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 Using center manifold theory, we have verified that both the pneumonia only submodel and the HIV/AIDS-pneumonia coepidemic model undergo backward bifurcations whenever and , 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 and the HIV/AIDS transmission rate play an important role to change the qualitative dynamics of HIV/AIDS and pneumonia coinfection. The pneumonia infection transmission rate gives rises to the possibility of backward bifurcation for HIV/AIDS and pneumonia coinfection if , 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 at and 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 [16]. 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, 69].

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, 1014]. 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, 1517]. 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 into twelve distinct classes as susceptible class to both HIV and pneumonia infections , pneumonia vaccine class , pneumonia-infected class , acute HIV-infected class , chronic HIV-infected class , AIDS patient class , acute HIV-pneumonia coepidemic class , chronic HIV-pneumonia coepidemic class , AIDS-pneumonia coepidemic class , pneumonia treatment class , HIV/AIDS treatment class entered from the three infection stages , , and , and HIV/AIDS-pneumonia coepidemic treatment class entered from , , and cases such that

The susceptible class acquires HIV at the standard incidence rate given by where is the modification rate that increases infectivity and is the HIV/AIDS contagion rate.

The susceptible class acquires pneumonia at the mass action incidence rate where is the modification rate that increases infectivity and is the pneumonia contagion rate.

To construct the complete coepidemic dynamical system, we have assumed the following: (i)A fraction of the population has been vaccinated before the disease outbreak at the portion of and fraction of population entered to the vulnerable class(ii)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 and from pneumonia-treated class in which those individuals who lose their temporary immunity by the rate(iii)Assume vaccination is not 100% effective, so vaccinated individuals also have a chance of being infected with proportion of the serotype not covered by the vaccine where (iv)Individuals in a given compartment are homogeneous(v)Assume no HIV transmission from and classes due to their reduced daily activities(vi)Individuals in each class are subject to natural mortality rate (vii)The human population is variable(viii)We assumed there is no dual-infection transmission simultaneously(ix)Assume HIV has no vertical transmission and pneumonia is not naturally recovered(x)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.

2.3. The HIV/AIDS-Pneumonia Coepidemic Dynamical System

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

With initial conditions,

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

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

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 in the bounded region given in (7).

Theorem 1. At the initial conditions (5), the solutions , , , , and of system (4) are nonnegative for all time .

Proof. Assume , , , , , , , , , , and ; then, for all , we have to prove that , , ,, , , , , , , , and .
Define: .
From the continuity of and , we deduce that . If , then positivity holds. But, if , or or or or or or or or or or or
Here, from the first equation of the model (4), we have
Using the method of integrating factor, we obtained where and from the definition of we see that , , and also the exponential function is always positive; then, the solution ; hence, . From the second equation of the model (4), we have and we have got , where and also, the exponential function always is positive; then, the solution; hence, . Similarly, all the remaining state variables ; hence, and ; hence, and ; hence, and ; hence, and ; hence, and ; hence, and ; hence, and ; hence, and ; hence, and ; hence . Thus, based on the definition of, it is not finite which means , and hence, all the solutions of system (2) are nonnegative.☐

Theorem 2. The region given by (7) is bounded in .

Proof. Using equation (6), since all the state variables are nonnegative by Theorem 1, in the absence of infections, we have got . By applying standard comparison theorem, we have got and integrating both sides gives where is some constant, and after some steps of calculations, we have got 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 which is given by

where the total population is and the HIV/AIDS single infection force of infection is given by with initial conditions , , , , and.

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 ==, which is given by

With initial conditions, , , , , total population , and pneumonia force of infection .

In the region, it is easy to show that the set 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 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 we have got

and such that .

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 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 where is the rate of appearance of new infection in compartment , is the transfer of infections from one compartment to another, and is the disease-free equilibrium point. Then, after a long calculation, we have got

Then, using Mathematica, we have got

The characteristic equation of the matrix is

Then, the spectral radius (effective reproduction number ) of of the pneumonia submodel (9) is . Here, 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) of the pneumonia submodel (9) is locally asymptotically stable if , 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 and Routh Hurwitz stability criteria. The Jacobian matrix of a dynamical system (9) at the disease-free equilibrium point is given by Then, the characteristic equation of the above Jacobian matrix is given by where .
After some steps, we have got or or if or . Therefore, since all the eigenvalues of the characteristics polynomial of the system (9) are negative if , 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 . Moreover, let 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 at an endemic equilibrium point. Setting the right-hand sides of the equations of model (9) to zero and we have got , and substitute and in to , we obtain and substitute and in, we obtain

Finally, substitute in to pneumonia submodel (9) force of infection as and letting, and , we have got ++ where , , and if .

Here, the nonzero equilibrium(s) of the model (9) satisfies so that the quadratic equation can be analyzed for the possibility of multiple equilibriums. It is worth noting that the coefficientis always positive and is positive (negative) if is less than (greater than) unity, respectively. Hence, we have established the following result.

Theorem 4. The pneumonia submodel (9) has the following: (i)Exactly one unique endemic equilibrium if (i.e.,>1)(ii)Exactly one unique endemic equilibrium if<0, and or (iii)Exactly two endemic equilibriums if (i.e., ), , and(iv)No endemic equilibrium otherwiseHere, 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; examples of the existence of backward bifurcation phenomenon in mathematical epidemiological models, and the causes, can be seen in [2, 9, 22, 2426]. The epidemiological consequence is that the classical epidemiological requirement of having the reproduction number 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 whenever the inequality 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, and such that. Furthermore, by using vector notation, pneumonia submodel (9) can be written in the form with

, as follows: with , then the method entails evaluating the Jacobian of system (17) at the DFE point, denoted by , and this gives us

Consider, next, the case when. Suppose, further, that is chosen as a bifurcation parameter.

Solving for from as and we have got and

After some steps of the calculation, we have got the eigenvalues of as or or or .

It follows that the Jacobian of (17) at the DFE, with , denoted by , 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

Eigenvectors of: for the case, it can be shown that the Jacobian of (29) at (denoted by ) has a right eigenvectors associated with the zero eigenvalue given by with values

Similarly, the left eigenvector associated with the zero eigenvalues at given by are .

After long calculations, the bifurcation coefficients and are obtained as where = and .

Thus, the bifurcation coefficient is positive if. Furthermore, .

Hence, from in Castillo-Chavez and Song [28], model (9) exhibits a backward bifurcation at whenever.

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 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 and hence ==

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

The basic reproduction number, denoted by , 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 , 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, , 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 is insufficient [23]. Due to its importance, researchers have sought to find ways of determining. Two important concepts in modeling outbreaks of infectious diseases are the basic reproduction number, universally denoted by 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 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 whereis the rate of appearance of new infection in compartment, is the transfer of infections from one compartment to another, and is the disease-free equilibrium point

After long detailed calculations, the transition matrix is given by and the transmission matrix is given by


Then, by using Mathematica, we have got

The characteristic equation of the matrix is given by

where ; then, the eigenvalues of are or or .

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