Stability analysis of within-host SARS-CoV-2/HIV coinfection model

Abstract

The world has been suffering from the coronavirus disease 2019 (COVID-19) since late 2019. COVID-19 is caused by a virus called the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The human immunodeficiency virus (HIV) coinfection with SARS-CoV-2 has been reported in many patients around the world. This has raised the alarm for the importance of understanding the dynamics of coinfection and its impact on the lives of patients. As in other pandemics, mathematical modeling is one of the important tools that can help medical and experimental studies of COVID-19. In this paper, we develop a within-host SARS-CoV-2/HIV coinfection model. The model consists of six ordinary differential equations. It depicts the interactions between uninfected epithelial cells, infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4⁺ T cells, infected CD4⁺ T cells, and free HIV particles. We confirm that the solutions of the developed model are biologically acceptable by proving their nonnegativity and boundedness. We compute all possible steady states and derive their positivity conditions. We choose suitable Lyapunov functions to prove the global asymptotic stability of all steady states. We run some numerical simulations to enhance the global stability results. Based on our model, weak CD4⁺ T cell immune response or low CD4⁺ T cell counts in SARS-CoV-2/HIV coinfected patient increase the concentrations of infected epithelial cells and SARS-CoV-2 viral load. This causes the coinfected patient to suffer from severe SARS-CoV-2 infection. This result agrees with many studies which showed that HIV patients are at greater risk of suffering from severe COVID-19 when infected. More studies are needed to understand the nature of SARS-CoV-2/HIV coinfection and the role of different immune responses during infection.

Keywords: COVID‐19; HIV; Lyapunov; SARS‐CoV‐2; coinfection; global stability.

FIGURE 1

The numerical simulations of model (1) for $\eta =0.9,\mu =1,\delta =5.36$, and $\theta =0.0001$ with three different sets of initial conditions. The uninfected steady state ${\mathrm{\Delta }}_0=(22.\mathrm{41,0},\mathrm{0,1000},\mathrm{0,0})$ is G.A.S [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2

The numerical simulations of model (1) for $\eta =0.55,\mu =1,\delta =5.36$, and $\theta =0.0016$ with three different sets of initial conditions. The single HIV‐infection steady state ${\mathrm{\Delta }}_H=(22.\mathrm{41,0},\mathrm{0,125},17.\mathrm{5,43}.75)$ is G.A.S [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3

The numerical simulations of model (1) for $\eta =2.9,\mu =0.02,\delta =0.03$, and $\theta =0.0001$ with three different sets of initial conditions. The single SARS‐CoV‐2‐infection steady state ${\mathrm{\Delta }}_V=(0.\mathrm{876,0}.\mathrm{0011,0}.\mathrm{0085,1010}.\mathrm{71,0},0)$ is G.A.S [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4

The numerical simulations of model (1) for $\eta =2.9,\mu =0.02,\delta =0.1$, and $\theta =0.0016$ with three different sets of initial conditions. The SARS‐CoV‐2/HIV coinfection steady state ${\mathrm{\Delta }}_{VH}=(0.\mathrm{375,0}.\mathrm{0084,0}.\mathrm{02,125},17.\mathrm{71,44}.28)$ is G.A.S [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5

The effect of decreasing $\mu$ on the concentrations of infected epithelial cells $Y(t)$ and SARS‐CoV‐2 particles $V(t)$. The parameters considered here are $\eta =2.9,\delta =0.1$, and $\theta =0.0016$ with initial conditions $\left(X(0),Y(0),V(0),S(0),W(0),H(0)\right)=(\mathrm{5,0}.\mathrm{0001,0}.\mathrm{0002,100},\mathrm{5,10})$ [Colour figure can be viewed at wileyonlinelibrary.com]