IP−IP∗+TP−TP∗2+A2IP−IP∗−IP∗lnIP/IP∗+A3TP−TP∗2 for positive constantsA1=1/2,A2=δP+2μ/β2 andA3=δP+2μ/2γ. Then, L is C1 on the interior of Ω2, EP∗=S∗,IP∗,TP∗ is the global minimum of L on Ω2, andLS∗,IP∗,TP∗=0.
Then, the time derivative ofL is given by
(29)dLdt=2A1S−S∗+IP−IP∗+TP−TP∗dS+IP+TPdt+A21−IP∗IPdIPdt+2A3TP−TP∗dTPdt,⟹dLdt=S−S∗+I−IP∗+TP−TP∗dS+IP+TPdt+δP+2μβ21−IP∗IPdIPdt+δP+2μγTP−TP∗dTPdt.Using expressions at the endemic equilibrium,(30)Λ=μS∗+IP∗+TP∗+δPIP∗,β2S∗=μ+γ+δP,0=μ+pγTP∗−γIP∗,we have got
(31)dLdt=S−S∗+I−IP∗+TP−TP∗μS∗+IP∗+TP∗+δPIP∗−μS+IP+TP−δPIP+δP+2μβ2IP−IP∗IPβ2IPS−β2k2S∗IP+δP+2μγTP−TP∗γIP−μ+pγTP+μ+pγTP∗−γIP∗,⟹dLdt=−μS−S∗+TP−TP∗2−δP+μIP−IP∗2−δP+2μμ+pγTP−TP∗2≤0.Hence,dL/dt is negative. Note that dL/dt=0 if and only if S=S∗,IP=IP∗ and TP=TP∗. Therefore, the largest compact invariant set inS,IP,TP∈Ω2:dL/dt=0 is the singleton {EP∗} set where EP∗ the endemic equilibrium point. Then, by LaSalle’s invariant principle, [24] then implies that EP∗is globally asymptotically stable in the interior of Ω2 if RP>1.


## 3.2.1. Disease-Free Equilibrium Point (DFE) of the Pneumonia-Only Submodel

The disease-free equilibrium point (DFE) of the system (18) is obtained by making the right hand side of the system as zero and setting all the infectious classes and treatment class to zero as IP=TP=0 we have got S0=Λ/μ such that EP0=Λ/μ,0,0.


## 3.2.2. The Reproduction Number of the Pneumonia-Only Submodel

We calculate the basic reproduction number denoted byRP using the van den Driesch and Warmouth next-generation matrix approach from [22]. The basic reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEP0/∂xj∂νiEP0/∂xj−1, whereFiis the rate of appearance of new infection in compartmenti, νi is the transfer of infections from one compartment ito another, and EP0 is the disease-free equilibrium point. The reproduction numberRPof the pneumonia-only dynamical system (18) is obtained by rearrange the differential equation of the dynamical system (18) above in terms of dXi/dt=Fi−vi=Fi−νi−−νi+. Then, after some calculations, we have got FV−1=β2Λ/μμ+γ+δP000 and the spectral radius (reproduction numberRP) of FV−1 of the pneumonia subdynamical system (18) is RP=β2Λ/μμ+γ+δP .


## 3.2.3. Local Stability of the Submodel Disease-Free Equilibrium Point

Theorem 6.
The disease-free equilibrium point of the pneumonia-only submodel is locally asymptotically stable (LAS) ifRP<1, otherwise unstable.Proof.
The local stability of the disease-free equilibrium of the system (18) can be studied from its Jacobian matrix at the disease-free equilibrium point EP0=S0,IP0,TP0=Λ/μ,0,0and Routh-Hurwitz stability criteria. Then, the Jacobian matrix of the dynamical system (18) at EP0=Λ/μ,0,0 is given by
(21)JEP0=−μβ2Λμpγ0−μ+γ+δP00γ−μ+pγ.Then, the characteristic equation of the above Jacobian matrix is given by(22)−μ−λβ2Λμpγ0β2Λμ−μ+γ+δP−λ00γ−μ+pγ−λ=0,⟹−μ−λβ2Λμ−μ+γ+δP−λ−μ+pγ−λ=0,⟹λ1=−μ,λ2=β2Λμ−μ+γ+δP,λ3=−μ+pγ,⟹λ2=β2Λμ−μ+γ+δP=μ+γ+δPβ2Λμμ+γ+δP−1,⟹λ2=μ+γ+δPRP−1<0ifRP<1.Therefore, since all the eigenvalues of the characteristics polynomial of the system (18) are negative for RP<1, the disease-free equilibrium point of the system (18) is locally asymptotically stable.


## 3.2.4. Existence of Endemic Equilibrium Point of the Pneumonia Submodel

Before investigating the global asymptotic stability of the DFE, it is instructive to determine the number of endemic equilibrium solutions of the model (18). Let an arbitrary endemic equilibrium point of the pneumonia-only dynamical system (18) be denoted byEP∗=S∗,IP∗,TP∗. Now, after some calculations, we have got a unique endemic equilibrium point EP∗=S∗,IP∗,TP∗ where S∗=d2Λm2+d2Λm3D1RP−1+D2pγRP−1/d2m2+d2m3D1RP−1μ+D1RP−1>0,IP∗=d2ΛD1RP−1/m2+m3D1RP−1>0 and TP∗=D2RP−1/d2m2+d2m3D1RP−1>0 ifRP>1 where D1=μ+γ+δPμ+pγμ/μ+γ+δPμ+pγ−pγ2 and D2=d2γΛD1.Theorem 7.
The model (18) has a unique endemic equilibrium point if RP>1.


## 3.2.5. Globally Asymptotically Stability (GAS) of the Disease-Free Equilibrium Point

Theorem 8.
The disease-free equilibrium pointEP0=S0,IP0,TP0=Λ/μ,0,0 of the dynamical system (18) is globally asymptotically stable if RP≤1, otherwise unstable.Proof.
Let the Lyapunov functionL:R+3⟶R+ is defined by: LS,IP,TP=a1IP
whereEP0=S0,IP0,TP0=Λ/μ,0,0 is disease-free equilibrium point where a1=1 is a positive constant.ThendL/dt=a1dIP/dt(23)⟹dLdt=a1dIPdt=a1λPS−μ+γ+δPIP=a1β2IPS−μ+γ+δPIP⟹dLdt=a1β2S−μ+γ+δPIP≤a1β2S0−μ+γ+δPIPsinceS≤S0=Λμ,(24)⟹dLdt≤a1β2Λμ−μ+γ+δPIP=a1μ+γ+δPβ2Λμμ+γ+δP−1,⟹dLdt≤a1μ+γ+δPRP−1=μ+γ+δPRP−1,⟹dL/dt≤0 if RP≤1. Hence, the disease-free equilibrium point EP0=Λ/μ,0,0 is globally asymptotically stable whenever RP≤1.


## 3.2.6. Local and Global Stabilities of Endemic Equilibrium Point

The following theorem studies the local stability of the endemic equilibrium. The result is obtained by means of the Routh-Hurwitz stability criteria.Theorem 9.
The endemic equilibrium point of the system (18) is locally asymptotically stable ifRP>1.Proof.
To show the local stability of the endemic equilibrium point, we use the method of the Jacobian matrix and Routh Hurwitz stability criteria. Then, the Jacobian matrix of the dynamical system (18) at the endemic equilibrium point EP∗=S∗,IP∗,TP∗ whereS∗=d2Λm2+d2Λm3D1RP−1+D2pγRP−1/d2m2+d2m3D1RP−1μ+D1RP−1,IP∗=d2ΛD1RP−1/m2+m3D1RP−1 and TP∗=D2RP−1/d2m2+d2m3D1RP−1,D2=d2γΛD1 for the system (18) whenever RP>1 is given by
(25)JEP∗=−μ−β2IP∗−β2S∗pγβ2IP∗β2S∗−d100γ−d2,
⟹JEP∗=A1A4pγA5A200γA3 whereA1=−μ−β2IP∗,A2=β2S∗−d1,A3=−d2,A4=−β2S∗, and A5=β2IP∗.Then, the characteristic equation of the above Jacobian matrix is given by(26)A1−λA4pγA5A2−λ00γA3−λ=0,⟹a3λ3+a2λ2+a1λ+a0=0wherea3=1,a2=−A1+A2+A3,a1=A1A2+A1A3+A2A3−A4A5, and a0=−A1A2A3+A3A4A5−pγ2A5.Here, we apply the necessary condition of Routh-Hurwitz stability criteria sincea3=1 is positive in sign, indicating all a2,a1, and a0 should be positive.Hence,a2=−A1+A2+A3=μ+β2IP∗−β2S∗+d1+d2(27)⟹a2=μd2m2+d2m3D1RP−1μ+D1RP−1+β2D2RP−1μ+D1RP−1−β2d2Λm2−β2d2Λm3D1RP−1−β2D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1+d1+d2d2m2+d2m3D1RP−1μ+D1RP−1>0ifRP>1,a1=A1A2+A1A3+A2A3−A4A5=μ+β2d2ΛD1RP−1m2+m3D1RP−1β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1−d1+d2μ+β2d2ΛD1RP−1m2+m3D1RP−1+β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1β2d2ΛD1RP−1m2+m3D1RP−1>0ifRP>1,(28)a0=−A1A2A3+A3A4A5−pγ2A5=μ+β2IP∗β2S∗−d1d2+d2β2S∗β2IP∗−β2IP∗pγ2=μ+β2d2ΛD1RP−1m2+m3D1RP−1β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1−d1d2+d2β2d2Λm2+d2Λm3D1RP−1+D2pγRP−1d2m2+d2m3D1RP−1μ+D1RP−1β2d2ΛD1RP−1m2+m3D1RP−1−β2d2ΛD1RP−1m2+m3D1RP−1pγ2>0ifRP>1.Thus, all the coefficients of the characteristic’s polynomiala0,a1,a2, anda3 have the same sign. Then, applying the Routh-Hurwitz criteria to determine the sign of the root without calculating their values of the root of the characteristics equationa3λ3+a2λ2+a1λ+a0=0 and we have got the Routh-Hurwitz array that has no sign change, indicating the roots of the characteristics equation of the dynamical system (18) are negatives. Hence, the endemic equilibrium point EP∗=S∗,IP∗,TP∗ of the dynamical system (18) is locally asymptotically stable.


## 3.2.7. Global Stability of Endemic Equilibrium of Pneumonia Submodel

Theorem 10.
IfRP>1 the unique endemic equilibrium pointEP∗=S∗,IP∗,TP∗ is globally asymptotically stable in the interior of Ω2.Proof.
Let us take the Lyapunov functionL:ℝ3+⟶ℝ byLS,IP,TP=A1S−S∗+IP−IP∗+TP−TP∗2+A2IP−IP∗−IP∗lnIP/IP∗+A3TP−TP∗2 for positive constantsA1=1/2,A2=δP+2μ/β2 andA3=δP+2μ/2γ. Then, L is C1 on the interior of Ω2, EP∗=S∗,IP∗,TP∗ is the global minimum of L on Ω2, andLS∗,IP∗,TP∗=0.
Then, the time derivative ofL is given by
(29)dLdt=2A1S−S∗+IP−IP∗+TP−TP∗dS+IP+TPdt+A21−IP∗IPdIPdt+2A3TP−TP∗dTPdt,⟹dLdt=S−S∗+I−IP∗+TP−TP∗dS+IP+TPdt+δP+2μβ21−IP∗IPdIPdt+δP+2μγTP−TP∗dTPdt.Using expressions at the endemic equilibrium,(30)Λ=μS∗+IP∗+TP∗+δPIP∗,β2S∗=μ+γ+δP,0=μ+pγTP∗−γIP∗,we have got
(31)dLdt=S−S∗+I−IP∗+TP−TP∗μS∗+IP∗+TP∗+δPIP∗−μS+IP+TP−δPIP+δP+2μβ2IP−IP∗IPβ2IPS−β2k2S∗IP+δP+2μγTP−TP∗γIP−μ+pγTP+μ+pγTP∗−γIP∗,⟹dLdt=−μS−S∗+TP−TP∗2−δP+μIP−IP∗2−δP+2μμ+pγTP−TP∗2≤0.Hence,dL/dt is negative. Note that dL/dt=0 if and only if S=S∗,IP=IP∗ and TP=TP∗. Therefore, the largest compact invariant set inS,IP,TP∈Ω2:dL/dt=0 is the singleton {EP∗} set where EP∗ the endemic equilibrium point. Then, by LaSalle’s invariant principle, [24] then implies that EP∗is globally asymptotically stable in the interior of Ω2 if RP>1.


## 3.3. Analysis of the HIV-Pneumonia Coinfection Model (3)

In this section, we analyze the main dynamical system (3). Epidemiologically, the HIV/AIDS-pneumonia coinfection dynamical system (3) will have four equilibrium points, namely, HIV/AIDS-pneumonia coinfection model disease-free equilibrium pointEHP0, HIV/AIDS-only endemic equilibrium pointEH∗, pneumonia-only endemic equilibrium point EP∗, and the endemic equilibrium point of the coexistence of HIV-pneumonia denoted by EHP∗.


### 3.3.1. Disease-Free Equilibrium Point of the HIV-Pneumonia Coinfection Model

The disease-free equilibrium pointEHP0=S0,H10,H20,H30,C10,C20,C30,TP0,TH0,T0 of the system (3) is obtained by setting all the infectious classes and treatment classes to zero and making the right hand side of the system as zero. Then, the disease-free equilibrium of the HIV/AIDS coinfection model is EHP0=Λ/μ,0,0,0,0,0,0,0,0,0.


### 3.3.2. The Basic Reproduction NumberRHP of the HIV-Pneumonia Coinfection Model

The basic reproduction number of the dynamical system (3) by applying the next-generation operator method [21, 22] is the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEHP0/∂xj∂νiEHP0/∂xj−1, where Fi is the rate of appearance of new infection in compartment i, νi is the transfer of infections from one compartment ito another, and EHP0 is the disease-free equilibrium point. Here, we obtained the following matrices:
(32)F=β2Λμ000β2ω1Λμβ2ω2Λμβ2ω3Λμ0000β1Λμβ1ρΛμ000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,V=m10000000000m2000000000−α1m3000000000−α2m40000000000m5000000000−α3m6000000000−α4m7000−γ000000m8000−γ1−γ2−γ30000μ00000−ε1−ε2−ε300μ.Then, using Mathematica, we have got the spectral radius (dominant eigenvalue) of the matrixFV−1 is RHP=maxΛβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2,Λβ2/μμ+γ+δP where RP=Λβ2/μμ+γ+δP is the basic reproduction number for pneumonia infection andRH=Λβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2 is the basic reproduction number for HIV/AIDS infection, and hence, RHP=maxRP,RH is the basic reproduction number of the HIV and pneumonia co-infection.


### 3.3.3. Local Stability of the Disease-Free Equilibrium Point

Theorem 11.
The disease-free equilibrium point of the model (3) above is locally asymptotically stable if RHP<1 and unstable ifRHP>1.Proof.
The Jacobian matrixJEHP0 of the model at EHP0 is given by
(33)JEHP0=−μβ2Λμ−β1Λμ−β1ρΛμ0−β2ω1Λμ−β2ω2Λμ−β2ω3Λμpγ000Z2000β2ω1Λμβ2ω2Λμβ2ω3Λμ00000Z3β1ρΛμ000000000α1Z40000000000α2Z500000000000Z60000000000α3Z70000000000α4Z8000γ0000000Z90000γ1γ2γ30000−μ000000ε1ε2ε300−μ.The eigenvalues of the above Jacobian matrix areλ1=λ2=λ3=−μ<0 or λ4=Z2=β2Λ/μ−μ+γ+δP or λ5=Z5=−μ+γ3+α3<0 or λ6=Z6=−μ+δP+ε1+α3<0 or λ7=Z7=−μ+δP+ε2+α4<0orλ8=Z8=−μ+δ2+ε3<0orλ9=Z9=−μ+pγ<0 or a2λ2+a1λ+a0=0 fora2=1,a1=−Z3−Z4anda0=Z3Z4−β1ρα1Λ/μ.Here,λ4=Z2=β2Λ/μ−μ+γ+δP=μ+γ+δPβ2Λ/μμ+γ+δP−1=μ+γ+δPRP−1⟹λ4=μ+γ+δPRP−1<0 ifRP<1.To check the remaining eigenvalues are negatives for the quadratic equationa2λ2+a1λ+a0=0, we can apply Routh-Hurwitz stability criteria since a2=1>0 both a1 and a0 should be positives.Now,a1=μ+γ2+α2+μ+γ1+α11−RH1⟹a1>0 if RH1<1 anda0=μ+γ2+α2μ+γ1+α11−β1ρα1Λ/μμ+γ2+α2μ+γ1+α1⟹a0=μ+γ2+α2μ+γ1+α11−RH2>0 if RH2<1.Hence, the last two eigenvalues are negatives ifRH<1 and thus all the eigenvalues are negatives if RHP=maxRH,RP<1. Thus, since all the eigenvalues are negatives, the disease-free equilibrium point of the HIV/AIDS-pneumonia coinfection dynamical system (3) above is locally asymptotically stable if
(34)RHP=maxRH,RP<1.


### 3.3.4. Existence of Endemic Equilibrium Point for the Full HIV/AIDS-Pneumonia Coinfection Model

The endemic equilibrium point (EEP) of the model (3) above is denoted by EHP∗=S∗,IP∗,H1∗,H2∗,H3∗,C1∗,C2∗,C3∗,TP∗,TH∗,T∗ which occurs when the disease persist in the community. From the analysis of the HIV-only submodel (7) and the pneumonia-only submodel (18), we have shown that there is no endemic equilibrium point if RH<1and RP<1 implies that there is no endemic equilibrium point if RHP<1 for the coinfection model. The endemic equilibrium of the system (3) above can be obtained as
(35)S∗=Λ+pγTP∗μ+λH∗+λP∗,IP∗=λP∗S∗μ+γ+δP,H1∗=λH∗S∗μ+γ1+α1+φ1λP∗,H2∗=α1H1∗μ+γ2+α2+φ2λP∗,H3∗=α2H2∗μ+γ3+δ1+φ3λP∗,C1∗=φ1λP∗H1∗μ+δP+ε1+α3,C2∗=φ2λP∗H2∗+α3C1∗μ+δP+ε2+α4,C3∗=φ3λP∗H3∗+α4C2∗μ+δ2+ε3(36)TP∗=γIP∗μ+pγ,TH∗=γ1H1∗+γ2H2∗+γ3H3∗μ,T=ε1C1∗+ε2C2∗+ε3C3∗μ.Summary of endemic equilibrium point. The explicit computation of the endemic equilibrium of the full model (3) given in equations (35) and (36) in terms of model parameters is difficult analytically; however, the model (3) above endemic equilibrium EHP∗=S∗,IP∗,H1∗,H2∗,H3∗,C1∗,C2∗,C3∗,TP∗,TH∗,T∗ exists ifRH>1andRP>1, i.e., RHP>1. We will give an explanation of E3∗ in our numerical simulations.


## 3.3.1. Disease-Free Equilibrium Point of the HIV-Pneumonia Coinfection Model

The disease-free equilibrium pointEHP0=S0,H10,H20,H30,C10,C20,C30,TP0,TH0,T0 of the system (3) is obtained by setting all the infectious classes and treatment classes to zero and making the right hand side of the system as zero. Then, the disease-free equilibrium of the HIV/AIDS coinfection model is EHP0=Λ/μ,0,0,0,0,0,0,0,0,0.


## 3.3.2. The Basic Reproduction NumberRHP of the HIV-Pneumonia Coinfection Model

The basic reproduction number of the dynamical system (3) by applying the next-generation operator method [21, 22] is the largest (dominant) eigenvalue (spectral radius) of the matrix: FV−1=∂FiEHP0/∂xj∂νiEHP0/∂xj−1, where Fi is the rate of appearance of new infection in compartment i, νi is the transfer of infections from one compartment ito another, and EHP0 is the disease-free equilibrium point. Here, we obtained the following matrices:
(32)F=β2Λμ000β2ω1Λμβ2ω2Λμβ2ω3Λμ0000β1Λμβ1ρΛμ000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,V=m10000000000m2000000000−α1m3000000000−α2m40000000000m5000000000−α3m6000000000−α4m7000−γ000000m8000−γ1−γ2−γ30000μ00000−ε1−ε2−ε300μ.Then, using Mathematica, we have got the spectral radius (dominant eigenvalue) of the matrixFV−1 is RHP=maxΛβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2,Λβ2/μμ+γ+δP where RP=Λβ2/μμ+γ+δP is the basic reproduction number for pneumonia infection andRH=Λβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2 is the basic reproduction number for HIV/AIDS infection, and hence, RHP=maxRP,RH is the basic reproduction number of the HIV and pneumonia co-infection.


## 3.3.3. Local Stability of the Disease-Free Equilibrium Point

Theorem 11.
The disease-free equilibrium point of the model (3) above is locally asymptotically stable if RHP<1 and unstable ifRHP>1.Proof.
The Jacobian matrixJEHP0 of the model at EHP0 is given by
(33)JEHP0=−μβ2Λμ−β1Λμ−β1ρΛμ0−β2ω1Λμ−β2ω2Λμ−β2ω3Λμpγ000Z2000β2ω1Λμβ2ω2Λμβ2ω3Λμ00000Z3β1ρΛμ000000000α1Z40000000000α2Z500000000000Z60000000000α3Z70000000000α4Z8000γ0000000Z90000γ1γ2γ30000−μ000000ε1ε2ε300−μ.The eigenvalues of the above Jacobian matrix areλ1=λ2=λ3=−μ<0 or λ4=Z2=β2Λ/μ−μ+γ+δP or λ5=Z5=−μ+γ3+α3<0 or λ6=Z6=−μ+δP+ε1+α3<0 or λ7=Z7=−μ+δP+ε2+α4<0orλ8=Z8=−μ+δ2+ε3<0orλ9=Z9=−μ+pγ<0 or a2λ2+a1λ+a0=0 fora2=1,a1=−Z3−Z4anda0=Z3Z4−β1ρα1Λ/μ.Here,λ4=Z2=β2Λ/μ−μ+γ+δP=μ+γ+δPβ2Λ/μμ+γ+δP−1=μ+γ+δPRP−1⟹λ4=μ+γ+δPRP−1<0 ifRP<1.To check the remaining eigenvalues are negatives for the quadratic equationa2λ2+a1λ+a0=0, we can apply Routh-Hurwitz stability criteria since a2=1>0 both a1 and a0 should be positives.Now,a1=μ+γ2+α2+μ+γ1+α11−RH1⟹a1>0 if RH1<1 anda0=μ+γ2+α2μ+γ1+α11−β1ρα1Λ/μμ+γ2+α2μ+γ1+α1⟹a0=μ+γ2+α2μ+γ1+α11−RH2>0 if RH2<1.Hence, the last two eigenvalues are negatives ifRH<1 and thus all the eigenvalues are negatives if RHP=maxRH,RP<1. Thus, since all the eigenvalues are negatives, the disease-free equilibrium point of the HIV/AIDS-pneumonia coinfection dynamical system (3) above is locally asymptotically stable if
(34)RHP=maxRH,RP<1.


## 3.3.4. Existence of Endemic Equilibrium Point for the Full HIV/AIDS-Pneumonia Coinfection Model

The endemic equilibrium point (EEP) of the model (3) above is denoted by EHP∗=S∗,IP∗,H1∗,H2∗,H3∗,C1∗,C2∗,C3∗,TP∗,TH∗,T∗ which occurs when the disease persist in the community. From the analysis of the HIV-only submodel (7) and the pneumonia-only submodel (18), we have shown that there is no endemic equilibrium point if RH<1and RP<1 implies that there is no endemic equilibrium point if RHP<1 for the coinfection model. The endemic equilibrium of the system (3) above can be obtained as
(35)S∗=Λ+pγTP∗μ+λH∗+λP∗,IP∗=λP∗S∗μ+γ+δP,H1∗=λH∗S∗μ+γ1+α1+φ1λP∗,H2∗=α1H1∗μ+γ2+α2+φ2λP∗,H3∗=α2H2∗μ+γ3+δ1+φ3λP∗,C1∗=φ1λP∗H1∗μ+δP+ε1+α3,C2∗=φ2λP∗H2∗+α3C1∗μ+δP+ε2+α4,C3∗=φ3λP∗H3∗+α4C2∗μ+δ2+ε3(36)TP∗=γIP∗μ+pγ,TH∗=γ1H1∗+γ2H2∗+γ3H3∗μ,T=ε1C1∗+ε2C2∗+ε3C3∗μ.Summary of endemic equilibrium point. The explicit computation of the endemic equilibrium of the full model (3) given in equations (35) and (36) in terms of model parameters is difficult analytically; however, the model (3) above endemic equilibrium EHP∗=S∗,IP∗,H1∗,H2∗,H3∗,C1∗,C2∗,C3∗,TP∗,TH∗,T∗ exists ifRH>1andRP>1, i.e., RHP>1. We will give an explanation of E3∗ in our numerical simulations.


## 4. Sensitivity Analysis of the Model Parameters and Numerical Simulations

Results of sensitivity analysis and the numerical simulation are given in this section, and the set of parameters used are given in Table3 below. The full model (3) is now simulated, using the parameter estimates in Table 3 (unless otherwise stated), to assess the potential impact of treatment strategies against pneumonia and HIV/AIDS coinfection, as follows.Table 3
Standard parameter values from literatures.
ParameterNominal valueSourceΛ0.0413∗N0Estimatedμ0.01Estimatedα1,α2,α3, α40.498, 0.08, 0.2885, and 0.3105, respectively[5], Assumedφ1,φ2,φ36, 7, and 10, respectivelyAssumedδP0.135[18]δ10.333Assumedδ20.42Assumedγ0.2[25]γ1,γ2,γ30.2, 0.15, and0.13, respectively[5, 25],Assumedε1,ε2,ε30.20, 0.201, and 0.230[25], Assumedβ1Variable[26]β2VariableAssumedρ,ω1,ω2,ω31.2, 1, 1, and 1Assumed


### 4.1. Sensitivity Analysis of the Model Parameters

Sensitivity is performed to identify the most dominant parameters for the spreading out as well as control of infection in the community. To go through sensitivity analysis, we follow the technique described in [9, 17]. Results of sensitivity analysis are given in this section, and the set of parameters used are given in Table 3 below where N0 is the total number of the assumed initial population under consideration in the model numerical simulation part.We are able to know how important each parameter is to the spread of the disease through sensitivity indices ofRHP to all different parameters. We carried out the sensitivity analysis to determine the model robustness to parameter values.The normalized forward sensitivity index of a variableRHPthat depends differentiably on a parameter p is defined as SIp=∂RHP/∂p∗p/RHP [9].These sensitivity indices allow us to determine the relative importance of different parameters in pneumonia and HIV/AIDS transmission and prevalence. The most sensitive parameter has the magnitude of the sensitivity index larger than that of all other parameters. We can calculate the sensitivity index in terms ofRH and RP since
(37)RHP=maxRH,RP.The sensitivity indices in terms ofRHP=RH=Λβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2and the sensitivity indices in terms of RHP=RP=Λβ2/μμ+γ+δP are given in Tables 4 and 5, respectively.Table 4
Sensitivity indices ofRHP=RH.
Sensitivity indexValueSIΛ=∂RH∂Λ×ΛRH+1SIβ1=∂RH/∂β1×β1/RH+1SIρ=∂RH∂ρ×ρRH+0.6134SIα1=∂RH/∂α1×α1/RH-0.0639SIμ=∂RH∂μ×μRH-0.3150SIγ1=∂RH/∂γ1×γ1/RH-0.1371SIγ2=∂RH/∂γ2×γ2/RH-0.0264SIα2=∂RH/∂α2×α2/RH-0.0141Table 5
Sensitivity indices ofRHP=RP.
Sensitivity indexValueSIΛ=∂RP∂Λ×ΛRP+1SIβ2=∂RP/∂β2×β2/RP]+1SIμ=∂RP∂μ×μRP-0.4421SIγ=∂RP/∂γ×γ/RP-0.6559SIδP=∂RP/∂δP×δP/RP-0.3852Using the parameter values in Table3, the sensitivity indexes are computed in Tables 4 and 5 as above.


### 4.2. Numerical Results and Discussion

Using the data provided in Table3 and different initial conditions, the numerical results are generated for the dynamics of model (3) using MATLAB numerical solver (ode45) which generate results for ode45. The ode45 was chosen because of its computational speed and increased level of accuracy for solving nonstiff ordinary differential equations and we simulated the model using different initial conditions for different values of β1 and β2. It shows the effect of β1 on RH and the effect of β2 onRP. Here, we have done numerical simulation for the HIV/AIDS-pneumonia coinfection dynamical system (3) for the purpose of verifying some of the analytical results and results which are difficult to be done analytically. This is done by using a set of parameter values in Table 3 above whose sources are mainly calculated from literatures in order to have realistic simulation results. In this analysis, we also discussed the effect of parameters change on the basic reproduction number graphically using MATLAB ode45 software.


#### 4.2.1. HIV/AIDS-Pneumonia Coinfection Model Simulation for Various Thresholds

Here, simulations are carried out to monitor the dynamics of the HIV/AIDS-pneumonia coinfection model (3) for different values of the associated reproduction thresholds RH and RP and we plot the graphs of the time verses infected population for different values of reproduction numbers.Figure2 above was plotted using MATLAB ode45 program under consideration of the basic reproduction numbers being less than a unity and shows the behavior of the infectious classes of the HIV/AIDS-pneumonia coinfection model (3) at RH<1 and RP<1 (i.e., RHP<1). The simulation given in Figure 2 above shows that each of infectious classes (IP,H1,H2,H3,C1,C2,C3) is converging to the disease-free equilibrium point of the model. This was obtained when RP=0.1445 at β2=0.0000079 and RH=0.1374 atβ1=0.00000289 with all other parameters are given as in Table 3. This indicates that the disease-free equilibrium point of the full HIV/AIDS-pneumonia coinfection model is globally asymptotically stable.Figure 2
Behavior of infected classes of the full HIV/AIDS-pneumonia coinfection model atRH<1 andRP<1 where β1=0.00000289 and β2=0.0000079.Figure3 was plotted using the values RP=7 at β2=0.00079 and RH=17 at β1=0.00029 with all other parameters are given in Table 3. The simulation in Figure 3 above shows the stability of the HIV/AIDS-pneumonia coinfection model endemic equilibrium point. From Figure 3 above, in the long run, the convergence of the solutions is observed at the values greater than 7 years. The plot shows that the HIV/AIDS-pneumonia coinfection model (3) endemic equilibrium point is locally asymptotically stable.Figure 3
The stability analysis of the endemic equilibrium of the HIV/AIDS-pneumonia coinfection model atβ1=0.00029 andβ2=0.00079.


#### 4.2.2. Reproduction Number Simulations with Variable Parameter Values

Here, we have taken parameters from Table3, and we have done numerical simulations of reproduction numbers with variable parameter values using ode45 method, and we obtained figures from Figures 4–9. Thus, Figure 4 shows the pneumonia-only submodel reproduction number simulation at variable treatment rate, and from the graph, we see that pneumonia infection dies out at pneumonia treatment rate γ>0.89. Figure 6 showed that pneumonia infection dies out wheneverβ2<0.00003. Figure 7 shows that in the long run the HIV/AIDS transmission decreases whenever β1<0.00002. Figure 8 shows the HIV/AIDS reproduction number simulation at variable acute-infected treatment rate, and from the graph, we see that the HIV/AIDS-only submodel reproduction number is less than unity whenever γ1>0.93 at β1=0.00015, and Figure 9 shows the HIV/AIDS reproduction number simulation at variable chronic-infected treatment rate, and from the graph, we see that the HIV/AIDS-only submodel reproduction number is less than unity whenever the γ2>0.75 at β1=0.00015. Similarly, using parameter values from Table 3, the numerical simulation in Figure 5 shows that comparison of the basic reproduction numbers RH and RP at different values of β1 and β2, respectively; also, from the simulations, we see that the basic reproduction number for HIV/AIDS is greater than the basic reproduction number of pneumonia-only infection where HIV/AIDS-only submodel reproduction number is also the basic reproduction number of the coinfection model. Also, RH<1 wheneverβ1<0.000006 andRP<1 ifβ2<0.000014. Antiretroviral therapy (ART) is used to suppress the HIV virus and stop the progression to AIDS stage, and WHO recommends the immediate use of ART soon after diagnosis especially at the early stage in order to prevent the onward transmission and pneumonia antibiotics are used to treat bacterial pneumonia.Figure 4
Pneumonia reproduction number. Simulation at variable treatment rate (γ).Figure 5
Graphs ofRH and RP at different values of β1 and β2, respectively.Figure 6
Pneumonia reproduction number. Simulation at variable transmission rateβ2.Figure 7
HIV/AIDS reproduction number simulation at variable transmission rate (β1).Figure 8
HIV reproduction number simulation at variable acute infection treatment rate (γ1).Figure 9
HIV/AIDS reproduction number simulation at variable chronic Infection treatment rate.


#### 4.2.3. Simulations of Infected Population with Various Treatment Rates

In this subsection, from the numerical simulation, we have shown the effects of treatment at each infected stages of the model (3). Simulations from Figures 10–16 show the effects of variations of treatment rates on the infected population. Figure 10 shows that when pneumonia treatment rateγ increases from 0.035 to 0.072 the pneumonia infection decreases. Figure 11 shows that when acute HIV infection treatment rateγ1 increases from 0.81 to 0.91 the acute HIV infection decreases. Figure 12 shows that when chronic HIV infection treatment rateγ2 increases from 0.59 to 0.89 the chronic HIV infection decreases. Figure 13 shows that when AIDS patients treatment rateγ3 increases from 0.061 to 0.192 the AIDS patients decreases. Figure 14 shows that when acute HIV/AIDS-pneumonia coinfection treatment rateε1 increases from 0.0.1 to 0.6 the acute HIV/AIDS-pneumonia coinfection decreases. Figure 15 shows that when chronic HIV/AIDS-pneumonia coinfection treatment rateε2 increases from 0.0.34 to 0.69 chronic HIV/AIDS-pneumonia coinfection decreases. Figure 16 shows that when AIDS-pneumonia coinfection treatment rateε3 increases from 0.32 to 0.75 AIDS-pneumonia coinfection decreases. Thus, our model considers treatment at every infection stages of the full HIV/AIDS-pneumonia coinfection model; all numerical simulation graphs from Figures 10–16 show those effects of treatment on the infected population in the corresponding compartments, and also, all simulated curves show that the infected population in the compartment decreases whenever the corresponding treatment rate increases which is the finding of this work.Figure 10
The population of pneumonia-infected individual’s response to treatment.Figure 11
The population of acute-infected Individual’s response to treatment.Figure 12
The population of chronic HIV-infected individual’s response to treatment.Figure 13
The population of AIDS patient’s response to treatment.Figure 14
The population of acute HIV-pneumonia coinfected individual’s response to treatment.Figure 15
The population of chronic HIV-pneumonia-infected individuals’ response to treatment.Figure 16
The population of AIDS-pneumonia coinfected Individuals’ response to treatment.


## 4.1. Sensitivity Analysis of the Model Parameters

Sensitivity is performed to identify the most dominant parameters for the spreading out as well as control of infection in the community. To go through sensitivity analysis, we follow the technique described in [9, 17]. Results of sensitivity analysis are given in this section, and the set of parameters used are given in Table 3 below where N0 is the total number of the assumed initial population under consideration in the model numerical simulation part.We are able to know how important each parameter is to the spread of the disease through sensitivity indices ofRHP to all different parameters. We carried out the sensitivity analysis to determine the model robustness to parameter values.The normalized forward sensitivity index of a variableRHPthat depends differentiably on a parameter p is defined as SIp=∂RHP/∂p∗p/RHP [9].These sensitivity indices allow us to determine the relative importance of different parameters in pneumonia and HIV/AIDS transmission and prevalence. The most sensitive parameter has the magnitude of the sensitivity index larger than that of all other parameters. We can calculate the sensitivity index in terms ofRH and RP since
(37)RHP=maxRH,RP.The sensitivity indices in terms ofRHP=RH=Λβ1/μμ+γ1+α1+Λρα1β1/μμ+γ1+α1μ+γ2+α2and the sensitivity indices in terms of RHP=RP=Λβ2/μμ+γ+δP are given in Tables 4 and 5, respectively.Table 4
Sensitivity indices ofRHP=RH.
Sensitivity indexValueSIΛ=∂RH∂Λ×ΛRH+1SIβ1=∂RH/∂β1×β1/RH+1SIρ=∂RH∂ρ×ρRH+0.6134SIα1=∂RH/∂α1×α1/RH-0.0639SIμ=∂RH∂μ×μRH-0.3150SIγ1=∂RH/∂γ1×γ1/RH-0.1371SIγ2=∂RH/∂γ2×γ2/RH-0.0264SIα2=∂RH/∂α2×α2/RH-0.0141Table 5
Sensitivity indices ofRHP=RP.
Sensitivity indexValueSIΛ=∂RP∂Λ×ΛRP+1SIβ2=∂RP/∂β2×β2/RP]+1SIμ=∂RP∂μ×μRP-0.4421SIγ=∂RP/∂γ×γ/RP-0.6559SIδP=∂RP/∂δP×δP/RP-0.3852Using the parameter values in Table3, the sensitivity indexes are computed in Tables 4 and 5 as above.


## 4.2. Numerical Results and Discussion

Using the data provided in Table3 and different initial conditions, the numerical results are generated for the dynamics of model (3) using MATLAB numerical solver (ode45) which generate results for ode45. The ode45 was chosen because of its computational speed and increased level of accuracy for solving nonstiff ordinary differential equations and we simulated the model using different initial conditions for different values of β1 and β2. It shows the effect of β1 on RH and the effect of β2 onRP. Here, we have done numerical simulation for the HIV/AIDS-pneumonia coinfection dynamical system (3) for the purpose of verifying some of the analytical results and results which are difficult to be done analytically. This is done by using a set of parameter values in Table 3 above whose sources are mainly calculated from literatures in order to have realistic simulation results. In this analysis, we also discussed the effect of parameters change on the basic reproduction number graphically using MATLAB ode45 software.


### 4.2.1. HIV/AIDS-Pneumonia Coinfection Model Simulation for Various Thresholds

Here, simulations are carried out to monitor the dynamics of the HIV/AIDS-pneumonia coinfection model (3) for different values of the associated reproduction thresholds RH and RP and we plot the graphs of the time verses infected population for different values of reproduction numbers.Figure2 above was plotted using MATLAB ode45 program under consideration of the basic reproduction numbers being less than a unity and shows the behavior of the infectious classes of the HIV/AIDS-pneumonia coinfection model (3) at RH<1 and RP<1 (i.e., RHP<1). The simulation given in Figure 2 above shows that each of infectious classes (IP,H1,H2,H3,C1,C2,C3) is converging to the disease-free equilibrium point of the model. This was obtained when RP=0.1445 at β2=0.0000079 and RH=0.1374 atβ1=0.00000289 with all other parameters are given as in Table 3. This indicates that the disease-free equilibrium point of the full HIV/AIDS-pneumonia coinfection model is globally asymptotically stable.Figure 2
Behavior of infected classes of the full HIV/AIDS-pneumonia coinfection model atRH<1 andRP<1 where β1=0.00000289 and β2=0.0000079.Figure3 was plotted using the values RP=7 at β2=0.00079 and RH=17 at β1=0.00029 with all other parameters are given in Table 3. The simulation in Figure 3 above shows the stability of the HIV/AIDS-pneumonia coinfection model endemic equilibrium point. From Figure 3 above, in the long run, the convergence of the solutions is observed at the values greater than 7 years. The plot shows that the HIV/AIDS-pneumonia coinfection model (3) endemic equilibrium point is locally asymptotically stable.Figure 3
The stability analysis of the endemic equilibrium of the HIV/AIDS-pneumonia coinfection model atβ1=0.00029 andβ2=0.00079.


### 4.2.2. Reproduction Number Simulations with Variable Parameter Values

Here, we have taken parameters from Table3, and we have done numerical simulations of reproduction numbers with variable parameter values using ode45 method, and we obtained figures from Figures 4–9. Thus, Figure 4 shows the pneumonia-only submodel reproduction number simulation at variable treatment rate, and from the graph, we see that pneumonia infection dies out at pneumonia treatment rate γ>0.89. Figure 6 showed that pneumonia infection dies out wheneverβ2<0.00003. Figure 7 shows that in the long run the HIV/AIDS transmission decreases whenever β1<0.00002. Figure 8 shows the HIV/AIDS reproduction number simulation at variable acute-infected treatment rate, and from the graph, we see that the HIV/AIDS-only submodel reproduction number is less than unity whenever γ1>0.93 at β1=0.00015, and Figure 9 shows the HIV/AIDS reproduction number simulation at variable chronic-infected treatment rate, and from the graph, we see that the HIV/AIDS-only submodel reproduction number is less than unity whenever the γ2>0.75 at β1=0.00015. Similarly, using parameter values from Table 3, the numerical simulation in Figure 5 shows that comparison of the basic reproduction numbers RH and RP at different values of β1 and β2, respectively; also, from the simulations, we see that the basic reproduction number for HIV/AIDS is greater than the basic reproduction number of pneumonia-only infection where HIV/AIDS-only submodel reproduction number is also the basic reproduction number of the coinfection model. Also, RH<1 wheneverβ1<0.000006 andRP<1 ifβ2<0.000014. Antiretroviral therapy (ART) is used to suppress the HIV virus and stop the progression to AIDS stage, and WHO recommends the immediate use of ART soon after diagnosis especially at the early stage in order to prevent the onward transmission and pneumonia antibiotics are used to treat bacterial pneumonia.Figure 4
Pneumonia reproduction number. Simulation at variable treatment rate (γ).Figure 5
Graphs ofRH and RP at different values of β1 and β2, respectively.Figure 6
Pneumonia reproduction number. Simulation at variable transmission rateβ2.Figure 7
HIV/AIDS reproduction number simulation at variable transmission rate (β1).Figure 8
HIV reproduction number simulation at variable acute infection treatment rate (γ1).Figure 9
HIV/AIDS reproduction number simulation at variable chronic Infection treatment rate.


### 4.2.3. Simulations of Infected Population with Various Treatment Rates

In this subsection, from the numerical simulation, we have shown the effects of treatment at each infected stages of the model (3). Simulations from Figures 10–16 show the effects of variations of treatment rates on the infected population. Figure 10 shows that when pneumonia treatment rateγ increases from 0.035 to 0.072 the pneumonia infection decreases. Figure 11 shows that when acute HIV infection treatment rateγ1 increases from 0.81 to 0.91 the acute HIV infection decreases. Figure 12 shows that when chronic HIV infection treatment rateγ2 increases from 0.59 to 0.89 the chronic HIV infection decreases. Figure 13 shows that when AIDS patients treatment rateγ3 increases from 0.061 to 0.192 the AIDS patients decreases. Figure 14 shows that when acute HIV/AIDS-pneumonia coinfection treatment rateε1 increases from 0.0.1 to 0.6 the acute HIV/AIDS-pneumonia coinfection decreases. Figure 15 shows that when chronic HIV/AIDS-pneumonia coinfection treatment rateε2 increases from 0.0.34 to 0.69 chronic HIV/AIDS-pneumonia coinfection decreases. Figure 16 shows that when AIDS-pneumonia coinfection treatment rateε3 increases from 0.32 to 0.75 AIDS-pneumonia coinfection decreases. Thus, our model considers treatment at every infection stages of the full HIV/AIDS-pneumonia coinfection model; all numerical simulation graphs from Figures 10–16 show those effects of treatment on the infected population in the corresponding compartments, and also, all simulated curves show that the infected population in the compartment decreases whenever the corresponding treatment rate increases which is the finding of this work.Figure 10
The population of pneumonia-infected individual’s response to treatment.Figure 11
The population of acute-infected Individual’s response to treatment.Figure 12
The population of chronic HIV-infected individual’s response to treatment.Figure 13
The population of AIDS patient’s response to treatment.Figure 14
The population of acute HIV-pneumonia coinfected individual’s response to treatment.Figure 15
The population of chronic HIV-pneumonia-infected individuals’ response to treatment.Figure 16
The population of AIDS-pneumonia coinfected Individuals’ response to treatment.


## 4.2.1. HIV/AIDS-Pneumonia Coinfection Model Simulation for Various Thresholds

Here, simulations are carried out to monitor the dynamics of the HIV/AIDS-pneumonia coinfection model (3) for different values of the associated reproduction thresholds RH and RP and we plot the graphs of the time verses infected population for different values of reproduction numbers.Figure2 above was plotted using MATLAB ode45 program under consideration of the basic reproduction numbers being less than a unity and shows the behavior of the infectious classes of the HIV/AIDS-pneumonia coinfection model (3) at RH<1 and RP<1 (i.e., RHP<1). The simulation given in Figure 2 above shows that each of infectious classes (IP,H1,H2,H3,C1,C2,C3) is converging to the disease-free equilibrium point of the model. This was obtained when RP=0.1445 at β2=0.0000079 and RH=0.1374 atβ1=0.00000289 with all other parameters are given as in Table 3. This indicates that the disease-free equilibrium point of the full HIV/AIDS-pneumonia coinfection model is globally asymptotically stable.Figure 2
Behavior of infected classes of the full HIV/AIDS-pneumonia coinfection model atRH<1 andRP<1 where β1=0.00000289 and β2=0.0000079.Figure3 was plotted using the values RP=7 at β2=0.00079 and RH=17 at β1=0.00029 with all other parameters are given in Table 3. The simulation in Figure 3 above shows the stability of the HIV/AIDS-pneumonia coinfection model endemic equilibrium point. From Figure 3 above, in the long run, the convergence of the solutions is observed at the values greater than 7 years. The plot shows that the HIV/AIDS-pneumonia coinfection model (3) endemic equilibrium point is locally asymptotically stable.Figure 3
The stability analysis of the endemic equilibrium of the HIV/AIDS-pneumonia coinfection model atβ1=0.00029 andβ2=0.00079.


## 4.2.2. Reproduction Number Simulations with Variable Parameter Values

Here, we have taken parameters from Table3, and we have done numerical simulations of reproduction numbers with variable parameter values using ode45 method, and we obtained figures from Figures 4–9. Thus, Figure 4 shows the pneumonia-only submodel reproduction number simulation at variable treatment rate, and from the graph, we see that pneumonia infection dies out at pneumonia treatment rate γ>0.89. Figure 6 showed that pneumonia infection dies out wheneverβ2<0.00003. Figure 7 shows that in the long run the HIV/AIDS transmission decreases whenever β1<0.00002. Figure 8 shows the HIV/AIDS reproduction number simulation at variable acute-infected treatment rate, and from the graph, we see that the HIV/AIDS-only submodel reproduction number is less than unity whenever γ1>0.93 at β1=0.00015, and Figure 9 shows the HIV/AIDS reproduction number simulation at variable chronic-infected treatment rate, and from the graph, we see that the HIV/AIDS-only submodel reproduction number is less than unity whenever the γ2>0.75 at β1=0.00015. Similarly, using parameter values from Table 3, the numerical simulation in Figure 5 shows that comparison of the basic reproduction numbers RH and RP at different values of β1 and β2, respectively; also, from the simulations, we see that the basic reproduction number for HIV/AIDS is greater than the basic reproduction number of pneumonia-only infection where HIV/AIDS-only submodel reproduction number is also the basic reproduction number of the coinfection model. Also, RH<1 wheneverβ1<0.000006 andRP<1 ifβ2<0.000014. Antiretroviral therapy (ART) is used to suppress the HIV virus and stop the progression to AIDS stage, and WHO recommends the immediate use of ART soon after diagnosis especially at the early stage in order to prevent the onward transmission and pneumonia antibiotics are used to treat bacterial pneumonia.Figure 4
Pneumonia reproduction number. Simulation at variable treatment rate (γ).Figure 5
Graphs ofRH and RP at different values of β1 and β2, respectively.Figure 6
Pneumonia reproduction number. Simulation at variable transmission rateβ2.Figure 7
HIV/AIDS reproduction number simulation at variable transmission rate (β1).Figure 8
HIV reproduction number simulation at variable acute infection treatment rate (γ1).Figure 9
HIV/AIDS reproduction number simulation at variable chronic Infection treatment rate.


## 4.2.3. Simulations of Infected Population with Various Treatment Rates

In this subsection, from the numerical simulation, we have shown the effects of treatment at each infected stages of the model (3). Simulations from Figures 10–16 show the effects of variations of treatment rates on the infected population. Figure 10 shows that when pneumonia treatment rateγ increases from 0.035 to 0.072 the pneumonia infection decreases. Figure 11 shows that when acute HIV infection treatment rateγ1 increases from 0.81 to 0.91 the acute HIV infection decreases. Figure 12 shows that when chronic HIV infection treatment rateγ2 increases from 0.59 to 0.89 the chronic HIV infection decreases. Figure 13 shows that when AIDS patients treatment rateγ3 increases from 0.061 to 0.192 the AIDS patients decreases. Figure 14 shows that when acute HIV/AIDS-pneumonia coinfection treatment rateε1 increases from 0.0.1 to 0.6 the acute HIV/AIDS-pneumonia coinfection decreases. Figure 15 shows that when chronic HIV/AIDS-pneumonia coinfection treatment rateε2 increases from 0.0.34 to 0.69 chronic HIV/AIDS-pneumonia coinfection decreases. Figure 16 shows that when AIDS-pneumonia coinfection treatment rateε3 increases from 0.32 to 0.75 AIDS-pneumonia coinfection decreases. Thus, our model considers treatment at every infection stages of the full HIV/AIDS-pneumonia coinfection model; all numerical simulation graphs from Figures 10–16 show those effects of treatment on the infected population in the corresponding compartments, and also, all simulated curves show that the infected population in the compartment decreases whenever the corresponding treatment rate increases which is the finding of this work.Figure 10
The population of pneumonia-infected individual’s response to treatment.Figure 11
The population of acute-infected Individual’s response to treatment.Figure 12
The population of chronic HIV-infected individual’s response to treatment.Figure 13
The population of AIDS patient’s response to treatment.Figure 14
The population of acute HIV-pneumonia coinfected individual’s response to treatment.Figure 15
The population of chronic HIV-pneumonia-infected individuals’ response to treatment.Figure 16
The population of AIDS-pneumonia coinfected Individuals’ response to treatment.


## 5. Conclusion

In this work, we formulated a mathematical model of eleven nonlinear differential equations on HIV/AIDS and pneumonia coinfection with the assumption of mass action incidence and incorporating treatment at each stage of the infection. We have shown the positivity and boundedness of the solutions of the model. The threshold parameterRHP was calculated and used to determine the conditions under which the HIV/AIDS and pneumonia could be transmitted and remained endemic in the population. We thus showed that three disease-free equilibrium pointsEH0,EP0,and EHP0, respectively, for the HIV submodel, pneumonia submodel, and full model are locally asymptotically stable when RHP<1, i.e., RH<1 and RP<1. We also showed that the population with both HIV/AIDS and pneumonia infection have three endemic equilibrium points EH∗,EP∗, and EHP∗respectively, for the HIVsub model, pneumonia submodel, and full model which were locally asymptotically stable when RHP>1, i.e., RH>1 or RP>1. Global stability analysis of the submodels disease-free equilibrium points was established whenever RHP<1, i.e., RH<1 and RP<1. To investigate, the effect of treatment at each infected compartment was considered for both the submodels and the full model, namely, treatment of pneumonia infection, treatment of acute HIV infection, treatment of chronic HIV infection, treatment of AIDS patients, treatment of acute HIV/AIDS-pneumonia coinfection, treatment of chronic HIV/AIDS-pneumonia coinfection, and treatment of AIDS-pneumonia coinfection. We have showed the most sensitive parameters of our model which can be epidemiologically controlled are the HIV/AIDS transmission rateβ1 and pneumonia transmission rate β2so it is reasonable to recommend the use of intervention strategy for HIV/AIDS transmission in makingβ1 less than 0.00014 and treatment for pneumonia transmission in makingβ2 less than 0.00006.Numerical simulations were used to compare the endemic scenarios showed by analytical results. From the numerical results, we obtained thatRH=17 and RP=7at β1=0.00029 andβ2=0.00079 and all other parameters used are from Table 3. We observed that treatment against a disease has the effect of reducing the progression rate of HIV infection to the AIDS stage and the disease prevalence. From the numerical results and discussion above, we would like to recommend the following to control the spread of HIV/AIDS and pneumonia coinfection. Thus, we can interpret the situation in an epidemiological manner that a society with some individuals infected with HIV/AIDS and without HIV/AIDS treatment is at risk of being coinfected with pneumonia which in turn creates socioeconomic effects if no intervention is implemented in time for either or both HIV/AIDS and pneumonia infection. We conclude that HIV/AIDS treatment for individuals with HIV/AIDS infections results in a significant reduction of the number of individuals progressing to AIDS stage and reduction of the coinfected individuals and reduction of the disease-induced death. Also, effective treatment of pneumonia for the coinfected individuals also reduced the number of individuals that progress to AIDS class. Here, we conclude that HIV/AIDS treatment for only HIV/AIDS individuals (HIV submodel) and coinfected individuals could be a better approach to studying the dynamics of HIV/AIDS and pneumonia and could be the best measure to reduce RHP and coinfection. Models which incorporate other protective measures such as vaccination for pneumonia infection, education of population, and using condom for HIV/AIDS infection may be considered for further research


## 6. Limitations of the Study

There was a lack of literatures about HIV/ADS and pneumonia coinfection and well-organized standard parameter values for the determination of model parameters.



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