HIV/AIDS-Pneumonia Codynamics Model Analysis with Vaccination and Treatment

Abstract

In this paper, we proposed and analyzed a realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections at each infection stage in a population. The model exhibits six equilibriums: HIV/AIDS only disease-free, pneumonia only disease-free, HIV/AIDS-pneumonia coepidemic disease-free, HIV/AIDS only endemic, pneumonia only endemic, and HIV/AIDS-pneumonia coepidemic endemic equilibriums. The HIV/AIDS only submodel has a globally asymptotically stable disease-free equilibrium if ℛ₁ < 1. Using center manifold theory, we have verified that both the pneumonia only submodel and the HIV/AIDS-pneumonia coepidemic model undergo backward bifurcations whenever ℛ₂ < 1  and ℛ₃ = max{ℛ₁, ℛ₂} < 1, respectively. Thus, for pneumonia infection and HIV/AIDS-pneumonia coinfection, the requirement of the basic reproduction numbers to be less than one, even though necessary, may not be sufficient to completely eliminate the disease. Our sensitivity analysis results demonstrate that the pneumonia disease transmission rate  β₂ and the HIV/AIDS transmission rate  β₁ play an important role to change the qualitative dynamics of HIV/AIDS and pneumonia coinfection. The pneumonia infection transmission rate β₂ gives rises to the possibility of backward bifurcation for HIV/AIDS and pneumonia coinfection if ℛ₃ = max{ℛ₁, ℛ₂} < 1, and hence, the existence of multiple endemic equilibria some of which are stable and others are unstable. Using standard data from different literatures, our results show that the complete HIV/AIDS and pneumonia coinfection model reproduction number is ℛ₃ = max{ℛ₁, ℛ₂} = max{1.386, 9.69 } = 9.69  at β₁ = 2 and β₂ = 0.2  which shows that the disease spreads throughout the community. Finally, our numerical simulations show that pneumonia vaccination and treatment against disease have the effect of decreasing pneumonia and coepidemic disease expansion and reducing the progression rate of HIV infection to the AIDS stage.

1. Introduction

HIV/AIDS remains a major global health problem affecting approximately 70 million people worldwide causing significant morbidity and mortality (WHO, 2018) [1]. Over two-thirds of HIV/AIDS-infected population throughout the world is living in the sub-Saharan African Region [1–6]. AIDS is a common individual immune system disease caused by human immunodeficiency virus (HIV), i.e., RNA retrovirus which has developed into a global pandemic since the first patient was identified in 1981, making it one of the most destructive epidemics in history. HIV attacks human white blood cells and is transmitted through open sex, needle sharing, infected blood, and at childbirth [3, 6–9].

Pneumonia is one of the leading airborne infectious diseases caused by microorganisms such as bacteria, viruses, or fungi. It has been the common cause of morbidity and mortality in adults, children under five years of age, and HIV-mediated immunosuppression worldwide, and it is a treatable respiratory lung infectious disease [5, 10–14]. In most prospective microbiology-based studies, bacteria especially Streptococcus bacteria are identified in 30-50% of pneumonia cases which are a leading cause of pneumonia in developing countries [13, 15–17]. However, over the past, our understanding about transmission of pneumonia is basically based on research from high-income western countries but the WHO, 2018 report assessed that from 9.5 million annual death worldwide, pneumonia and other respiratory infections cause about 2 million child deaths yearly in developing countries [14, 18].

A coepidemic is the coexistence of two or more infections on a single individual at the population level [19]. HIV/AIDS-associated opportunistic infectious diseases are more common or more dangerous because of HIV immunosuppression [10].

Mathematical and statistical models of infectious diseases have, historically, provided useful insight into the transmission dynamics and control of infectious diseases [14]. Mathematical models have been used to investigate the dynamics of single infections and coepidemics, and HIV/AIDS-pneumonia is among the diseases that infect a large number of individuals worldwide [10, 17, 20, 21].

Babaei et al. [8] developed and analyzed a simple mathematical model for the interaction between drug addiction and the contagion of HIV/AIDS in Iranian prisons. They analyzed the stability of drug addiction and HIV/AIDS models separately with no medical treatment and investigated the impact of rehabilitating treatments on the control of HIV/AIDS spread in prisons, and finally, the reproduction numbers are compared in cases where there is no cure or some treatment methods are available. From their analysis, we have shown that their treatment methods for addiction withdrawal have a direct impact on the decrement and control of HIV/AIDS infection in prisons. Kizito et al. [13] constructed and discussed a mathematical model of treatment and vaccination impacts on pneumococcal pneumonia transmission dynamics. They found that, with treatment and vaccination combined, pneumonia can be eradicated; however, with treatment intervention alone, pneumonia remains in the population. Bakare and Nwozo [22] construct and analyzed a mathematical model for malaria–schistosomiasis coinfection. They have calculated the basic reproduction numbers and discussed the stability of equilibrium points of the model. They have shown the region where their model state variables become both mathematically and epidemiologically well-posed. They showed the model did not undergo backward bifurcation. Their mathematical modeling analysis result shows that intervention strategy suppresses the human-mosquito contact rate and human-snail contact rate to achieve malaria–schistosomiasis coepidemic free community. Shah et al. [3] formulated and analyzed a mathematical model for HIV/AIDS-TB coinfection considering HIV-infected population, and they found that medication plays a vital role in controlling the spread of the disease.

Limited mathematical modeling research analysis has been conducted on HIV/AIDS-pneumonia coepidemics, for prevention and controlling of the disease transmission with controlling and prevention mechanisms; however, theoretical sources such as [10, 15, 20, 21] show the coexistence of HIV/AIDS-pneumonia. For our new research article, we reviewed only two published HIV/AIDS-pneumonia coepidemic model articles. Nthiiri et al. [5] constructed mathematical modeling on HIV/AIDS-pneumonia coinfection with maximum protection against single HIV/AIDS, and pneumonia infections were their basic concern. They did not consider maximum protection against coinfection. In their model analysis, we have found that when protection is maximum, the number of HIV/AIDS and pneumonia cases is going down. Teklu and Mekonnen [6] constructed a deterministic mathematical model and analyzed it both mathematically and numerically. Our model considered treatment at each infection stage of the coinfection model, and we found that when the treatment rate increases, the number of infectious population at each infection stage decreases. Our model did not consider pneumonia vaccination.

We are motivated by the above studies especially the HIV/AIDS-pneumonia coexistence in the community; therefore, in this study, we considered the three center for disease control and prevention (CDC) stages of the HIV infection which are acute HIV infection, chronic HIV infection, and AIDS stage; we presented and analyzed a mathematical model describing the transmission dynamics of HIV/AIDS and pneumonia coinfection in a population where treatment for HIV/AIDS and both vaccination and treatment for pneumonia are available, respectively, in the community. Our model will be used to evaluate the effect of treatment at every infection stage of the HIV/AIDS only model, pneumonia only model, HIV/AIDS-pneumonia coinfection model, and effect of vaccination for pneumonia only model as control strategies for minimizing incidences of coinfections in the target population. The paper is organized as follows. The model is formulated in Section 2 and is analyzed in Section 3. Sensitivity analysis and numerical simulation were carried out in Section 4. Finally, discussion, conclusion, and recommendation of the study are carried out in Sections 5 and 6, respectively.

2. Mathematical Model Formulation

2.1. Assumptions and Descriptions

According to CDC the three HIV/AIDS infection stages, we divide the human population N(t) into twelve distinct classes as susceptible class to both HIV and pneumonia infections Y₁(t), pneumonia vaccine class Y₂ (t) , pneumonia-infected class Y₃(t), acute HIV-infected class Y₄(t), chronic HIV-infected class Y₅(t), AIDS patient class Y₆(t), acute HIV-pneumonia coepidemic class Y₇(t), chronic HIV-pneumonia coepidemic class Y₈(t), AIDS-pneumonia coepidemic class Y₉(t), pneumonia treatment class Y₁₀(t), HIV/AIDS treatment class Y₁₁(t) entered from the three infection stages Y₄(t), Y₅(t), and Y₆(t), and HIV/AIDS-pneumonia coepidemic treatment class Y₁₂(t) entered from Y₇(t), Y₈(t), and Y₉(t) cases such that

$$\begin{array}{c}N\left(t\right)=Y_1\left(t\right)+Y_2\left(t\right)+Y_3\left(t\right)+Y_4\left(t\right)+Y_5\left(t\right)+Y_6\left(t\right)+Y_7\left(t\right)+Y_8\left(t\right)+Y_9\left(t\right)+Y_{10}\left(t\right)+Y_{11}\left(t\right)+Y_{12}\left(\text{t}\right).\end{array}$$ (1)

The susceptible class acquires HIV at the standard incidence rate given by

$$\begin{array}{c}{\lambda }_{HC}\left(t\right)=\frac{{\beta }_1}{N}\left(Y_4\left(t\right)+{\rho }_1{Y}_5\left(t\right)+{\rho }_2{Y}_7\left(t\right)+{\rho }_3{Y}_8\left(t\right)\right),\end{array}$$ (2)

where ρ₃ ≥ ρ₂ ≥ ρ₁ ≥ 1 is the modification rate that increases infectivity and β₁ is the HIV/AIDS contagion rate.

The susceptible class acquires pneumonia at the mass action incidence rate

$$\begin{array}{c}{\lambda }_{PC}\left(t\right)={\beta }_2\left(Y_3\left(t\right)+{\omega }_1{Y}_7\left(t\right)+{\omega }_2{Y}_8\left(t\right)+{\omega }_3{Y}_9\left(t\right)\right),\end{array}$$ (3)

where ω₃ > ω₂ > ω₁ is the modification rate that increases infectivity and β₂ is the pneumonia contagion rate.

To construct the complete coepidemic dynamical system, we have assumed the following:

1. A fraction of the population has been vaccinated before the disease outbreak at the portion of π and (1 − π) fraction of population entered to the vulnerable class

2. The susceptible class is increased from the vaccinated class in which those individuals who are vaccinated but did not respond to vaccination with the waning rate of τ and from pneumonia-treated class in which those individuals who lose their temporary immunity by the rate θ

3. Assume vaccination is not 100% effective, so vaccinated individuals also have a chance of being infected with proportion ϵ  of the serotype not covered by the vaccine where 0 ≤ ϵ < 1

4. Individuals in a given compartment are homogeneous

5. Assume no HIV transmission from Y₆(t) and Y₉(t) classes due to their reduced daily activities

6. Individuals in each class are subject to natural mortality rate d

7. The human population is variable

8. We assumed there is no dual-infection transmission simultaneously

9. Assume HIV has no vertical transmission and pneumonia is not naturally recovered

10. No permanent immunity for pneumonia-infected individuals and become susceptible again after treatment

2.2. Schematic Diagram of the HIV/AIDS-Pneumonia Coepidemic Model

In this subsection using parameters in Table 1, variable definitions in Table 2, and the model assumptions and descriptions given in (2.1), the schematic diagram for the transmission of HIV/AIDS-pneumonia coepidemic is given by the diagram.

Table 1.

Descriptions of model parameters.

Parameter	Interpretations
d	Natural mortality rate
Λ	Human recruitment rate
δ 1	Development rate from acute HIV to chronic HIV infection
δ 2	Development rate from chronic HIV to AIDS stage
ϵ	The proportion of the serotype not covered by the vaccine
θ	Immunity loss rate
ψ 1	Alteration rate indicating acute HIV infection is more vulnerable to pneumonia
ψ 2	Alteration rate indicating chronic HIV infection is more vulnerable to pneumonia
ψ 3	Alteration rate indicating AIDS patient is more vulnerable to pneumonia
λ HC	HIV/AIDS standard incidence rate
λ PC	Pneumonia mass action incidence rate
δ 3	Development rate from acute HIV-pneumonia to chronic HIV-pneumonia coepidemics
δ 4	Development rate from chronic HIV-pneumonia to AIDS-pneumonia coepidemics
d P	Pneumonia death rate
d A	AIDS death rate
d AP	AIDS-pneumonia death rate
κ	Pneumonia infection treatment rate
κ 1	Acute HIV infection treatment rate
τ 1	Vaccination waning rate
κ 2	Chronic HIV infection treatment rate
κ 3	AIDS patients treatment rate
σ 1	Acute HIV-pneumonia coepidemic treatment rate
σ 2	Chronic HIV-pneumonia coepidemic treatment rate
σ 3	AIDS-pneumonia coepidemic treatment rate
β 1	Transmission rate of HIV
β 2	Transmission rate of pneumonia

Table 2.

Definitions of variables.

Variables	Definitions
Y 1	Vulnerable to both HIV and pneumonia class
Y 2	Pneumonia-vaccinated class
Y 3	Pneumonia-infected class
Y 4	Acute HIV-infected class
Y 5	Chronic HIV-infected class
Y 6	AIDS patients class
Y 7	Acute HIV-pneumonia coepidemic class
Y 8	Chronic HIV-pneumonia coepidemic class
Y 9	AIDS-pneumonia coepidemic class
Y 10	HIV/AIDS treatment class
Y 11	Pneumonia treatment class
Y 12	Coepidemics treatment class

2.3. The HIV/AIDS-Pneumonia Coepidemic Dynamical System

From Figure 1, the HIV/AIDS and pneumonia coinfection dynamical system is given by

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{d{Y}_1}{dt}=\left(1-\pi \right)\mathrm{\Lambda }+{\tau }_1{Y}_2+\theta Y_{10}-\left(d+{\lambda }_{HC}+{\lambda }_{PC}\right)Y_1,\\ \frac{d{Y}_2}{dt}=\pi \mathrm{\Lambda }-ϵ{\lambda }_{PC}{Y}_2-\left(d+{\tau }_1+{\lambda }_{HC}\right)Y_2,\end{array}\\ \frac{d{Y}_3}{dt}=ϵ{\lambda }_{PC}{Y}_2+{\lambda }_{PC}{Y}_1-\left(\nu {\lambda }_{HC}+d+\kappa +d_P\right)Y_3,\\ \frac{d{Y}_4}{dt}={\lambda }_{HC}{Y}_1+{\lambda }_{HC}{Y}_2-\left(d+{\kappa }_1+{\delta }_1+{\psi }_1{\lambda }_{PC}\right)Y_4,\end{array}\\ \frac{d{Y}_5}{dt}={\delta }_1{Y}_4-\left(d+{\kappa }_2+{\delta }_2+{\psi }_2{\lambda }_{PC}\right)Y_5,\\ \frac{d{Y}_6}{dt}={\delta }_2{H}_2-\left(d+{\kappa }_3+d_A+{\psi }_3{\lambda }_{PC}\right)Y_6,\end{array}\\ \frac{d{Y}_7}{dt}={\psi }_1{\lambda }_{PC}{Y}_4+\nu {\lambda }_{H\text{C}}{Y}_3-\left(d+d_P+{\sigma }_1+{\delta }_3\right)Y_7,\\ \frac{d{Y}_8}{dt}={\psi }_2{\lambda }_{PC}{Y}_5+{\delta }_3{Y}_7-\left(d+d_P+{\sigma }_2+{\delta }_4\right)Y_8,\end{array}\\ \frac{d{Y}_9}{dt}={\psi }_3{\lambda }_{PC}{Y}_6+{\delta }_4{Y}_8-\left(\text{d}+d_{AP}+{\sigma }_3\right)Y_9,\\ \frac{d{Y}_{10}}{dt}=\kappa Y_3-\left(d+\theta \right)Y_{10},\end{array}\\ \frac{d{Y}_{11}}{dt}={\kappa }_1{Y}_4+{\kappa }_2{Y}_5+{\kappa }_3{Y}_6-d{Y}_{11},\\ \frac{d{Y}_{12}}{dt}={\sigma }_1{Y}_7+{\sigma }_2{Y}_8+{\sigma }_3{Y}_9-d{Y}_{12}.\end{array}\end{array}$$ (4)

Figure 1.

Flowchart of the HIV/AIDS-pneumonia coinfection model (4) where λ_HC and λ_PC are given in (2) and (3), respectively.

With initial conditions,

$$\begin{array}{c}{Y}_1\left(0\right)>0,\\ Y_2\left(0\right)\ge 0,\\ Y_3\left(0\right)\ge 0,\\ Y_4\left(0\right)\ge 0,\\ Y_5\left(0\right)\ge 0,\\ Y_6\left(0\right)\ge 0,\\ Y_7\left(0\right)\ge 0,\\ Y_8\left(0\right)\ge 0,\\ Y_9\left(0\right)\ge 0,\\ Y_{10}>0,\\ Y_{11}>0,\\ Y_{12}\left(0\right)\ge 0.\end{array}$$ (5)

The sum of all the differential equations in (4) is

$$\begin{array}{c}\frac{dN}{dt}=\mathrm{\Lambda }-dN-\left(d_P{Y}_3+d_A{Y}_6+d_P{Y}_7+d_P{Y}_8+d_{AP}{Y}_9\right),\end{array}$$ (6)

2.4. Positivity and Boundedness of the Solutions of the Model (4)

The model is mathematically analyzed by proving various theorems and algebraic computation dealing with different quantitative and qualitative attributes. Since the system deals with human populations which cannot be negative, we need to show that all the state variables are always nonnegative well as the solutions of system (4) remain positive with positive initial conditions (5) in the bounded region

$$\begin{array}{c}\Omega =\left\{\left(Y_1,Y_2,Y_3,Y_4,Y_5,Y_6,Y_7,Y_8,Y_9,Y_{10},Y_{11},Y_{12}\right)\in {ℝ}_{+}^{12},N\le \frac{\mathrm{\Lambda }}{d}\right\}.\end{array}$$ (7)

Here, in order for the model (4) to be epidemiologically well-posed, it is important to show that each state variable defined in Table 2 with positive initial conditions (5) is nonnegative for all time t > 0 in the bounded region given in (7).

Theorem 1 . —

At the initial conditions (5), the solutions Y₁(t), Y₂(t), Y₃(t), Y₄(t), Y₅(t),  Y₆(t), Y₇(t), Y₈(t), Y₉(t), Y₁₀(t), Y₁₁(t), and Y₁₂(t) of system (4) are nonnegative for all time  t > 0.

Proof —

Assume Y₁(0) > 0, Y₂(0) > 0, Y₃(0) > 0, Y₄(0) > 0, Y₅(0) > 0, Y₆(0) > 0, Y₇(0) > 0, Y₈(0) > 0, Y₉(0) > 0, Y₁₀(0) > 0, Y₁₁(0), and Y₁₂(0) > 0; then, for all t > 0, we have to prove that Y₁ (t) > 0, Y₂(t) > 0, Y₃(t) > 0, Y₄(t) > 0, Y₅(t) > 0, Y₆(t) > 0, Y₇(t) > 0, Y₈(t) > 0, Y₉(t) > 0,  Y₁₀(t) > 0, Y₁₁(t) > 0, and Y₁₂(t) > 0.

Define: τ = sup{t > 0 : Y₁ (t) > 0, Y₂(t) > 0, Y₃(t) > 0,  Y₄(t) > 0, Y₅(t) > 0, Y₆(t) > 0, Y₇(t) > 0, Y₈(t) > 0, Y₉(t) > 0, Y₁₀(t) > 0, Y₁₁(t) > 0, and Y₁₂(t) > 0}.

From the continuity of Y₁(t), Y₂(t), Y₃(t), Y₄(t),  Y₅(t), Y₆(t), Y₇(t), Y₈(t), Y₉(t), Y₁₀(t), Y₁₁(t), and Y₁₂(t)(t), we deduce that τ > 0. If τ = +∞, then positivity holds. But, if 0 < τ < +∞, Y₁(τ) = 0 or Y₂(τ) = 0 or Y₃(τ) = 0 or Y₄(τ) = 0 or Y₅(τ) = 0 or Y₆(τ) = 0 or Y₇(τ) = 0 or Y₈(τ) = 0 or Y₉(τ) = 0 or Y₁₀(τ) = 0 or Y₁₁(τ) = 0 or Y₁₂(0) = 0.

Here, from the first equation of the model (4), we have dY₁/dt = (1 − π)Λ + θY₁₀ + τY₂ − (d + λ_HC + λ_PC)Y_1.

Using the method of integrating factor, we obtained Y₁(τ) = M₁Y₁(0) + M₁∫₀^τexp^{∫(d + λ_Hc(t) + λ_Pc(t))dt}((1 − π)Λ + θY₁₀(t) + τY₂(t))dt > 0 where M₁ = exp^{−(dτ + ∫₀τ(λ_HC(w) + λ_PC(w)dw)} > 0, Y₁(0) > 0, and from the definition of τ, we see that Y₂(t) > 0,  Y₁₀(t) > 0, and also the exponential function is always positive; then, the solution Y₁(τ) > 0; hence, Y₁(τ) ≠ 0. From the second equation of the model (4), we have dY₂/dt = πΛ − (d + τ₁ + ϵλ_Pc + λ_Hc)Y₂ and we have got Y₂(τ) = M₁Y₂(0) + M₁∫₀^τexp^{∫(d + τ₁ + ϵλ_Pc(t) + λ_Hc(t))dt}(πΛ)dt > 0, where M₁ = exp^{−(dτ + τ₁τ + ∫₀τ(λ_Hc(w) + ϵλ_Pc(w)dw)} > 0, Y₂(0) > 0, and also, the exponential function always is positive; then, the solution Y₂(τ) > 0; hence, Y₂(τ) ≠ 0. Similarly, all the remaining state variables Y₃(τ) > 0; hence, Y₃(τ) ≠ 0 and Y₄(τ) > 0; hence, Y₄(τ) ≠ 0 and Y₅(τ) > 0; hence, Y₅(τ) ≠ 0 and Y₆(τ) > 0; hence, Y₆(τ) ≠ 0 and Y₇(τ) > 0; hence, Y₇(τ) ≠ 0 and Y₈(τ) > 0; hence, Y₈(τ) ≠ 0 and Y₉(τ) > 0; hence, Y₉(τ) ≠ 0 and Y₁₀(τ) > 0; hence, Y₁₀(τ) ≠ 0 and Y₁₁(τ) > 0; hence, Y₁₁(τ) ≠ 0 and Y₁₂(τ) > 0; hence Y₁₂(τ) ≠ 0. Thus, based on the definition of τ, it is not finite which means τ = +∞, and hence, all the solutions of system (2) are nonnegative.☐

Theorem 2 . —

The region Ω given by (7) is bounded in ℝ₊¹².

Proof —

Using equation (6), since all the state variables are nonnegative by Theorem 1, in the absence of infections, we have got dN/dt ≤ Λ − dN. By applying standard comparison theorem, we have got ∫(dN/(Λ − dN)) ≤ ∫dt and integrating both sides gives −(1/d)ln(Λ − dN) ≤ t + c where c is some constant, and after some steps of calculations, we have got 0 ≤ N (t) ≤ Λ/d which means all possible solutions of system (4) with positive initial conditions given in (5) enter in the bounded region (6).☐

3. The Mathematical Model Analysis

Before we analyze the HIV/AIDS-pneumonia coinfection model (4), we need to gain some background about the HIV/AIDS-only submodel and pneumonia-only submodel transmission dynamics.

3.1. HIV/AIDS Submodel Analysis

We have the HIV/AIDS submodel of (4) when Y₂ = Y₃ = Y₇ = Y₈ = Y₉ = Y₁₀ = Y₁₂ = 0 which is given by

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{d{Y}_1}{dt}=\mathrm{\Lambda }-\left(d+{\lambda }_H\right)Y_1,\\ \frac{d{Y}_4}{dt}={\lambda }_H{Y}_1-\left(d+{\kappa }_1+{\delta }_1\right)Y_4,\\ \frac{d{Y}_5}{dt}={\delta }_1{Y}_4-\left(d+{\kappa }_2+{\delta }_2\right)Y_5,\end{array}\\ \frac{d{Y}_6}{dt}={\delta }_2{Y}_5-\left(d+{\kappa }_3+d_A\right)Y_6,\\ \frac{d{Y}_{11}}{dt}={\kappa }_1{Y}_4+{\kappa }_2{Y}_5+{\kappa }_3{Y}_6-d{Y}_{11},\end{array}\end{array}$$ (8)

where the total population is N₁(t) = Y₁(t) + Y₄(t) + Y₅(t) + Y₆(t) + Y₁₁(t) and the HIV/AIDS single infection force of infection is given by λ_H = (β₁/N₁)(Y₄ + ρ₁Y₅) with initial conditions Y₁(0) > 0, Y₄(0) ≥ 0, Y₅(0) ≥ 0, Y₆(0) ≥ 0, and Y₁₁(0) ≥ 0.

Here, the detailed HIV/AIDS submodel model analysis is given in [6].

3.2. Pneumonia Submodel Analysis

From model (4), we have got the pneumonia submodel at Y₄ = Y₅ = Y₆ = Y₇= Y₈=Y₉ = Y₁₁ = Y₁₂ = 0, which is given by

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{d{Y}_1}{dt}=\left(1-\pi \right)\mathrm{\Lambda }+{\tau }_1{Y}_2+\theta Y_{10}-\left(d+{\lambda }_P\right)Y_1,\\ \frac{d{Y}_2}{dt}=\pi \mathrm{\Lambda }-ϵ{\lambda }_P{Y}_2-\left(d+{\tau }_1\right)Y_2,\end{array}\\ \frac{d{Y}_3}{dt}=ϵ{\lambda }_P{Y}_2+{\lambda }_4{Y}_1-\left(d+\kappa +d_P\right)Y_3,\\ \frac{d{Y}_{10}}{dt}=\kappa Y_3-\left(d+\theta \right)Y_{10}.\end{array}\end{array}$$ (9)

With initial conditions, Y₁(0) > 0, Y₂(0) ≥ 0, Y₃(0) ≥ 0, Y₁₀(0) ≥ 0, total population  N₂(t) = Y₁(t) + Y₂(t) + Y₃(t) + Y₁₀(t), and pneumonia force of infection  λ_P = β₂Y₃(t).

In the region${\Omega }_2=\left\{\begin{array}{c}\left(Y_1,Y_2,Y_3,Y_{10}\right)\in {{ℝ}^4}_{+},N_2\le \Lambda /d\end{array}\right\}$, it is easy to show that the set Ω₂ is positively invariant and a global attractor of all positive solutions of submodel (9). Hence, it is sufficient to consider the dynamics of model (9) in Ω₂ as epidemiologically and mathematically well-posed.

3.2.1. Disease-Free Equilibrium Point of the Pneumonia Submodel

The disease-free equilibrium point of the pneumonia submodel is obtained by making the right-hand side of the system (15) as zero and setting the infectious class and treatment class to zero as Y₃ = Y₁₀ = 0 we have got

Y ₁ ⁰ = Λ(d + τ₁) − Λπd/d(d + τ₁) and Y₂⁰ = Λπ/(d + τ₁) such that E₂⁰ = (Y₁⁰, Y₂⁰, Y₃⁰, Y₁₀⁰) = ((Λ(d + τ₁) − Λπd/d(d + τ₁)), (Λπ/d + τ₁), 0, 0).

3.2.2. The Effective Reproduction Number of the Pneumonia Submodel

The effective reproduction number measures the average number of new infections generated by a typically infectious individual in a community when some strategies are in place, like vaccination or treatment. We calculate the effective reproduction number  ℛ₂ using the van den Driesch and Warmouth next-generation matrix approach [23]. The Effective reproduction number is the largest (dominant) eigenvalue (spectral radius) of the matrix FV⁻¹ = [∂ℱ_i(E₂⁰)/∂x_j][∂ν_i(E₂⁰)/∂x_j]⁻¹ where ℱ_i is the rate of appearance of new infection in compartment i, ν_i is the transfer of infections from one compartment i to another, and E₂⁰ is the disease-free equilibrium point. Then, after a long calculation, we have got

$$\begin{array}{c}F=\left[\begin{array}{cc}\frac{{\beta }_2ϵ\mathrm{\Lambda }\pi d+{\beta }_2\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)-{\beta }_2\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)}& 0\\ 0& 0\end{array}\right],\\ V=\left[\begin{array}{cc}d+\kappa +d_P& 0\\ -\kappa & d+\theta \end{array}\right].\end{array}$$ (10)

Then, using Mathematica, we have got

$$\begin{array}{c}{V}^{-1}=\left[\begin{array}{cc}\frac{1}{d+\kappa +d_P}& 0\\ \frac{\gamma }{\left(\theta +d\right)\left(d+\kappa +d_P\right)}& \frac{1}{\theta +d}\end{array}\right],\\ F{V}^{-1}=\left[\begin{array}{cc}\frac{{\beta }_2ϵ\mathrm{\Lambda }\pi d+{\beta }_2\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)-{\beta }_2\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)\left(d+\kappa +d_P\right)}& 0\\ 0& 0\end{array}\right].\end{array}$$ (11)

The characteristic equation of the matrix FV⁻¹ is

$$\begin{array}{c}\left|\begin{array}{cc}\frac{{\beta }_2ϵ\mathrm{\Lambda }\pi d+{\beta }_2\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)-{\beta }_2\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)\left(d+\kappa +d_P\right)}-\lambda & 0\\ 0& 0-\lambda \end{array}\right|=0.\end{array}$$ (12)

Then, the spectral radius (effective reproduction number ℛ₂) of FV⁻¹ of the pneumonia submodel (9) is  ℛ₂ = (β₂ϵΛπd + β₂Λ(d + τ₁) − β₂Λπd)/(d(d + τ₁)(d + κ + d_P)). Here, ℛ₂ is the effective reproduction number for pneumonia infection.

3.2.3. Local and Global Stability of the Disease-Free Equilibrium Point

Theorem 3 . —

The disease-free equilibrium point (DFE) E₂⁰ of the pneumonia submodel (9) is locally asymptotically stable if ℛ₂ < 1, otherwise unstable.

Proof —

The local stability of the disease-free equilibrium of the system (9) can be studied from its Jacobian matrix at the disease-free equilibrium point E₂⁰ = ((Λ(d + τ₁) − Λπd)/(d(d + τ₁)), Λπ/(d + τ₁ ), 0, 0) and Routh Hurwitz stability criteria. The Jacobian matrix of a dynamical system (9) at the disease-free equilibrium point is given by

$$\begin{array}{c}J\left(E_2^0\right)=\left(\begin{array}{cccc}-d& {\tau }_1& \frac{-{\beta }_2\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)+{\beta }_2\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)}& \theta \\ 0& -\left(d+{\tau }_1\right)& \frac{-ϵ{\beta }_2\mathrm{\Lambda }\pi }{\text{d}+{\tau }_1}& 0\\ 0& 0& \frac{{\beta }_2ϵ\mathrm{\Lambda }\pi d+{\beta }_2\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)-{\beta }_2\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)}-\left(d+\kappa +d_P\right)& 0\\ 0& 0& \kappa & -\left(d+\theta \right)\end{array}\right).\end{array}$$ (13)

Then, the characteristic equation of the above Jacobian matrix is given by

$$\begin{array}{c}\left|\begin{array}{cccc}-d-\lambda & {\tau }_1& \frac{-{\beta }_2\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)+{\beta }_2\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)}& \theta \\ 0& -\left(d+{\tau }_1\right)-\lambda & \frac{-ϵ{\beta }_2\mathrm{\Lambda }\pi }{\text{d}+{\tau }_1}& 0\\ 0& 0& M-\lambda & 0\\ 0& 0& \kappa & -\left(d+\theta \right)-\lambda \end{array}\right|=0,\end{array}$$ (14)

where M = ((β₂ϵΛπd + β₂Λ(d + τ₁) − β₂Λπd)/(d(d + τ₁))) − (d + κ + d_P).

After some steps, we have got λ₁ = −d < 0 or λ₂ = −(d + τ₁) < 0 or λ₃ = (d + κ + d_P)[ℛ₂ − 1] < 0 if ℛ₂ < 1 or λ₄ = −(d + θ) < 0. Therefore, since all the eigenvalues of the characteristics polynomial of the system (9) are negative if ℛ₂ < 1, the disease-free equilibrium point of the pneumonia submodel is locally asymptotically stable.☐

3.2.4. Existence of EEP for the Pneumonia Submodel

Let an arbitrary endemic equilibrium point of pneumonia-only dynamical system (9) be denoted by E₂^∗ = (Y₁^∗, Y₂^∗, Y₃^∗, Y₁₀^∗). Moreover, let λ_P^∗ = β₂Y₃^∗ be the associated pneumonia mass action incidence rate (“force of infection”) at an equilibrium point. To find conditions for the existence of an arbitrary equilibrium point(s) for which pneumonia infection is endemic in the population, the equations of model (9) are solved in terms of the force of infection rate λ_P^∗ = β₂Y₃^∗ at an endemic equilibrium point. Setting the right-hand sides of the equations of model (9) to zero and we have got Y₂^∗ = πΛ/(ϵλ_P^∗ + d + τ₁), Y₁₀^∗ = κY₃^∗/(d + θ) and substitute Y₂^∗ and Y₁₀^∗  in to Y₁^∗, we obtain Y₁^∗ = ((1 − π)Λ + τ₁Y₂^∗ + θT_P^∗)/(d + λ_P^∗) = ((1 − π)Λ/(d + λ_P^∗)d + λ_P^∗) + (πΛτ₁/(ϵλ_P^∗ + d + τ₁)(d + λ_P^∗)) + (θγY₃^∗/(d + θ)(d + λ_P^∗)) and substitute Y₂^∗ and Y₁^∗ in Y₃^∗, we obtain

$$\begin{array}{c}{Y}_3^{\ast }=\frac{\pi \mathrm{\Lambda }ϵ{\lambda }_P^{\ast }\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)}{\left(d+\kappa +{\text{d}}_P\right)\left(ϵ{\lambda }_P^{\ast }+d+{\tau }_1\right)\left[\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)-\theta \kappa {\lambda }_P^{\ast }\right]}+\frac{\left(1-\pi \right)\mathrm{\Lambda }{\lambda }_P^{\ast }\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)}{\left(d+\kappa +d_P\right)\left(d+{\lambda }_P^{\ast }\right)\left[\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{{\lambda }_P}^{\ast }\right)-\theta \kappa {\lambda }_P^{\ast }\right]}+\frac{\pi \mathrm{\Lambda }{\tau }_1{\lambda }_P^{\ast }\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)}{\left(d+\kappa +d_P\right)\left(ϵ{\lambda }_P^{\ast }+d+{\tau }_1\right)\left(d+{\lambda }_P^{\ast }\right)\left[\left(d+\gamma +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)-\theta \kappa {\lambda }_P^{\ast }\right]}.\end{array}$$ (15)

Finally, substitute Y₃^∗ in to pneumonia submodel (9) force of infection λ_P^∗ = β₂Y₃^∗ as

$$\begin{array}{c}{\lambda }_P^{\ast }={\beta }_2{Y}_3^{\ast }=\frac{{\beta }_2\pi \mathrm{\Lambda }ϵ{\lambda }_P^{\ast }\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)}{\left(ϵ{\lambda }_P^{\ast }+d+{\tau }_1\right)\left[\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)-\theta \kappa {\lambda }_P^{\ast }\right]}+\frac{{\beta }_2\left(1-\pi \right)\mathrm{\Lambda }{\lambda }_P^{\ast }\left(d+\theta \right)}{\left[\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)-\theta \kappa {\lambda }_P^{\ast }\right]}+\frac{{\beta }_2\pi \mathrm{\Lambda }{\tau }_1{\lambda }_P^{\ast }\left(d+\theta \right)}{\left(ϵ{\lambda }_P^{\ast }+\kappa +{\tau }_1\right)\left[\left(d+\kappa +d_P\right)\left(d+\theta \right)\left(d+{\lambda }_P^{\ast }\right)-\theta \kappa {\lambda }_P^{\ast }\right]},\end{array}$$ (16)

and letting m₁ = d + κ + d_P, m₂ = d + τ₁, and m₃ = d + θ, we have got a₂λ_P^∗2+a₁λ_P^∗+a₀ = 0 where a₂ = m₁m₃ϵ − θκϵ > 0,  a₁ = m₁m₃dϵ + m₁m₂m₃ − m₂θκ − β₂Λm₃ϵ, and a₀ = m₁m₂m₃μ[1 − ℛ₂] > 0 if ℛ₂ < 1.

Here, the nonzero equilibrium(s) of the model (9) satisfies f(λ_P^∗) = a₂λ_P^∗2 + a₁λ_P^∗ + a₀ = 0 so that the quadratic equation can be analyzed for the possibility of multiple equilibriums. It is worth noting that the coefficient a₂ is always positive and a₀ is positive (negative) if ℛ_P is less than (greater than) unity, respectively. Hence, we have established the following result.

Theorem 4 . —

The pneumonia submodel (9) has the following:

1. Exactly one unique endemic equilibrium if a₀ < 0 (i.e., ℛ₂>1)

2. Exactly one unique endemic equilibrium if a₁<0, and a₀ = 0 or a₁² − 4a₂a₀ = 0

3. Exactly two endemic equilibriums if a₀ > 0 (i.e., ℛ₂ < 1), a₁ < 0, and a₁² − 4a₂a₀ > 0

4. No endemic equilibrium otherwise

Here, item (iii) shows the happening of the backward bifurcation in pneumonia submodel (9), i.e., the locally asymptotically stable disease-free equilibrium point coexists with a locally asymptotically stable endemic equilibrium point if ℛ₂ < 1; examples of the existence of backward bifurcation phenomenon in mathematical epidemiological models, and the causes, can be seen in [2, 9, 22, 24–26]. The epidemiological consequence is that the classical epidemiological requirement of having the reproduction number ℛ₂ to be less than one, even though necessary, is not sufficient for the effective control of the disease. The existence of the backward bifurcation phenomenon in submodel (9) is now explored.

3.2.5. Bifurcation Analysis

It is instructive to explore the possibility of backward bifurcation in model (9).

Theorem 5 . —

Model (9) exhibits backward bifurcation at ℛ₂ = 1 whenever the inequality D₁ > D₂ holds.

Here, we apply the center manifold theory in [27]; however, to apply this theory, the following simplification and change of variables are made.

Let Y₁ = x₁,Y₂ = x₂, Y₃ = x₃, and Y₁₀ = x₄ such that N₂ = x₁ + x₂ + x₃ + x₄. Furthermore, by using vector notation X = (x₁, x₂, x₃, x₄)^T, pneumonia submodel (9) can be written in the form dX/dt = F(X) with

F = (f₁, f₂, f₃, f₄)^T, as follows:

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{{dx}_1}{dt}=f_1=\left(1-\pi \right)\mathrm{\Lambda }+{\tau }_1{x}_2+\theta x_4-\left(d+{\lambda }_P\right)x_1,\\ \frac{{dx}_2}{dt}=f_2=\pi \mathrm{\Lambda }-\left(ϵ{\lambda }_P+d+{\tau }_1\right)x_2,\end{array}\\ \frac{{dx}_3}{dt}=f_3=ϵ{\lambda }_P{x}_2+{\lambda }_P{x}_1-\left(d+\kappa +d_P\right)x_3,\\ \frac{{dx}_4}{dt}=f_4=\kappa x_3-\left(d+\theta \right)x_4,\end{array}\end{array}$$ (17)

with λ_P = β₂x₃, then the method entails evaluating the Jacobian of system (17) at the DFE point E₂⁰, denoted by J(E₂⁰), and this gives us

$$\begin{array}{c}J\left(E_2^0\right)=\left(\begin{array}{cccc}-d& {\tau }_1& \frac{-{\beta }_2\mathrm{\Lambda }\left(d+{\tau }_1\right)+{\beta }_2\mathrm{\Lambda }\pi d}{d\left(d+{\tau }_1\right)}& \theta \\ 0& -\left(d+{\tau }_1\right)& \frac{-ϵ{\beta }_2\mathrm{\Lambda }\pi }{\mu +{\tau }_1}& 0\\ 0& 0& \frac{{\beta }_2ϵ\mathrm{\Lambda }\pi d+{\beta }_2\mathrm{\Lambda }\left(d+{\tau }_1\right)-{\beta }_2\mathrm{\Lambda }\pi \mu }{d\left(d+{\tau }_1\right)}-\left(d+\kappa +d_P\right)& 0\\ 0& 0& \kappa & -\left(d+\theta \right)\end{array}\right).\end{array}$$ (18)

Consider, next, the case when ℛ_P = 1. Suppose, further, that β₂ = β^∗  is chosen as a bifurcation parameter.

Solving for β₂ from ℛ₂ = 1 as ℛ₂ = β₂ϵΛπd + β₂Λ(d + τ₁) − β₂Λπd/d(d + τ₁)(d + κ + d_P) = 1 and we have got β₂ = β^∗ = d(d + τ₁)(d + κ + d_P)/ϵΛπd + Λ(d + τ₁) − Λπd and

$$\begin{array}{c}{J}_{{\beta }^{\ast }}=\left(\begin{array}{cccc}-d& {\tau }_1& \frac{-{\beta }^{\ast }\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)+{\beta }^{\ast }\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)}& \theta \\ 0& -\left(d+{\tau }_1\right)& \frac{-ϵ{\beta }^{\ast }\mathrm{\Lambda }\pi }{\text{d}+{\tau }_1}& 0\\ 0& 0& \frac{{\beta }^{\ast }ϵ\mathrm{\Lambda }\pi d+{\beta }^{\ast }\mathrm{\Lambda }\left(\text{d}+{\tau }_1\right)-{\beta }^{\ast }\mathrm{\Lambda }\pi \text{d}}{d\left(\text{d}+{\tau }_1\right)}-\left(d+\kappa +d_P\right)& 0\\ 0& 0& \gamma & -\left(\mu +\theta \right)\end{array}\right).\end{array}$$ (19)

After some steps of the calculation, we have got the eigenvalues of J_β^∗ as λ₁ = −d or λ₂ = −(d + τ₁) or λ₃ = 0 or λ₄ = −(d + θ).

It follows that the Jacobian J(E₂⁰) of (17) at the DFE, with β₂ = β^∗, denoted by J_β^∗, has a simple zero eigenvalue with all the remaining eigenvalues having a negative real part. Hence, the center manifold theory [27] can be used to analyze the dynamics of model (9). In particular, Theorem 2 of Castillo-Chavez and Song [28] will be used to show that model (9) undergoes backward bifurcation at ℛ₂ = 1

Eigenvectors of J_β^∗: for the case ℛ₂ = 1, it can be shown that the Jacobian of (29) at β₂ = β^∗ (denoted by J_β^∗) has a right eigenvectors associated with the zero eigenvalue given by u = (u₁, u₂, u₃, u₄)^T with values

$$\begin{array}{c}{u}_1=\left[\frac{-ϵ{\beta }^{\ast }\mathrm{\Lambda }\pi d{\tau }_1\left(d+\theta \right)-{\beta }^{\ast }\mathrm{\Lambda }{\left(\text{d}+{\tau }_1\right)}^2\left(d+\theta \right)+{\beta }^{\ast }\mathrm{\Lambda }\pi \text{d}\left(\text{d}+{\tau }_1\right)\left(d+\theta \right)+\theta \kappa d{\left(\text{d}+{\tau }_1\right)}^2}{d^2{\left(\text{d}+{\tau }_1\right)}^2}\right]u_3,\\ u_2=-\frac{ϵ{\beta }^{\ast }\mathrm{\Lambda }\pi }{{\left(\text{d}+{\tau }_1\right)}^2}{u}_3,\\ u_3=u_3>0,\\ u_4=\frac{\kappa }{d+\theta }{u}_3.\end{array}$$ (20)

Similarly, the left eigenvector associated with the zero eigenvalues at β₂ = β^∗ given by v = (v₁, v₂, v₃, v₄)^T are v₁ = v₂ = v₄ = 0, v₃ = v₃ > 0.

After long calculations, the bifurcation coefficients a and b are obtained as a = D₁ − D₂ where D₁=β^∗Λπd(d + τ₁)(d + θ) + θκd(d + τ₁)²/d²(d + τ₁)², and  D₂ = (ϵβ^∗Λπdτ₁(d + θ) + β^∗Λ(d + τ₁)²(d + θ)/d²(d + τ₁)²) + ϵ(ϵβ^∗Λπ/(d + τ₁)²).

Thus, the bifurcation coefficient a is positive if D₁ > D₂. Furthermore,  b = v₃u₂u₃(Λ(d + τ₁) − Λπd/d(d + τ₁)) > 0.

Hence, from in Castillo-Chavez and Song [28], model (9) exhibits a backward bifurcation at ℛ₂ = 1 whenever D₁ > D₂.

3.3. Analysis of the Full HIV/AIDS-Pneumonia Coinfection

Having analyzed the dynamics of the two submodels, that is, HIV/AIDS submodel (8) and the pneumonia submodel (9), the complete HIV/AIDS-pneumonia coinfection model (4) is now considered (the analysis is done in the positively invariant region Ω given in (7)).

3.3.1. Disease-Free Equilibrium Point of the HIV/AIDS-Pneumonia Coinfection

The disease-free equilibrium point of model (4) is obtained by setting all the infectious classes and treatment classes to zero such that  Y₃ = Y₄ = Y₅ = Y₆ = Y₇ = Y₈ = Y₉ = Y₁₀ = Y₁₁ = Y₁₂ = 0 and hence E₃⁰== (Λ(d + τ₁) − Λπd/d(d + τ₁), Λπ/(d + τ₁ ), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).

3.3.2. Effective Reproduction Number of the HIV/AIDS-Pneumonia Coinfection

The basic reproduction number, denoted by ℛ₀, is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [6, 23, 28]. For simple classical models if ℛ₀ < 1, then it means that on average, an infected individual infects less than one susceptible over the course of its infectious period and the disease cannot grow. If however, ℛ₀ > 1, then an infected individual infects more than one susceptible over the course of its infectious period and the disease will persist. For more complicated models with several infected compartments, this simple heuristic definition of ℛ₀ is insufficient [23]. Due to its importance, researchers have sought to find ways of determining ℛ₀. Two important concepts in modeling outbreaks of infectious diseases are the basic reproduction number, universally denoted by ℛ₀ and the generation time (the average time from symptom onset in a primary case to symptom onset in a secondary case), which jointly determine the likelihood and speed of epidemic outbreaks [29].

Here, we calculated the HIV/AIDS-pneumonia coinfection effective reproduction number ℛ₃ of model (4) using the van den Driesch and Warmouth next-generation matrix approach [23]. The effective reproduction number is the largest (dominant) eigenvalue (spectral radius) of the matrix FV⁻¹ = [∂ℱ_i(E₃⁰)/∂x_j][∂ν_i(E₃⁰)/∂x_j]⁻¹ where ℱ_i is the rate of appearance of new infection in compartment i , ν_i is the transfer of infections from one compartment i to another, and E₃⁰ is the disease-free equilibrium point E₃⁰ = (Λ(d + τ₁) − Λπd/d(d + τ₁), Λπ/d + τ₁, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).

After long detailed calculations, the transition matrix F is given by

$$\begin{array}{c}\text{F}=\left[\begin{array}{cccccccccc}A& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {\beta }_1& {\beta }_1{\rho }_1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],\end{array}$$ (21)

and the transmission matrix V is given by

$$\begin{array}{c}V=\left[\begin{array}{cccccccccc}{D}_1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& D_2& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -{\delta }_1& D_3& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -{\delta }_2& D_4& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& D_5& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -{\delta }_3& D_6& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -{\delta }_4& D_7& 0& 0& 0\\ -\kappa & 0& 0& 0& 0& 0& 0& \theta & 0& 0\\ 0& -{\kappa }_1& -{\kappa }_2& -{\kappa }_3& 0& 0& 0& 0& d& 0\\ 0& 0& 0& 0& -{\sigma }_1& -{\sigma }_2& -{\sigma }_3& 0& 0& d\end{array}\right],\end{array}$$ (22)

where D₁ = d + κ + d_P, D₂ = d + κ₁ + δ₁, D₃ = d + κ₂ + δ₂, D₄ = d + κ₃ + d_A, D₅ = d + d_P + σ₁ + δ₃, D₆ = d + d_P + σ₂ + δ₄, and D₇ = d + d_AP + ε₃.

Then, by using Mathematica, we have got

$$\begin{array}{c}F{V}^{-1}=\left[\begin{array}{cccccccccc}\frac{\left(\pi ϵ\mathrm{\Lambda }{\beta }_2/d+{\tau }_1\right)+{\beta }_2\left[-\pi \mathrm{\Lambda }d+\mathrm{\Lambda }\left[d+{\tau }_1\right]/d\left[d+{\tau }_1\right]\right]}{D_1}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& \frac{{\beta }_1}{D_2}+\frac{{\delta }_1{\beta }_1{\rho }_1}{D_2{D}_3}& \frac{{\beta }_1{\rho }_1}{D_3}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right].\end{array}$$ (23)

The characteristic equation of the matrix FV⁻¹ is given by

$$\begin{array}{c}\left|\begin{array}{cccccccccc}\left(A_1-\lambda \right)& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& \left(B-\lambda \right)& \frac{{\beta }_1{\rho }_1}{D_3}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& \left(0-\lambda \right)& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \left(0-\lambda \right)& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \left(0-\lambda \right)& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& \left(0-\lambda \right)& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \left(0-\lambda \right)& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \left(0-\lambda \right)& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \left(0-\lambda \right)& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& \left(0-\lambda \right)\end{array}\right|=0,\end{array}$$ (24)

where A₁ = (πϵΛβ₂/(d + τ₁)) + β₂[(−πΛd + Λ[d + τ₁]/d[d + τ₁])]/D₁, B = (β₁/D₂) + (δ₁β₁ρ₁/D₂D₃);  then, the eigenvalues of FV⁻¹ are λ₁ = β₂ϵπdΛ + β₂Λ[d + τ₁] − β₂πΛd/D₁(d + τ₁) or λ₂ = (β₁/D₂) + δ₁β₁ρ₁/D₂D₃  or λ₃ = λ₄ = λ₅ = λ₆ = λ₇ = λ₈ = λ₉ = λ₁₀ = 0.

Thus, the effective reproduction number of the HIV/AIDS-pneumonia coinfection dynamical system (4) is the dominant eigenvalue of the matrix FV⁻¹ which is given by

ℛ ₃ = max{λ₁, λ₂} = max{β₂ϵπdΛ + β₂Λ[d + τ₁] − β₂πΛd/D₁(d + τ₁), (β₁/D₂) + (δ₁β₁ρ₁/D₂D₃)}. Here,  ℛ₂ = β₂ϵπdΛ + β₂Λ[d + τ₁] − β₂πΛd/(d + κ + d_P)(d + τ₁) is the effective reproduction number for pneumonia-only infected individual and ℛ₁ = (β₁/(d + κ₁ + δ₁)) + (β₁ρ₁δ₁/(d + κ₁ + δ₁)(d + κ₂ + κ₂)) is the basic reproduction for HIV/AIDS-only infected individual.

Here, ℛ₁ represent the basic reproduction number for HIV/AIDS submodel,  ℛ₂ and ℛ₃ are the effective reproduction numbers for the pneumonia submodel and HIV/AIDS-pneumonia coinfection model, respectively. The following three outcomes are possible: (i) for ℛ₁ < 1, the HIV/AIDS submodel disease-free steady state E₁ is globally stable in the region Ω₁, and HIV is not spreading in the community; (ii) for ℛ₂ < 1, then E₂ is not globally stable in the region Ω₂, and pneumonia may spread through the community; (iii) for ℛ₃ < 1, the steady state E₃ is not globally stable in the region Ω, and HIV/AIDS-pneumonia coinfection may spread through the community.

Note that none of the parameters corresponding to coinfection treatment (i.e., σ₁ or σ₂ or σ₃) are present in the expression for ℛ₃, indicating no impact of treating coinfected population on ℛ₃.

3.3.3. Locally Asymptotically Stability of the Disease-Free Equilibrium (DFE)

Theorem 6 . —

The disease-free equilibrium of model (4) above is locally asymptotically stable if ℛ₃ < 1, otherwise unstable.

Proof —

The Jacobian matrix J(E₃⁰) of model (4) at E₃⁰ is given by

$$\begin{array}{c}J\left(E_3^0\right)=\left[\begin{array}{cccccccccccc}-d& {\tau }_1& -{\beta }_2{Y}_1^0& -\frac{{\beta }_1}{N^0}{Y}_1^0& -\frac{{\beta }_1}{N^0}{\rho }_1{Y}_1^0& 0& 0& 0& \theta & 0& 0& 0\\ 0& -d-{\tau }_1& -{\beta }_2ϵY_2^0& -\frac{{\beta }_1}{N^0}{Y}_2^0& -\frac{{\beta }_1}{N^0}{\rho }_1{Y}_2^0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& Z_1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& Z_2& \frac{{\beta }_1}{N^0}{\rho }_1{Y}_1^0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\delta }_1& Z_3& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\delta }_2& Z_4& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& Z_5& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\delta }_3& Z_6& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\delta }_4& Z_7& 0& 0& 0\\ 0& 0& \kappa & 0& 0& 0& 0& 0& 0& Z_8& 0& 0\\ 0& 0& 0& {\kappa }_1& {\kappa }_2& {\kappa }_3& 0& 0& 0& 0& -d& 0\\ 0& 0& 0& 0& 0& 0& {\sigma }_1& {\sigma }_2& {\sigma }_3& 0& 0& -d\end{array}\right],\end{array}$$ (25)

where Z₁ = β₂ϵY₂⁰ + β₂Y₁⁰ − (d + κ + d_P), Z₂ = (β₁/N⁰)Y₁⁰ − (d + κ₁ + δ₁),  Z₃ = −(d + κ₂ + δ₂),  Z₄ = −(d + κ₃ + d_A), Z₅ = −(d + d_P + σ₁ + δ₃), Z₆ = −(d + d_P + σ₂ + δ₄), Z₇ = −(d + d_AP + σ₃), and Z₈ = −(d + θ).

Then, the characteristic equation of the Jacobian matrix J (E₃⁰) is given by

$$\begin{array}{c}\left|\begin{array}{cccccccccccc}-d-\lambda & {\tau }_1& -{\beta }_2{Y}_1^0& -\frac{{\beta }_1}{N^0}{Y}_1^0& -\frac{{\beta }_1}{N^0}{\rho }_1{Y}_1^0& 0& 0& 0& 0& \theta & 0& 0\\ 0& -d-{\tau }_1-\lambda & -{\beta }_2ϵY_2^0& -\frac{{\beta }_1}{N^0}{Y}_2^0& -\frac{{\beta }_1}{N^0}{\rho }_1{Y}_2^0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& Z_1-\lambda & 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& Z_2-\lambda & \frac{{\beta }_1}{N^0}{\rho }_1{Y}_1^0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\delta }_1& Z_3-\lambda & 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\delta }_2& Z_4-\lambda & 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& Z_5-\lambda & 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\delta }_3& Z_6-\lambda & 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\delta }_4& Z_7-\lambda & 0& 0& 0\\ 0& 0& \kappa & 0& 0& 0& 0& 0& 0& Z_8-\lambda & 0& 0\\ 0& 0& 0& {\kappa }_1& {\kappa }_2& {\kappa }_3& 0& 0& 0& 0& -d-\lambda & 0\\ 0& 0& 0& 0& 0& 0& {\sigma }_1& {\sigma }_2& {\sigma }_3& 0& 0& -d-\lambda \end{array}\right|=0.\end{array}$$ (26)

After detailed calculations, we have got that

λ ₁ = λ₂ = λ₃ = −d < 0 or  λ₄ = (d + κ + d_P)[ℛ₂ − 1] < 0 if ℛ₂ < 1 or λ₅ = −(d + τ₁) < 0 or λ₆ = −(d + κ₃ + d_A) < 0 or λ₇ = −(d + d_P + σ₁ + δ₃) < 0 or λ₈ = −(d + d_P + σ₂ + δ₄) < 0 or λ₉ = −(d + d_AP + σ₃) < 0 or λ₁₀ = −(d + θ) < 0 or a₂λ² + a₁λ + a₀ = 0 for a₂ = 1, a₁ = (d + κ₂ + δ₂) + (d + κ₁ + δ₁)[1 − (Y₁⁰/N⁰)ℛ_Y4] > 0 if ℛ_Y4 < 1 and a₀ = (d + κ₂ + δ₂)(d + κ₁ + δ₁)[1 − (Y₁⁰/N⁰)ℛ_Y5) > 0 if ℛ_Y5 < 1.

Then, by applying Routh-Hurwitz stability criteria since a₂ = 1 > 0, a₁ > 0, and a₀ > 0, all the eigenvalues of the Jacobian matrix are negative if ℛ₁ < 1 and ℛ₂ < 1, i.e., ℛ₃ = max{ℛ₁, ℛ₂} < 1. Thus, the disease-free equilibrium point (DFE) of HIV/AIDS-pneumonia coinfection model (4) is locally asymptotically stable if

$$\begin{array}{c}{\mathcal{R}}_3=\max \left\{{\mathcal{R}}_1,{\mathcal{R}}_2\right\}<1.\end{array}$$ (27)

☐

3.3.4. Existence of Endemic Equilibrium Point (EEP) for the Full Model

The endemic equilibrium point (EEP) of full model (4) is denoted by E₃^∗ = (Y₁^∗, Y₂^∗, Y₃^∗, Y₄^∗, Y₅^∗, Y₆^∗, Y₇^∗, Y₈^∗, Y₉^∗, Y₁₀^∗, Y₁₁^∗, Y₁₂^∗) which occurs when the disease persists in the community. From the analysis of HIV/AIDS-only submodel (8) and the pneumonia-only submodel from (9), we have shown that there is no endemic equilibrium point if ℛ₁ < 1 and there is/are an endemic equilibrium point(s) if ℛ₂ < 1 implies that there is/are endemic equilibrium point(s) if ℛ₃ < 1 for the coinfection model and hence there is a bifurcation point for the full model. The endemic equilibrium of system (4) is obtained as

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}{Y}_1^{\ast }=\frac{\left(1-\pi \right)\mathrm{\Lambda }+{\tau }_1{Y}_2^{\ast }+\theta Y_{10}^{\ast }}{d+{\lambda }_{HC}^{\ast }+{\lambda }_{PC}^{\ast }},Y_2^{\ast }=\frac{\pi \mathrm{\Lambda }}{ϵ{\lambda }_{PC}^{\ast }+\left(d+{\tau }_1+{\lambda }_{HC}^{\ast }\right)},Y_3^{\ast }=\frac{ϵ{\lambda }_{PC}^{\ast }{Y}_2^{\ast }+{\lambda }_{PC}^{\ast }{Y}_1^{\ast }}{\nu {\lambda }_{HC}^{\ast }+d+\kappa +d_P},\\ Y_4^{\ast }=\frac{{\lambda }_{HC}^{\ast }{Y}_1^{\ast }}{d+{\kappa }_1+{\delta }_1+{\psi }_1{\lambda }_{PC}^{\ast }},Y_5^{\ast }=\frac{{\delta }_1{H}_1^{\ast }}{d+{\kappa }_2+{\kappa }_2+{\psi }_2{\lambda }_{PC}^{\ast }},Y_6^{\ast }=\frac{{\delta }_2{Y}_5^{\ast }}{d+{\kappa }_3+d_A+{\delta }_3{\lambda }_{PC}^{\ast }},\end{array}\\ Y_7^{\ast }=\frac{{\psi }_1{\lambda }_{PC}^{\ast }{Y}_4^{\ast }+\nu {\lambda }_{HC}^{\ast }{Y}_3^{\ast }}{d+d_P+{\sigma }_1+{\delta }_3},Y_8^{\ast }=\frac{{\delta }_2{\lambda }_{PC}^{\ast }{Y}_5^{\ast }+{\delta }_3{Y}_7^{\ast }}{d+d_P+{\sigma }_2+{\delta }_4},Y_9^{\ast }=\frac{{\psi }_3{\lambda }_{PC}^{\ast }{Y}_6^{\ast }+{\delta }_4{Y}_8^{\ast }}{d+d_{AP}+{\kappa }_3},\\ Y_{10}^{\ast }=\frac{\kappa Y_3^{\ast }}{d+\theta },Y_{11}^{\ast }=\frac{{\kappa }_1{Y}_4^{\ast }+{\kappa }_2{Y}_5^{\ast }+{\kappa }_3{Y}_6^{\ast }}{d},\text{and\hspace{0.17em}}{Y}_{12}^{\ast }=\frac{{\kappa }_1{Y}_7^{\ast }+{\kappa }_2{Y}_8^{\ast }+{\kappa }_3{Y}_9^{\ast }}{d}.\end{array}\end{array}$$ (28)

Summary of endemic equilibrium points: the explicit computation of the endemic equilibrium of coinfection model (4) given in (28) in terms of model parameters is difficult analytically; however, model (4) endemic equilibriums correspond to the following:

1. E₄^∗ = (Y₁^∗, 0, Y₄^∗, Y₅^∗, Y₆^∗, 0, 0, 0, 0, 0, Y₁₁^∗, 0), if ℛ₁ > 1 is the pneumonia free (HIV) endemic equilibrium point. The analysis of the equilibrium E₁^∗ is similar to the endemic equilibrium E₁^∗ in model (7)

2. E₅^∗ = (Y₁^∗, Y₂^∗, Y₃^∗, 0, 0, 0, 0, 0, 0, Y₁₀^∗, 0, 0), if ℛ₂ > 1 is the HIV/AIDS free (pneumonia) endemic equilibrium point. The analysis of the equilibrium E₅^∗ is similar to the endemic equilibrium E₂^∗ in equation (9)

3. E₆^∗ = (Y₁^∗, Y₂^∗, Y₃^∗, Y₄^∗, Y₅^∗, Y₆^∗, Y₇^∗, Y₈^∗, Y₉^∗, Y₁₀^∗, Y₁₁^∗, Y₁₂^∗) is the HIV/AIDS-pneumonia coinfection endemic equilibrium point. It exists when each component of E₆^∗ in equation (28) is positive and summarizes the existence of the endemic equilibrium points in the following theorem

3.3.5. Bifurcation Analysis

The threshold quantity  ℛ₃ = max{ℛ₁, ℛ₂}  is the effective reproduction number of the system (4) where ℛ₁ and ℛ₂ are defined as above.

Theorem 7 . —

Model (4) exhibits the phenomenon of backward bifurcation at ℛ₃ = 1 whenever the inequality G₁ > G₂ holds.

The phenomenon of backward bifurcation can be proved with the concept of the center manifold theory [10, 27] on coepidemic model (4). To apply this theory, the following simplification and change of variables are made.

Let Y₁ = x₁, Y₂ = x₂, Y₃ = x₃, Y₄ = x₄, Y₅ = x₅, Y₆ = x₆, Y₇ = x₇, Y₈ = x₈, Y₉ = x₉, Y₁₀ = x₁₀, Y₁₁ = x₁₁, and Y₁₂ = x₁₂ so that N = x₁ + x₂ + x₃ + x₄+x₅,+x₆+x₇,+x₈+x₉+x₁₀+x₁₁+x₁₂.

Further, by using vector notation  X = (x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀, x₁₁, x₁₂)^T, complete model (4) can be written in the form dX/dt = F(X) with F = (f₁, f₂, f₃, f₄, f₅, f₆, f₇, f₈, f₉, f₁₀, f₁₁, f₁₂)^T, as follows:

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{{dx}_1}{dt}=f_1=\left(1-\pi \right)\mathrm{\Lambda }+{\tau }_1{x}_2+\theta x_{10}-\left(d+{\lambda }_{HC}+{\lambda }_{PC}\right)x_1,\\ \frac{{dx}_2}{dt}=f_2=\pi \mathrm{\Lambda }-ϵ{\lambda }_{PC}{x}_2-\left(d+{\tau }_1+{\lambda }_{HC}\right)x_2,\end{array}\\ \frac{{dx}_3}{dt}=f_3=ϵ{\lambda }_{PC}{x}_2+{\lambda }_{PC}{x}_1-\left(\nu {\lambda }_{HC}+d+\kappa +d_P\right)x_3,\\ \frac{{dx}_4}{dt}=f_4={\lambda }_{HC}{x}_1-\left(d+{\kappa }_1+{\delta }_1+\psi {\lambda }_{PC}\right)x_4,\end{array}\\ \frac{{dx}_5}{dt}=f_5={\delta }_1{x}_4-\left(d+{\kappa }_2+{\kappa }_2+{\psi }_2{\lambda }_{PC}\right)x_5,\\ \frac{{dx}_6}{dt}=f_6={\delta }_2{x}_5-\left(d+{\kappa }_3+d_A+{\psi }_3{\lambda }_{PC}\right)x_6,\end{array}\\ \frac{{dx}_7}{dt}=f_7={\psi }_1{\lambda }_{PC}{x}_4+\nu {\lambda }_{HC}{x}_3-\left(d+d_P+{\sigma }_1+{\delta }_3\right)x_7,\\ \frac{{dx}_8}{dt}=f_8={\psi }_2{\lambda }_{PC}{x}_5+{\delta }_3{x}_7-\left(d+d_P+{\sigma }_2+{\delta }_4\right)x_8,\end{array}\\ \frac{{dx}_9}{dt}=f_9={\psi }_3{\lambda }_{PC}{x}_6+{\delta }_4{x}_8-\left(d+d_{AP}+{\kappa }_3\right)x_9,\\ \frac{{dx}_{10}}{dt}=f_{10}=\kappa x_3-\left(d+\theta \right)x_{10},\end{array}\\ \frac{{dx}_{11}}{dt}=f_{11}={\kappa }_1{x}_4+{\kappa }_2{x}_5+{\kappa }_3{x}_6-d{x}_{11},\\ \frac{{dx}_{12}}{dt}=f_{12}={\kappa }_1{x}_7+{\kappa }_2{x}_8+{\kappa }_3{x}_9-d{x}_{12},\end{array}\end{array}$$ (29)

with λ_HC = β₁/N[x₄ + ρ₁x₅ + ρ₂x₇ + ρ₂x₈] where ρ₃ ≥ ρ₂ ≥ ρ₁ ≥ 1 and λ_PC = β₂[x₃ + ω₁x₇ + ω₂x₈ + ω₃x₉]  where ω₃ ≥ ω₂ ≥ ω₁ ≥ 1; then, the method entails evaluating the Jacobian of system (29) at the DFE E₃⁰, denoted by J(E₃⁰), and this gives us

$$\begin{array}{c}J\left(E_3^0\right)=\left(\begin{array}{cccccccccccc}-d& {\tau }_1& F_1& F_2& F_3& 0& F_4& F_5& F_6& \theta & 0& 0\\ 0& F_7& F_8& F_9& F_{10}& 0& F_{11}& F_{12}& F_{13}& 0& 0& 0\\ 0& 0& F_{14}& 0& 0& 0& F_{15}& F_{16}& F_{17}& 0& 0& 0\\ 0& 0& 0& F_{18}& F_{19}& 0& F_{20}& F_{21}& 0& 0& 0& 0\\ 0& 0& 0& {\delta }_1& F_{22}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\delta }_2& F_{23}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& F_{24}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\delta }_3& F_{25}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& d_4& F_{26}& 0& 0& 0\\ 0& 0& \kappa & 0& 0& 0& 0& 0& 0& F_{27}& 0& 0\\ 0& 0& 0& {\kappa }_1& {\kappa }_2& {\kappa }_3& 0& 0& 0& 0& -d& 0\\ 0& 0& 0& 0& 0& 0& {\kappa }_1& {\kappa }_2& {\kappa }_3& 0& 0& -d\end{array}\right),\end{array}$$ (30)

where F₁ = −β₂Y₁⁰, F₂ = −β₁(Y₁⁰/(Y₁⁰ + Y₂⁰)), F₃ = −β₁ρ₁(Y₁⁰/(Y₁⁰ + Y₂⁰)), F₄ = −β₁ρ₂(Y₁⁰/(Y₁⁰ + Y₂⁰)) − β₂ω₁Y₁⁰, F₅ = −β₁ρ₃(Y₁⁰/(Y₁⁰ + Y₂⁰)) − β₂ω₂Y₁⁰, F₆ = −β₂ω₃Y₁⁰,  F₇ = −(d + τ₁), F₈ = −ϵβ₂Y₂⁰, F₉ = −β₁(Y₂⁰/(Y₁⁰ + Y₂⁰)), F₁₀ = −β₁ρ₁(Y₂⁰/(Y₁⁰ + Y₂⁰)),

F ₁₁ = −ϵβ₂ω₁V_P⁰ − β₁ρ₂(Y_P⁰/(Y₁⁰ + Y_P⁰)), F₁₂ = −ϵβ₂ω₂V_P⁰ − β₁ρ₃(Y₂⁰/(Y₁⁰ + Y₂⁰)), F₁₃ = −ϵβ₂ω₃Y₂⁰, F₁₄ = ϵβ₂Y₂⁰ + β₂Y₁⁰ − (d + κ + d_P), F₁₅ = ϵβ₂ω₁Y₂⁰ + β₂ω₁Y₁⁰, F₁₆ = ϵβ₂ω₂Y₂⁰ + β₂ω₂Y₁⁰, F₁₇ = ϵβ₂ω₃Y₂⁰ + β₂ω₃Y₁⁰, F₁₈ = β₁(Y₁⁰/(Y₁⁰ + Y₂⁰)) − (d + κ₁ + δ₁), F₁₉ = β₁ρ₁(Y₁⁰/(Y₁⁰ + Y₂⁰)), F₂₀ = β₁ρ₂(Y₁⁰/(Y₁⁰ + Y₂⁰)), F₂₁ = β₁ρ₃(Y₁⁰/(Y₁⁰ + Y₂⁰)), F₂₂ = −(d + κ₂ + δ₂), F₂₃ = −(d + κ₃ + d_A), F₂₄ = −(d + d_P + σ₁ + δ₃), F₂₅ = −(d + d_P + σ₂ + δ₄), F₂₆ = −(d + d_AP + σ₃), and  F₂₇ = −(d + θ).

Without loss of generality, consider the case when ℛ₂ > ℛ₁, and ℛ₃ = 1, so that ℛ₂ = 1. Furthermore, let β₂ = β^∗ is chosen as a bifurcation parameter. Solving for β₂ from ℛ₂ = 1 as ℛ₂ = β₂ϵΛπd + β₂Λ(d + τ₁) − β₂Λπd/d(d + τ₁)(d + κ + d_P) = 1, we have got the value β^∗ = β₂ = d(d + τ₁)(d + κ + d_P)/ϵΛπd + Λ(d + τ₁) − Λπd.

After solving the Jacobian J(E₃⁰) of the system (29) at the DFE, with β₂ = β^∗, we obtained the eigenvalues as λ₁ = −d < 0 or λ₂ = −(d + τ₁) < 0 or λ₃ = 0  or λ₄ = −d < 0 or λ₅ = −d < 0 or λ₆ = −(d + θ) < 0 or λ₇ = −(d + d_AP + κ₃) < 0 or λ₈ = −(d + κ₃ + d_A) < 0 or λ₉ = −(d + d_P + σ₁ + δ₃) < 0 or λ₁₀ = −(d + d_P + σ₁ + δ₄) < 0 or

$$\begin{array}{c}{a}_2{\lambda }^2+a_1\lambda +a_0=0,\end{array}$$ (31)

where a₂ = 1 > 0, a₁ = (d + κ₁ + δ₁)[1 − (Y₁⁰/(Y₁⁰ + Y₄⁰))ℛ_Y4] + (d + κ₂ + δ₂) > 0 if ℛ_Y4 < 1, and a₀ = a₀ = (d + κ₁ + δ₁)(d + κ₂ + δ₂)[1 − (Y₁⁰/(Y₁⁰ + Y₂⁰))ℛ₁] > 0 if ℛ₁ < 1.

Equation (31) has/have no positive root/s whenever ℛ₁ < 1, and hence, both eigenvalues are negative. It follows that the Jacobian J(E₃⁰) of (29) at the DFE, with β₂ = β^∗, denoted by J_β^∗, has a simple zero eigenvalue (with all other eigenvalues having negative real part). Hence, the center manifold theory [27] can be used to analyze the dynamics of model (4). In particular, the Castillo-Chavez and Song theorem [28] will be used to show that model (4) undergoes backward bifurcation at ℛ_P = 1.

Eigenvectors of J_β^∗: for the case when ℛ_P = 1, the right eigenvectors of the Jacobian of (29) at β₂ = β^∗ (denoted by J_β^∗) associated with the zero eigenvalue given by u = (u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀, u₁₁, u₁₂)^T are u₁ = ((−τ₁F₈F₂₇ + F₁F₇F₂₇ − θγF₇)/dF₇F₂₇)u₃, u₂ = −(F₈/F₇)u₃, u₃ = u₃ > 0, u₁₀ = (−κ/F₂₇)u₃ , and u₄ = u₅ = u₆ = u₇ = u₈ = u₉ = u₁₁ = u₁₂ = 0.

The left eigenvectors associated with the zero eigenvalue at β₂ = β₂^∗ satisfying v.w = 1 given by v = (v₁, v₂, v₃, v₄, v₅, v₆, v₇, v₈, v₉, v₁₀, v₁₁, v₁₂) are v₁ = v₂ = v₄ = v₅ = v₆ = v₁₀ = v₁₁ = v₁₂=0, v₃ = v₃ > 0, v₇ = ((−δ₃δ₄F₁₇ + δ₃F₁₆F₂₆ − F₁₅F₂₅F₂₆)/F₂₄F₂₅F₂₆)v₃, v₈ = (δ₄F₁₇ − F₁₆F₂₆)/F₂₅F₂₆v₃, and v₉ = −(F₁₇/F₂₆)v₃.

After going through detailed computations and simplification, we have the following bifurcation coefficients a and b as

$$\begin{array}{c}a=2v_3{u}_1{u}_3\frac{{\partial }^2{f}_3\left(0,0\right)}{\partial x_1\partial x_3}+2v_3{u}_2{u}_3\frac{{\partial }^2{f}_3\left(0,0\right)}{\partial x_2\partial x_3}=2{\beta }_2^{\ast }{v}_3{u}_3\left[u_1+ϵu_2\right].\end{array}$$ (32)

⟹a = 2β₂^∗v₃u₃²[G₁ − G₂] where G₁=θκ/d(d + θ) and G₂ = ϵβ^∗τ₁V_P⁰ + β^∗(d + τ₁)Y₁⁰ + ϵβ^∗dY₂⁰/d(d + τ₁). Thus, the bifurcation coefficient a is positive whenever G₁ > G₂. Furthermore, b = v₃u₃(∂²f₂(0, 0)/∂x₃∂β₂) = v₃u₃(ϵY₂⁰ + Y₁⁰) > 0.

Hence, it follows from in Castillo-Chavez and Song [28] that model (4) exhibits a backward bifurcation at ℛ₃ = ℛ₂ = 1 whenever G₁ > G₂.

Theorem 8 . —

1. Model (4) will undergo backward bifurcation if a = G₁ > G₂ > 0

2. Model (4) will undergo forward bifurcation if a = G₁ > G₂ < 0

4. Sensitivity and Numerical Analysis

4.1. Sensitivity Analysis

Definition. The normalized forward sensitivity index of a variable ℛ₃ that depends differentiably on a parameter p is defined as SI(p) = (∂ℛ₃/∂p)∗(p/ℛ₃) [18].

Sensitivity indices allow us to examine the relative importance of different parameters in pneumonia and HIV/AIDS spread 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 of ℛ₁ and ℛ₂ since ℛ₃ = max{ℛ₁, ℛ₂}. Sensitivity analysis results and the numerical simulation are given in this section with parameters values given in Table 3 where N⁰ is the total number of the initial population of complete model (4).

Table 3.

Parameter values used for the full HIV/AIDS-pneumonia coepidemic model simulation.

Parameter	Value	Source
Λ	0.0413∗N0	Estimated
d	0.02	Estimated
δ 1	0.498	[7]
δ 2	0.08	[7]
δ 3	0.2885	[6]
δ 4	0.3105	[6]
ψ 1	1.1	Assumed
ψ 2	1.2	Assumed
ψ 3	1.4	Assumed
ν	1	Assumed
d P	0.1	[16]
θ	0.1	[18]
d A	0.333	[6]
π	0.2	[18]
τ 1	0.0025	[18]
ϵ	0.002	[18]
d AP	0.42	Assumed
κ	0.2	[18]
κ 1	0.2	[7]
κ 2	0.15	[7]
κ 3	0.13	Assumed
σ 1	0.498	[7]
σ 2	0.08	[7]
σ 3	0.230	Assumed
β 1	Variable	[6]
β 2	Variable	[6]
ρ 1, ρ2, ρ3, ω1, ω2, ω3	1.2,1,1,1,1,1	Assumed

Using the values of parameters in Table 3, the sensitivity indices are calculated in Tables 4 and 5.

Table 4.

Sensitivity indices of ℛ₃ = ℛ₁.

Sensitivity index	Values
SI(β1)	+1
SI(ρ1)	+0.6134
SI(δ1)	- 0.0639
SI(d)	-0.3150
SI(κ1)	-0.1371
SI(κ2)	-0.0264
SI(δ2)	-0.0141

Table 5.

Sensitivity indices of ℛ₃ = ℛ₂.

Sensitivity index	Values
SI(Λ)	+1
SI(β2)	+1
SI(d)	-0.4421
SI(κ)	-0.6559
SI(dP)	-0.3852
SI(π)	-0.3852
SI(ϵ)	-0.3852
SI(τ1)	-0.3852

In this paper, with parameter values in Table 3, we have got ℛ₁ = 1.386 at β₁ = 2 implies HIV/AIDS spreads in the community and also we have got the indices as shown in Table 4. Here, sensitivity analysis shows that the human recruitment rate Λ and HIV/AIDS spreading rate β₁ have the highest impact on the basic reproduction number of HIV/AIDS (ℛ₁).

Similarly, with parameter values in Table 3, we have got ℛ₂ = 9.69 at β₂ = 0.2 imply that pneumonia spreads throughout the community and also we have got the indices as shown in Table 4. Here, sensitivity analysis shows that the foremost sensitive positive parameters are the human recruitment rate Λ and the pneumonia spreading rate β₂. The foremost sensitive negative parameter is treatment rate of pneumonia (κ) which is inversely related to the effective reproduction number ℛ₂, i.e., a smaller amount of increase in this parameter value will lead to a greater amount of reduction in the effective reproduction number while a smaller amount of decrement will cause a big increment in the basic reproduction number. Epidemiologically, the most sensitive parameters to ℛ₁ and ℛ₂ which can be controlled through interventions and preventions are found to be β₁ and β₂, respectively.

4.2. Numerical Analysis

In this section, numerical simulation is performed for complete HIV/AIDS-pneumonia coepidemic model (4). With ode45, we have checked the effect of some parameters in the spreading as well as for the control of pneumonia only, HIV/AIDS only, and coepidemic of HIV/AIDS and pneumonia. The parameter values put forward in Table 3 are used for numerical simulation. In the numerical simulation part, we investigated the stability of the endemic equilibrium point of complete model (4), parameter effects on the reproduction numbers, and the impact of treatment mainly on dually infected individuals in the community.

4.2.1. Local Stability of Endemic Equilibrium Point of Complete Model (4)

Figure 2 shows that in the long run (after 50 years), the solutions of dynamical system (4) will be converging to its endemic equilibrium point, i.e., the endemic equilibrium point is locally asymptotically stable whenever

$$\begin{array}{c}{\mathcal{R}}_3=\max \left\{{\mathcal{R}}_1,{\mathcal{R}}_2\right\}=\max \left\{1.386,9.69\right\}=9.69>1.\end{array}$$ (33)

Figure 2.

Local stability of endemic equilibrium point of the coepidemic model (4) whenever ℛ₁ = 1.386 at β₁ = 2 and ℛ₂ = 9.69 at β₂ = 0.2.

4.2.2. Effect of Parameters on the Threshold Parameter ℛ₂

In this subsection, as we see in Figure 3, we have investigated the effect of pneumonia vaccination portion π on the pneumonia effective reproduction number  ℛ₂. The figure reflects that when the value of π increases, the pneumonia effective reproduction number is going down, and whenever the value of π > 0.64 imply ℛ₂ < 1. Therefore, public policymakers must concentrate on maximizing the values of pneumonia vaccination portion π to prevent and control pneumonia spreading.

Figure 3.

Effect of pneumonia vaccination on ℛ₂.

In this subsection, as we see in Figure 4, we have investigated the effect of pneumonia spreading rate β₂ on the pneumonia effective reproduction number ℛ₂ by keeping the other rates as in Table 3. Figure 4 reflects that when the value of β₂ increases, the pneumonia effective reproduction number ℛ₂ increases, and whenever the value of β₂ < 0.022 implies ℛ₂ < 1. Therefore, public policymakers must concentrate on minimizing the values of pneumonia spreading rate β₂ to minimize pneumonia effective reproduction number ℛ₂.

Figure 4.

Effect of pneumonia transmission on ℛ₂.

4.2.3. Effect of Pneumonia Treatment Rate on Infectious Population

In this subsection, as we see in Figure 5, we have investigated the effect of κ in decreasing the number of pneumonia-only infectious populations. The figure reflects that when the values of κ increase, the number of pneumonia-only infectious population is going down. Therefore, public policymakers must concentrate on maximizing the values of treatment rate κ to pneumonia disease.

Figure 5.

Effect of treatment on pneumonia-infected population.

4.2.4. Effect of Treatment Rates on HIV/AIDS Infectious Population

In this subsection, as we see in Figures 6–8, respectively, we have investigated the effects of κ₁, κ₂, and κ₃ in decreasing the number of acute HIV only, chronic HIV only, and AIDS-infected population, respectively. The figures reflect that when the values of κ₁, κ₂, and κ₃ increase, the number of acute HIV only, chronic HIV only, and AIDS-infected population is going down, respectively. Therefore, public policymakers must concentrate on maximizing the values of treatment rate of individuals to HIV/AIDS infection.

Figure 6.

Effect of treatment on acute HIV-infected population at β₁ = 0.5.

Figure 7.

Effect of treatment on chronic HIV-infected population at β₁ = 0.5.

Figure 8.

Effect of treatment on AIDS patients at β₁ = 0.5.

4.2.5. Effect of HIV/AIDS Transmission Rate on Coinfectious Population

In this section, we see in Figure 9 the effect of the spreading rate of HIV/AIDS β₁ on the acute HIV-pneumonia coepidemic population Y₇. The figure reflects that as the value of the transmission rate (β₁) of HIV/AIDS is increased, the coepidemic population increases, which means the expansion of coepidemic of HIV/AIDS-pneumonia will increase. To control coepidemic of HIV/AIDS-pneumonia, decreasing the spreading rate of HIV/AIDS is important. Therefore, stakeholders must concentrate on decreasing the spreading rate of HIV/AIDS by using the treatment and appropriate method of prevention mechanism to bring down the expansion of coepidemic in the community.

Figure 9.

Effect of β₁ on acute HIV-pneumonia coepidemic population.

4.2.6. Effect of Treatment Rates on HIV/AIDS-Pneumonia Coepidemic Population

In this subsection, as we see in Figures 10–12, we have investigated the effects of treatment rates σ₁, σ₂, and σ₃ in decreasing the number of acute HIV and pneumonia, chronic HIV and pneumonia, and AIDS and pneumonia coinfectious population, respectively. The figures reflect that when the values of σ₁, σ₂, and σ₃ increase, the number of acute HIV-pneumonia, chronic HIV-pneumonia, and AIDS-pneumonia coepidemic population is going down, respectively. Therefore, public policymakers must concentrate on maximizing the values of treatment rates of HIV/AIDS-pneumonia coepidemic population.

Figure 10.

Effect of treatment on acute HIV and pneumonia coepidemic.

Figure 11.

Effect of treatment on chronic HIV and pneumonia.

Figure 12.

Effect of treatment on AIDS and pneumonia coepidemic.

5. Discussion

In Section 1, we reviewed and introduced the epidemiology of HIV/AIDS, pneumonia, and HIV/AIDS-pneumonia coepidemic. In Section 2, we construct the compartmental HIV/AIDS-pneumonia coepidemic dynamical system using an ordinary differential equation and we partitioned it into twelve distinct compartments. In Section 3, we analyzed the model qualitatively. To study the qualitative behavior of complete model (4), first, we split the complete model into two, which are HIV/AIDS-only and pneumonia-only models. The qualitative behaviors, i.e., the positivity of future solutions of the models, boundedness of the dynamical system, disease-free equilibrium points, basic reproduction numbers, endemic equilibriums, stability analysis of disease-free equilibrium points, stability analysis of endemic equilibrium points, bifurcations analysis of pneumonia-only model and the complete HIV/AIDS-pneumonia coepidemic model, and sensitivity analysis of reproduction numbers of HIV/AIDS-only and pneumonia-only model, are analyzed in their respective order, and numerically, we experimented on the stability of endemic equilibrium point of the HIV/AIDS-pneumonia coepidemic model, effect of basic parameters in the expansion or control of pneumonia only, HIV/AIDS only, and HIV/AIDS-pneumonia coepidemic infections and parameter effects on the infected population. From the result, we conclude that increasing both the pneumonia treatment rate and pneumonia vaccination portion rate has a great contribution to bringing down pneumonia infection as well as the coepidemic in the community. Similarly, increasing the HIV/AIDS treatment rates also has a contribution to minimizing the expansion of HIV/AIDS infection. The coepidemic treatment rates also influence minimizing coepidemic population if its value is increased. The other result obtained in this section is that decreasing the transmission rates has a great influence of controlling coepidemic in the population.

6. Conclusion

A realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections are available at each stage of the infection in a population constructed and analyzed. We have shown the positivity and boundedness of the complete HIV/AIDS-pneumonia coepidemic model. Using center manifold theory, we have shown that the pneumonia-only infection and the complete HIV/AIDS-pneumonia coepidemic models undergo the phenomenon of backward bifurcation whenever their corresponding effective reproduction numbers are less than one. The complete model has a disease-free equilibrium that is locally asymptotically stable whenever the maximum of the reproduction numbers of the two submodels described above is less than one. Numerical simulation shows that the complete HIV/AIDS-pneumonia coepidemic model endemic equilibrium point is locally asymptotically stable when its effective reproduction number is greater than one. These results have important public health implications, as they govern the elimination and/or persistence of the two diseases in a community. By analyzing the various associated reproduction numbers, we have shown that the impact of some parameters changes on the associated reproduction numbers to give future recommendations for the stakeholders in the community. From the numerical result, we have got the complete model reproduction number is ℛ₃ = max{ℛ₁, ℛ₂} = max{1.386, 9.69 } = 9.69  at β₁ = 2 and β₂ = 0.2. From our numerical result, we recommend that public policymakers must concentrate on maximizing the values of pneumonia vaccination portion and treatment rate of individuals to pneumonia disease. Finally, some of the main epidemiological findings of this study include pneumonia vaccination and treatment against disease has the effect of decreasing the pneumonia and coepidemic disease expansion and prevalence and reducing the progression rate of HIV infection to the AIDS stage and the HIV/AIDS prevalence.

6.1. Limitation of the Study

Due to conflict in our country Ethiopia, it is difficult to incorporate experimental data in the study.

Data Availability

Data used to support the findings of this study are included in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

All authors have read and approved the final manuscript.