Global Dynamics of a Within-Host COVID-19/AIDS Coinfection Model with Distributed Delays

Azoz, S. A.; Elaiw, A. M.; Ramadan, E.; Al Agha, A. D.; Raezah, Aeshah A.

doi:https://doi.org/10.1155/2022/9129187

Journal of Mathematics

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Abstract Introduction Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9129187 | https://doi.org/10.1155/2022/9129187

Global Dynamics of a Within-Host COVID-19/AIDS Coinfection Model with Distributed Delays

S. A. Azoz,¹A. M. Elaiw,^2,3E. Ramadan,¹A. D. Al Agha,⁴and Aeshah A. Raezah⁵

Academic Editor: Chang Phang

Received11 Jul 2022

Revised07 Nov 2022

Accepted22 Nov 2022

Published22 Dec 2022

Abstract

Acquired immunodeficiency syndrome (AIDS) is a spectrum of conditions caused by infection with the human immunodeficiency virus (HIV). Among people with AIDS, cases of COVID-19 have been reported in many countries. COVID-19 (coronavirus disease 2019) is caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). In this manuscript, we are going to present a within-host COVID-19/AIDS coinfection model to study the dynamics and influence of the coinfection between COVID-19 and AIDS. The model is a six-dimensional delay differential equation that describes the interaction between uninfected epithelial cells, infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4⁺ T cells, infected CD4⁺ T cells, and free HIV-1 particles. We demonstrated that the proposed model is biologically acceptable by proving the positivity and boundedness of the model solutions. The global stability analysis of the model is carried out in terms of the basic reproduction number. Numerical simulations are carried out to investigate that if COVID-19/AIDS coinfected individuals have a poor immune response or a low number of CD4⁺ T cells, then the viral load of SARS-CoV-2 and the number of infected epithelial cells will rise. On the contrary, the existence of time delays can rise the number of uninfected CD4⁺ T cells and uninfected epithelial cells, thus reducing the viral load within the host.

1. Introduction

In December 2019, the first case of the emergence of the severe acute respiratory syndrome coronavirus 2 (COVID-19) occurred in Wuhan, China. In March 2020, the World Health Organization (WHO) declared COVID-19 a worldwide epidemic. Globally, as of 27 April 2022, over 500 million people were infected with COVID-19, including 6 million deaths [1]. Old age and its accompanying symptoms such as diabetes, heart disease, and high blood pressure are considered risk factors for developing severe COVID-19 infection and are associated with a high death rate [2, 3]. Some other risk factors are associated when infection with COVID-19 occurs in people with chronic diseases such as acquired immunodeficiency syndrome (AIDS) [4]. In 2020, there were 37.7 million persons living with HIV-1 (PLWH) worldwide; HIV-1 causes acquired immunodeficiency syndrome (AIDS) with 680,000 of them dying from HIV-1-related diseases, and only 73% of them were on antiretroviral medication (ART) [5]. Because their immune systems are impaired, PLWH who do not receive ART or whose condition is poorly managed could be more susceptible to developing COVID-19. If infected with COVID-19, such people are at a greater risk of developing acute symptoms and dying. The coinfection cases are challenging due to the scarcity of data on the outcomes and consequences of SARS-CoV-2 infection in HIV-1 positive individuals [3, 6, 7].

HIV-1 and SARS-CoV-2 are both RNA viruses. SARS-CoV-2 attacks upper respiratory epithelial cells, and the virus generated by infected cells goes down to the lower airway, infecting bronchial and alveolar epithelial cells [4, 8]. On the other hand, HIV-1 targets CD4⁺ lymphocytes, which are the immune system’s most plentiful white blood cells (referred to as CD4⁺ T cells). A great effort is being made in many areas of the world to create measures to battle these viruses and study their biological and immunological features and clinical outcomes. Some of these studies indicate that COVID-19 pandemic has caused disruptions in HIV-1 care facilities in many countries [9, 10]. However, it is unclear whether people infected with HIV-1 having an increased incidence of COVID-19 and significant clinical signs, despite a controversial suggestion that antiretroviral therapy or HIV-1-related immunosuppression could protect HIV-1 infected people from severe COVID-19. A number of HIV-1 and SARS-CoV-2 coinfection cases have been documented throughout the world [11, 12]. Most studies of COVID-19/AIDS coinfection reported that there is a lack of clarity on what constitutes the primary illness and what constitutes comorbidity in the context of coinfection. Few studies inferred that SARS-CoV-2 infection does not increase the course of HIV-1 infection in PLWH nor does HIV-1 infection have an impact on COVID-19 infection course in PLWH [13–15]. However, Wang et al. [16] published a case report of an HIV-1/COVID-19 patient with such a lower CD4⁺ T cell number, and as a result, the patient had a prolonged COVID-19 course and decreased antibody levels. Moreover, COVID-19/AIDS coinfection has been observed to cause pneumonia problems more frequently than COVID-19 alone [17]. This study aims to give a comprehensive picture of SARS-CoV-2 infection in persons having HIV-1/AIDS.

Mathematical models that consist of a system of differential equations have proven their effectiveness in studying the interactions between viruses and their hosts and the common interactions between diseases (see e.g., [12, 18–24]). HIV within-host models have been widely investigated and great results have been reached [18, 19, 25–28]. On the other side, SARS-CoV-2 within-host modeling has received less attention ([24, 29–32]). Some coinfection models between SARS-CoV-2 and other viruses have been developed. For example, Pinky and Dobrovolny [33] used a within-host model to investigate SARS-CoV-2 coinfections with several viruses types such as influenza A virus (IAV), parainfluenza virus (PIV), and human rhinovirus (HRV). In fact, the models of coinfection are essential to grasp the coinfection dynamics between SARS-CoV-2 and HIV, to assist the experimental studies and save time, and to develop effective treatments for coinfected people. Ahmed et al. [34] created a fractional epidemiological model to analyze the pandemic scenario in numerous HIV and COVID-19 affected countries, including South Africa and Brazil. Then, to the best of our knowledge, the first ordinary differential within-host SARS-CoV-2/HIV coinfection system is presented by Al Agha et al. [20]. The formulation of their model is based on Nowak and Bangham’s model that was used widely to model HIV monoinfection and SARS-CoV-2 monoinfection. Al Agha et al. used the same principals to model SARS-CoV-2/HIV coinfection and connect the two infections together. The model is formulated as follows:where , , and represent the healthy epithelial cells, infected epithelial cells, and free SARS-CoV-2 particles, respectively, whilst , , and depict healthy CD4⁺ T cells, infected CD4⁺ T cells, and free HIV particles concentrations at time , respectively. Epithelial cells are recruited at rate and turned into infected cells at pace . Infected produce SARS-CoV-2 particles at rate . CD4⁺ T cells are recruited at rate , eliminate infected epithelial cells at a proportion , and proliferate at rate . HIV particles infect CD4⁺ T cells at rate . The infected cells produces HIV at rate . The components , , , , , and die at rates , , , , , and , respectively. Then, Elaiw et al. [21] adopted the same previous model with the addition of the effect of latent cells, and then, they made a comprehensive study of the proposed model. Ringa et al. [22] presented a new mathematical model for COVID-19 and HIV/AIDS. The dynamics of the full model is driven by that of its submodels. Also, they studied the impact of intervention measures by incorporating it into the model using time-dependent controls.

Most of the previous publications are the assumptions that cells produce viruses immediately after they are infected. It is commonly observed that in many biological processes, a time delay is inevitable. For HIV-1 infection, it roughly takes about one day for a newly infected cell to become productive and then to be able to produce new virus particles. Therefore, mathematicians have frequently used different types of delays to make biological models more realistic. In [26, 28, 35], HIV models with time delay were introduced, whilst modeling and analysis of COVID-19 based on a time delay dynamic model are presented in [12, 23]. Although there are some publications that combine the coinfection between viruses in the presence of time delay, there are still no models of coinfection between SARS-CoV-2 and HIV with time delay. Due to the decisive role of time delays in dynamic systems, the objective of this work is to expand model (1) to accommodate distributed delays. This can help comprehend the coinfection dynamics between SARS-CoV-2 and HIV-1 from a different perspective. A continuous distribution function is used to represent the delay in case of distributed time delay. This makes distributed delays more realistic than discrete time delays which presume that each individual in the population has the same delay period. Thus, we have investigated a model with six delay differential equations, and we have established the solutions nonnegativity and boundedness, listed the prospective equilibrium points and the conditions of existence, discussed the global stability of the equilibria, and examined time delay impact on the model’s dynamics. The document includes the following sections: the model is presented in Section 2. Section 3 confirms the basic properties of the model. Section 4 exhibits the global properties of the model. Section 5 lists the numerical simulations. Finally, Section 6 debates the results and some potential next directions.

2. COVID-19/AIDS Coinfection Model with Distributed Delay

In this section, we extend model (1) by considering a variety of distributed time delays as follows:

Thus, we have a system of six delay differential equations where , , , , , and stand for the concentrations of uninfected epithelial cells, infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4⁺ T cells, infected CD4⁺ T cells, and HIV-1 particles at time , respectively. A scheme describing the coinfection between SARS-CoV-2 and HIV in host without time delay is shown in Figure 1. The factor designates the likelihood that uninfected epithelial cells were in touch with SARS-CoV-2 particles at time survived time units, and infection occurs at time . The term simulates the probability of new immature SARS-CoV-2 particles at time survived time units and mature at time . Moreover, the factor symbolizes the probability that uninfected CD4⁺ T cells contacted by HIV-1 particles at time survived time units and become infected at time . The term represents the probability that new immature HIV-1 particles at time persisted time units and mature at time , where , and are the positive constants. The delay parameter is a random variable picked from probability distribution functions during time interval . The functions () satisfy andwhere . Let us denote the following model:where . This implies that . The initial conditions of model (2) are specified as follows:where , and such that is a positive constant. Here, is the Banach space of fading memory type [36]. Therefore, using the standard theory of differential equations with infinitely distributed delays [37, 38], model (2) with initial constraints (3) has a single solution.

3. Basic Characteristics

This section proves that model (2) solutions are non-negative and ultimately bounded. Additionally, it computes whole potential equilibria and the threshold numbers.

3.1. Non-Negativity and Boundedness

Proposition 1. All of model (2) solutions with beginning conditions (3) are non-negative and eventually bounded.

Proof. Starting with model (2) first equation, we obtain , which yields that for all . From fourth equation of the model, we get ; then, for all . Furthermore, the rest of the model equations give us the following model:For all , as a result of the recursive argument, we obtain for all . Hence, system (2) solutions with initial conditions (3) realize for all non-negative values of .
Now, we establish the boundedness of the model’s solutions. Based on model (2) first equation, we gain , where . We define the following model:Then, we get the following model:where . This implies that , where . Since and are non-negative, then and , where . Using model (2) third equation, we get the following model:This implies that , where .
We define the following model:Following that, we get the following model:where . Thus, we have , where. Finally, the last equation of model (2) gives the following model:Thus, , where . Accordingly, the following region is positively invariant with regard to model (2).

3.2. Equilibrium Points

This subsection displays all of model (2) possible equilibrium states and deduces four threshold parameters that determine the equilibria existence.

We solve the following set of algebraic equations to calculate the following model equilibrium points:

From last equation of model (4), we have . Then, we substitute in the fifth equation, and we get the following model:

So, we have two possibilities:

Then, doing the same for second and third equations, we obtain another two possibilities as follows:

Equations (16) and (17) provide us with four possibilities. Accordingly, model (4) has four equilibrium points:(i)Uninfected equilibrium , where and (ii)The HIV-1 monoinfection equilibrium , where It follows that and only when . Thus, we have the following model: Here, . Here, is the basic reproduction number for HIV-1 infection. It sets start of HIV-1 infection in host body. We note that and if . Therefore, exists when .(iii)SARS-CoV-2 monoinfection equilibrium , where satisfies the following equation: To prove that equation (21) has a positive root, we introduce a function as follows: Then, we have the following equation: Here, . This implies that when . In addition, we find that It follows that there exists such that . From equation (20), we get , and . As a result, we deduce that exists when . Here, is the basic reproduction number for SARS-CoV-2 infection. It defines start of SARS-CoV-2 infection in host body.(iv)COVID-19/AIDS coinfection equilibrium , where

It follows that and only when . On the other hand, and only when .

Thus, we can rewrite the components of as follows:where

Therefore, exists when and . Here, and are the threshold parameters that mark the COVID-19/AIDS coinfection incidence.

The threshold parameters are defined as follows:

For simplicity, the contractions listed will be used in the parts that followand

4. Global Properties

We demonstrate the global asymptotic stability of all equilibria in this section by building Lyapunov functions using the approach described in [39]. We define , where can be any variable for the model.

Theorem 1. Globally asymptotically stable (G.A.S) of equilibrium is satisfied when and .

Proof. Take a Lyapunov function as follows:Clearly, for all and . Calculating along the solutions of system (2) gives the following equation:Adding up the terms in equation (32), we obtain the following equation:Using and , we obtain the following equation:Since and , we get for all . Also, when , and . Set and the largest invariant subset (L.I.S) of by . Then, the model solutions converge to . The set contains elements with , and , and hence, . The second and last equations of the model (2) give the following equation:Thus, we get for all . Then, , and using Lyapunov–LaSalle asymptotic stability theorem [40–42], is G.A.S.

In the following theorems, we need to use the equalities:

Furthermore,

Theorem 2. If and , the equilibrium is G.A.S.

Proof. Consider a Lyapunov function as follows:Differentiating , we obtain the following equation:Summing the terms of equation (39), we obtain the following equation:By utilizing the equilibrium conditions for , we get the following equation:Then, we obtain the following equation:Using the equalities given by equation (36) in case of , we get the following equation:Therefore, equation (43) becomesSince , we find that for all . Also, when , , , , and . Model (2) solutions converge to the L.I.S of . The set contains elements with ; then, . Second equation of system (2) implieswhich gives for all . Therefore, and is G.A.S according to Lyapunov–LaSalle asymptotic stability theorem [40–42].

Theorem 3. If and , then the equilibrium is G.A.S.

Proof. We introduce a Lyapunov function as follows:Differentiating , we obtain the following equation:By collecting the terms of equation (47), we have the following equation:By using the equilibrium conditions for ,we obtain the following equation:Using the equalities given by equation (37) in case of , we get the following equation:Therefore, equation (51) becomesIf , then does not exist since and . This implies thatTherefore, we get for all . Hence, we have , and therefore, for all . In addition, when , , , , and . Solutions of the model (2) that converge to is the L.I.S of . The set has elements with , and thus, . Using system (2) last equation, we get the following equation:Yield for all values of . Therefore, and is G.A.S according to Lyapunov–LaSalle asymptotic stability theorem [40–42].

Theorem 4. If and , then the equilibrium is G.A.S.

Proof. We consider a Lyapunov function as follows:By differentiating , we obtain the following equation:Collecting terms of equation (56) gives the following equation:By using the equilibrium conditions for ,we get the following equation:Using the equalities given by equations (36) and (37) in case of , we get the following equation:Therefore, equation (60) becomes the following equation:Since , then for the positive values of . Moreover, when , , , , , and . The model trajectories that converge to be the L.I.S of . Hence, and is G.A.S according to Lyapunov–LaSalle stability theorem.

All equilibria of model (2) with the existence conditions and global stability constraints are summarized in Table 1.

5. Numerical Simulations

We execute numerical simulations in this part to enhance the outcomes of Theorems 1–4. Moreover, the impact of time delays on system dynamical behavior will be tested. To transform a model with distributed time delay (2) to a discrete one, we choose a Dirac delta function as a specific formula of kernel as follows:

Then, we get the following equation:

Thus, model (2) is reduced as follows:

For model (64), the threshold parameters are given by the following equation:

To solve system (18) numerically, we change some parameters values whilst assigning fixed estimate to the rest parameters (Table 2). We modify the parameters , and to test the conclusions of Theorems 1–4. Furthermore, to test the impact of the time delays upon COVID-19/AIDS dynamics, delays parameters , and have been changed.

5.1. Stability of Equilibrium Points

During this part, we choose delay parameters as follows: , , , and . Additionally, we select three distinct starting conditions of the model (18): Initial-1: , , , , , and , Initial-2: , , , , , and , Initial-3: , , , , , and .

Here, and it is optional to pick these values. Moreover, the initial conditions are split into three groups to provide global stability for any starting conditions. To dissolve system (18), we utilize MATLAB solver dde23. Based on equilibrium points , , , and global stability explained in Theorems 1–4, the simulations are divided into four cases. In these instances, we change values of , , , and of system (18). Other parameters values are set and recorded in Table 2. The four scenarios are detailed as follows:(i)Case 1 (stability of ): we take , , , and . The thresholds in this case are given by and . In harmony with Theorem 1, the equilibrium is G.A.S (Figure 2). This is the best case scenario when the person is free of SARS-CoV-2 and HIV-1 infection.(ii)Case 2 (stability of ): we get , , , and . This provides us with and . According to Theorem 2, the equilibrium is G.A.S (Figure 3). This simulates the situation in which a person has HIV-1 infection with depressed CD4⁺ T cell levels, but SARS-CoV-2 infection is not present.(iii)Case 3 (stability of ): we select , , , and . This gives and . In this situation, the system solutions converge globally to equilibrium . This result accords with Theorem 3 (Figure 4). This scenario simulates the case of a person infected with SARS-CoV-2 but not HIV-1 infection.(iv)Case 4 (stability of ): we consider , , , and . This implies that , , and . In agreement with Theorem 4, the equilibrium is G.A.S (Figure 5). In this case, COVID-19/AIDS coinfection occurs, where an HIV-1 patient gets infected with SARS-CoV-2. CD4⁺ T cells, which are the main target of HIV-1, are recruited to eliminate SARS-CoV-2 infection from the body. However, if the patient has low CD4⁺ T cell counts, the clearance of SARS-CoV-2 may not be achieved. This can cause severe infection and death.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

5.2. Impact of Time Delays on COVID-19/AIDS Dynamics

Here, we adjust parameters of delay and set the parameters values and . Since and offered by equation (65) rely on , varying parameters will convert stability of the equilibria. We consider the following cases: (D.P.S1) (D.P.S2) and (D.P.S3) and

With the above values, we solve model (64) with given initial conditions:

Initial-3: () = (15, 0.002, 0.003, 300, 15, 20).

The inclusion of time delays can increase the number of uninfected epithelial and CD4+ T cells while diminish the number of other compartments, as shown in Figure 6. Table 3 shows the values and for selected values of , . Clearly, and decrease when are increased, and accordingly, the stability of can be changed. Let us compute the critical value of the time delay that changes the stability of . Without loss of generality, we let the parameters and , and write and as functions of and , respectively, as follows:

(a)

(b)

(c)

(d)

(e)

(f)

To compel basic reproduction numbers and to verify and , respectively, we choose the following equations:

And

Therefore, if and , then is G.A.S. Computing and gives and , respectively. It follows(i)If and , then , , and is G.A.S.(ii)If or , then , , and will lose its stability.

6. Discussion

Coinfection between COVID-19/AIDS has become a serious problem during COVID-19 pandemic. Mathematical modeling represents a main tool in helping experimental studies understand new diseases. We studied a within-host COVID-19/AIDS coinfection model with distributed delays in this paper. The model explores the contacts between healthy epithelial cells, infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4⁺ T cells, infected CD4⁺ T cells, and free HIV-1 particles. There are four equilibrium points for the model with the following listed properties:(a)Uninfected equilibrium : its existence is permanent and it is G.A.S if and . This represents the situation of a person without SARS-CoV-2 or HIV-1 infections.(b)The HIV-1 monoinfection equilibrium exists if , and it is G.A.S if . At this point, the person has only HIV-1 infection, but he is not infected by SARS-CoV-2.(c)SARS-CoV-2 monoinfection equilibrium is appeared when , and if , then it is G.A.S. It is the instance of a person who is suffering from SARS-CoV-2 infection only.(d)COVID-19/AIDS coinfection equilibrium exists and G.A.S if and . In this case, the patient suffers from COVID-19/AIDS coinfection.

The numerical and theoretical results were found to be in agreement. The time delays increase the concentrations of uninfected epithelial and CD4⁺ T cells, while they decrease concentrations of free SARS-CoV-2 and HIV-1 particles. Thus, parameters of delay can be examined and used in developing effective treatments for COVID-19/AIDS coinfected patients. Moreover, the model with distributed delays confirmed the effect observed in [20] that low numbers of CD4⁺ T cells can increase the risk of severe SARS-CoV-2 infection in coinfected patient. Thus, our model can be used to estimate the parameters required to get rid of SARS-CoV-2 in HIV-1 patients. Also, a bifurcation analysis can be executed in order to get a deeper understanding of the stability changes. Furthermore, the work can be developed by finding a better approximation of all parameters in model (2) through fitting with real data. We will keep these points in mind for future projects. [46].

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under grant number RGP.2/154/43.

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Copyright © 2022 S. A. Azoz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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