HIV dynamics in a periodic environment with general transmission rates

1.   Introduction

Human immunodeficiency virus (HIV) gradually destroys various types of blood cells, significantly weakening the immune system. Although antiretroviral drugs exist, their effectiveness is often limited, and without treatment, the virus can progress to acquired immunodeficiency syndrome (AIDS). HIV infections are caused by one of two retroviruses: HIV-1 or HIV-2. HIV-1 is the predominant cause of HIV infections globally, while HIV-2 is more common in West Africa. Another retrovirus, human T-lymphotropic virus (HTLV), although less prevalent, can also cause severe illness. HIV primarily targets and gradually depletes $CD4^{+}$ lymphocytes, a type of white blood cell critical to the body's defense against foreign cells, infections, and cancer. As HIV reduces these cells, the immune system becomes increasingly vulnerable to a wide range of opportunistic infections. Consequently, the majority of complications associated with HIV, including mortality, result from these secondary infections rather than the HIV infection itself.

Mathematical modeling plays a crucial role in understanding infectious diseases like HIV and predicting their long-term behavior. By simulating the evolution of key variables, these models provide valuable insights into the dynamics of the disease. In this context, the model serves as a complementary tool, augmenting our understanding of the complex interactions within the system rather than attempting to replace real-world observations. A significant body of research has focused on the mathematical modeling of HIV dynamics, particularly the interaction between HIV and T-lymphocytes. These models often employ nonlinear ordinary differential equations to capture the complexity of the system. In ^[1], Liu and Jiang studied the dynamics of a higher-order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration. In ^[2], Naik et al. studied a dynamical fractional-order HIV-1 model by considering the chaotic behavior. In ^[3], Di Mascio et al. proposed and analyzed a mathematical model for the long-term control of viremia in HIV-1 infected patients treated with antiretroviral therapy. In ^[4], Kumar et al. studied a fractional model of HIV-1 infection with the effect of antiviral drug therapy. In ^[5], Ullah et al. proposed a fractional-order model describing HIV-1 transmission under the influence of antiviral drug treatment.

Seasonality is known to have a profound impact on the dynamics of several epidemics, with many displaying periodic behavior. This periodicity can be attributed to factors such as varying contact rates between uninfected and infected individuals, or it may occur autonomously ^[6,7,8,9]. Several studies ^{[10,11,12,13,14]} have explored the impact of seasonality on various epidemics, including HIV and chikungunya virus transmission. Recently, there has been a growing emphasis on studying HIV models from a within-host perspective to obtain a deeper understanding of HIV infections, not only in the time-fixed models that gained traction, but also in those considering periodic/seasonal effects. The intricate dynamics of viral infections within host organisms present a compelling area of study, particularly when examining the interplay between various biological and environmental factors that can influence infection outcomes. These factors, which include periodic effects and periodic treatments, can significantly impact the replication of viruses and their interactions with the host, ultimately shaping the course of the infection. While circadian rhythms, the natural cycles of biological processes that occur roughly every 24 hours, serve as a prime example of how timing can regulate physiological functions such as immune responses, other periodic phenomena, such as seasonal variations in contact rates or vaccination programs, can also play a crucial role in disease transmission dynamics. In ^[15], Wang and Song studied a mathematical model for HIV infection with periodic solutions. In ^[16], the authors examined the influence of periodic variations on HIV transmission while in ^[17], the authors focused on HIV infection dynamics with three routes of transmission with linear transmission rates in a periodic environment.

In this study, we refine the modeling of HIV dynamics by incorporating three distinct routes of transmission and adopting general nonlinear transmission rates within a seasonal environment, thereby introducing greater realism into the model. The basic reproduction number $\mathcal{R}_0$ is derived using an integral operator. Our analysis reveals that the virus-free periodic trajectory remains globally stable when $\mathcal{R}_0 < 1$, whereas the virus persists periodically when $\mathcal{R}_0 > 1$. These theoretical results are substantiated by comprehensive numerical simulations. The paper is structured as follows: In Section 2, we introduce a system of nonlinear ordinary differential equations that models the dynamics of HIV transmission through three distinct routes in a seasonal environment, where the transmission rates are expressed in general nonlinear forms. We demonstrate that the virus-free periodic solution is globally asymptotically stable when $\mathcal{R}_0 < 1$, and that the virus persists when $\mathcal{R}_0 > 1$. Section 3 provides several numerical examples that support our theoretical findings. Finally, the concluding remarks of our study are presented in Section 4.

2.   Mathematical model development

The mathematical model proposed here is a generalization of the one presented in ^[17], which is a compartmental model describing the transfer between different compartments. We consider the variables $S, L$, and $P$ to represent the numbers of susceptible, latently infected, and productively infected cells, respectively. Similarly, the variables $V$ and $C$ denote the numbers of free virions (HIV-1 particles) and T-lymphocytes, respectively. The infected cells are subdivided into two compartments based on their status: $L$ or $P$. The variation in the number of infected cells depends on the number of target cells and the incidence rates. The three routes of infection are given by $\sigma_1 \varphi_1(V) S$, $\sigma_2 \varphi_2(L) S$, and $\sigma_3 \varphi_3(P)S$, corresponding to infection from free virions, latently infected cells, and productively infected cells, respectively.

given an initial condition with non-negative values $(S^0, L^0, P^0, V^0, C^0)\in \mathbb{R}_+^5\;$. The significance of the model's parameters are given in Table 1.

Note that the incidence rates ($\varphi_1(V)$, $\varphi_2(L)$, and $\varphi_3(P)$), the neutralization rate ($\varphi_4(P)$) and the T-Lymphocyte impairment rate ($\varphi_5(P)$) are all continuous, increasing functions that pass through the origin. Thus, we assume that these functions ($\varphi_1(V)$, $\varphi_2(L)$, $\varphi_3(P)$, $\varphi_4(P)$, and $\varphi_5(P)$) satisfy certain assumptions. Furthermore, we assume that the death rates of the cells are distinct and depend on the cell status.

Assumption 2.1. ● All the model's parameters are $\omega$-periodic nonnegative functions.

● $\varphi_1$, $\varphi_2$, $\varphi_3$, $\varphi_4$, and $\varphi_5$ are continuous increasing functions such that

● $d_s (t) \leq d_l(t) \leq d_p(t), \; \forall \; t\in \mathbb{R}_+$.

Let $C(t)$ be a continuous, $n \times n$ matrix function, $\omega$-periodic, irreducible, and cooperative. Let $\xi_C(t)$ be the solution of

and $r(\xi_C(\omega))$ the spectral radius of $\xi_C(\omega)$ having positive elements $\forall t > 0$. By applying the famous theory of Perron-Frobenius ^[18], one can deduce that $\xi_C(\omega)$ has the principal eigenvalue $r(\xi_C(\omega))$. Therefore, we need to use the following lemma several times.

Lemma 2.2 (^[19]). The ordinary differential equation (2.2) admits the solution $\xi(t) = x(t)e^{\ell t}$ where $\ell = \dfrac{1}{\omega} \ln(r(\xi_C(\omega)))$ and the function $x(t)$ is positive and $\omega$-periodic.

Consider the one-dimensional equation

such that the initial condition $S^0 \in \mathbb{R}_+$. Equation (2.3) has a unique $\omega$-periodic globally attractive solution denoted by $\Lambda^*(t)$ satisfying $\Lambda^*(t) > 0$ for all $t > 0$. As a result, model (2.1) allows for a unique virus-free periodic trajectory denoted $\mathcal{A}_0(t) = (\Lambda^*(t), 0, 0, 0, 0)$.

For any continuous $\omega$-periodic variable $\varphi(t)$, we denote $\varphi^u = \max\limits_{t\in[0, \omega)} \varphi(t)$, $\varphi^l = \min\limits_{t\in[0, \omega)} \varphi(t)$, and $d(t) = \min\limits_{t\geq 0}(d_v(t), d_c(t))$.

Proposition 2.3. $\Omega^u = \left\{(S, L, P, V, C)\in \mathbb{R}_+^5\; /\; S+L+P \leq \Lambda^u; V + C \leq (\eta_2^u+\eta_3^u) \dfrac{\Lambda^u}{d^l} \right\}$ is compact, positive, invariant, and an attractor of every solution of system (2.1) such that we have

Proof. By summing the first three equations of system (2.1), we obtain

and

□

2.1.   Disease-free periodic trajectory

By using the theory of Wang and Zhao ^[20], we can define the basic reproduction number $\mathcal{R}_0$ by rewriting system (2.1) in the following suitable form: Let

and

Our goal is to satisfy Assumptions (A1)–(A7) of ^[20]. Through the new variables' order, (2.1) will be written as

Therefore, Assumptions (A1)–(A5) in ^[20] are already satisfied. (2.5) admits a virus-free periodic trajectory $X^*(t) = \left(0, 0, 0, \Lambda^*(t), 0 \right)^T$. Let

and

where $h_i(t, X(t))$ and $X_i(t)$ are the $i$-th components of $h(t, X(t))$ and $X(t)$, respectively. We can easily obtain that

Then, $r(\phi_M(\omega)) < 1$. Then, the disease-free trajectory $X^*(t)$ is asymptotically stable inside $\Omega_s$, where

and then Assumption (A6) of ^[20] is also verified.

Let us define the matrix functions ${\bf Z}(t)$ and ${\bf W}(t)$ given by

and

such that $\mathcal Z_i(t, X(t))$ and $\mathcal W_i(t, X(t))$ are the $i$-th components of $\mathcal Z(t, X(t))$ and $\mathcal W(t, X(t))$, respectively. By a simple calculation, we obtain

and

The expression $\dfrac{d}{dt} H(t_1, t_2) = -{\bf W}(t_1)H(t_1, t_2)$ with $t_1 \geq t_2$ and $H(t_1, t_1) = I_3$ admits a $3\times 3$ matrix solution denoted by $H(t_1, t_2)$. Then, Assumption (A7) of ^[20] is also verified.

Let us define the linear operator $K: C_\omega \to C_\omega$ as

where $C_\omega$ is the Banach space of $\omega$-periodic functions $\mathbb{R} \mapsto \mathbb{R}^3$, equipped with $\|.\|_\infty$ as its norm. Therefore, the basic reproduction number $\mathcal{R}_0$ is expressed as the spectral radius of the operator $K$:

Furthermore, according to the theory in [20, Theorem 2.2], we have the following results.

Theorem 2.4. [20, Theorem 2.2]

● $\mathcal{R}_0 < 1 \; \Leftrightarrow\; r(\phi_{Z-W} (\omega)) < 1$.

● $\mathcal{R}_0 = 1 \; \Leftrightarrow\; r(\phi_{Z-W} (\omega)) = 1$.

● $\mathcal{R}_0 > 1 \; \Leftrightarrow\; r(\phi_{Z-W} (\omega)) > 1$.

Thus, the local asymptotic stability of $\mathcal{A}_0(t)$ is conditional to the satisfaction of the condition where $\mathcal{R}_0 < 1$; else, it will be unstable if $\mathcal{R}_0 > 1$.

Theorem 2.5. The global asymptotic stability of the disease-free solution, $\mathcal{A}_0(t)$, is conditional to the satisfaction of the condition where $\mathcal{R}_0 < 1$, and it will be unstable if $\mathcal{R}_0 > 1$.

Proof. According to Theorem 2.4, the local asymptotic stability of $\mathcal{A}_0(t)$ is conditional to $\mathcal{R}_0 < 1$. Therefore, we have to show that $\mathcal{A}_0(t)$ is a globally attractive solution for the case where $\mathcal{R}_0 < 1$. By reference to the limit (2.4) in Lemma 2.3, $\forall\; \varsigma_1 > 0$, $\exists \, T_1 > 0$ satisfying $S(t)+L(t)+P(t) \leq \Lambda^*(t)+\varsigma_1$, $\forall \, t > T_1$. Then, $S(t)\leq \Lambda^*(t)+\varsigma_1$, and

$\forall \, t > T_1$. Let us consider the matrix

By using Theorem 2.4, we have $r(\varphi_{Z-W} (\omega)) < 1$, and then we can choose $\varsigma_1 > 0$ small enough to satisfy $r(\varphi_{Z-W+\varsigma_1 M_2}(\omega)) < 1$, and we consider the following system:

According to Lemma 2.2 and the comparison principle, we can prove that $\exists\, y(t)$, an $\omega$-periodic positive function $y_1(t)$ that satisfies $x(t)\leq y(t) e^{k_1t}$, where

and

Hence, $\lim\limits_{t\to \infty} L(t) = \lim\limits_{t\to \infty} P(t) = \lim\limits_{t\to \infty} V(t) = 0$, and then $\lim\limits_{t\to \infty} C(t) = 0$. Furthermore, according to Eq (2.4), we deduce that $\lim\limits_{t\to \infty} (S(t) - \Lambda^*(t)) = 0$. We conclude the global attractivity of $\mathcal{A}_0(t)$, enabling us to finalize the proof. □

2.2.   HIV-infected periodic trajectory

Consider the Poincaré map $Q: \mathbb{R}^5_+ \to \mathbb{R}^5_+$ applied to system (2.1) where $Y_0 \mapsto w(\omega, Y^0)$ and $w(t, Y^0)$ is a trajectory of system (2.1) such that $w(0, Y^0) = Y^0\in \mathbb{R}_+^4$ is the initial condition. Let us define the sets $\Gamma = \left\{ (S, L, P, V, C) \in \mathbb{R}^5_+\right\}$, $\Gamma_0 = Int(\mathbb{R}_+^5)$, and $\partial \Gamma_0 = \Gamma \setminus \Gamma_0$. By using Lemma 2.3, it is easy to see that $\Gamma$ and $\Gamma_0$ are positively invariant and that $Q$ is point dissipative. Let us consider

Before applying the uniform persistence theory ^[19,21], we have to demonstrate that

On the one hand, we have $M_\partial\supseteq \left\{(S, 0, 0, 0, 0), \; S \geq 0 \right\}$, and it remains to be shown that $M_\partial \setminus \left\{(S, 0, 0, 0, 0), \; S \geq 0 \right\} = \emptyset$. Let

Once $P^0 = 0$ and $0 < L^0$, $L(t) > 0$, $\forall \, t > 0$. Therefore, we obtain $\dot P(t)_{|t = 0} = \eta_1(0) L^0 > 0$. Once $P^0 > 0$ and $L^0 = 0$, $P(t) > 0$ and $S(t) > 0$, $\forall \, t > 0$. Then, $\forall \, t > 0$, one has

$\forall \, t > 0$, which means that $(S(t), L(t), P(t), V(t), C(t)) \not\in \partial \Gamma_0$ for $0 < t$. Eq (2.10) follows directly since $\Gamma_0$ is positively invariant, as established in Proposition 2.3. Subsequently, $\exists \, (\Lambda^*(0), 0, 0, 0, 0)$, a unique fixed point of $Q$ in $M_\partial$, and the HIV will persist.

Theorem 2.6. If $\mathcal{R}_0 > 1$, then (2.1) admits at least a positive periodic solution. Furthermore, $\exists \, \varrho > 0$ that satisfies $\forall \; (S^0, L^0, P^0, V^0, C^0) \in \mathbb{R}_+ \times Int(\mathbb{R}^5_+)$,

Proof. We aim in this proof to use the theory in reference [21, Theorem 3.1.1] to demonstrate the uniform persistence of the Poincaré map $Q$ respecting $(\Gamma_0, \partial \Gamma_0)$, which allows us to prove the uniform persistence of the trajectories of system (2.1) respecting $(\Gamma_0, \partial \Gamma_0)$. Note that $r(\varphi_{Z-W}(\omega)) > 1$ according to Theorem 2.4. Then, we can choose a constant $\varsigma_2 > 0$ such that $r(\varphi_{Z-W-\varsigma_2 M_2} (\omega)) > 1$. Consider the perturbed dynamics

The Poincaré map $Q$ admits a unique fixed point $\bar S_{\alpha}^0$ that is continuous with respect to $\alpha$. Thus, one can choose $\alpha > 0$ satisfying $\bar S_{\alpha}(t) > \bar S(t)-\varsigma_2, \;$ $\forall \; t > 0$. Let us denote $M_1 = (\bar S^0, 0, 0, 0, 0)$. Since each solution of the dynamics is continuous with respect to the initial condition, then $\exists \, \alpha^*$ satisfies $\forall \; (S^0, L^0, P^0, V^0, C^0) \in \Gamma_0$ with $\|(S^0, L^0, P^0, V^0, C^0)- M_1\|\leq \alpha^*$, and we obtain that

By using the contradiction process, we will demonstrate that

Assume that $\limsup_{n\to \infty} d(Q^n(S^0, L^0, P^0, V^0, C^0), M_1) < \alpha^*$ for some $(S^0, L^0, P^0, V^0, C^0)\in \Gamma_0$. In particular, assume that $d(Q^n(S^0, L^0, P^0, V^0, C^0), M_1) < \alpha^*$, $\forall n > 0$. Therefore, we get

for all $n > 0$ and $0 \leq t \leq \omega$. For any $t\geq 0$, assume that $t = n\omega +t_1$, where $t_1 \in [0, \omega)$ and $n\leq \dfrac{t}{\omega}$ is the greatest integer of $\dfrac{t}{\omega}$. Then, we get

Let

Then, $0 \leq L(t), P(t), V(t) \leq \alpha$, $\forall \; t \geq 0$, and

The Poincaré map $Q$ has a fixed point $\bar S_{\alpha}^0$ which is globally attractive such that $\bar S_{\alpha}(t) > \bar S(t)-\varsigma_2$. Then, there exists a constant $T_2 > 0$ satisfying

Therefore, $\forall \; t > T_2$,

As $r(\varphi_{Z-W-\varsigma_2 M_2} (\omega)) > 1$, there exists an $\omega$-periodic solution $y(t)$ that satisfies $J(t) \geq e^{k_2 t}y(t)$ and

Then, $\lim\limits_{t\to \infty} P(t) = \infty$, and this is impossible since the trajectory is bounded, and so (2.12) is satisfied. The weak uniform persistence of $Q$ is verified with respect to $(\Gamma_0, \partial \Gamma_0)$. According to Proposition 2.3, the map $Q$ admits a global attractor, and then $M_1 = (\bar S^0, 0, 0, 0, 0)$ is invariant in $\Gamma$ and $W^s(M_1) \cap \Gamma_0 = \emptyset$. All solutions inside $M_\partial$ tend towards $M_1$, which is acyclic in $M_\partial$. By using the results in [21, Theorem 1.3.1], we deduce that the map $Q$ is uniformly persistent with respect to $(\Gamma_0, \partial \Gamma_0)$. Furthermore, when using [21, Theorem 1.3.6], the map $Q$ has a fixed point $(\tilde S^0, \tilde L^0, \tilde P^0, \tilde V^0, \tilde C^0) \in \Gamma_0$ such that $(\tilde S^0, \tilde L^0, \tilde P^0, \tilde V^0, \tilde C^0) \in R_+ \times Int(R^4_+)$. Our goal now is to demonstrate that $\tilde S^0 > 0$. We shall use the contradiction technique by assuming that $\tilde S^0 = 0$. According to system (2.1), $\tilde S(t)$ fulfills

with $\tilde S^0 = \tilde S(m\omega) = 0, m = 1, 2, 3, \cdots$. By using Lemma 2.3, $\forall\; \varsigma_3 > 0$, $\exists \, T_3 > 0$ satisfying

Then, one gets

for $t\geq T_3$. Therefore, $\exists \, \bar m$ satisfying $m \omega > T_3$, $\forall \, m > \bar m$. According to the comparison principle, one obtains

$\tilde S(m\omega) > 0$, $\forall \, m > \bar m$ which contradicts the fact that $\tilde S(m\omega) = 0$. Therefore, $\tilde S^0$ should satisfy $\tilde S^0 > 0$, and $(\tilde S^0, \tilde L^0, \tilde P^0, \tilde V^0, \tilde C^0)$ is an $\omega$-periodic solution of (2.1). □

3.   Numerical investigation

The goal of this section is to give several numerical tests that confirm the obtained theoretical results. The incidence rates were modeled by Monod-type functions as follows:

where $\varphi_i^{max}$ and $k_i$, $i = 1, \cdots, 5$ are nonnegative constants. Note that $\varphi_i$, $i = 1, \cdots, 5$ are continuous and increasing functions. The $\omega$-periodic functions were modeled by a well-known form given by

where $a_0 \geq 0$ is the baseline value, $0 < a_1 \leq 1$ is the magnitude of the periodic variation, and $0 \leq \Theta \leq 1$ is the phase.

The seasonal cycles frequencies $\Lambda_1$, $d_{s1}$, $d_{l1}$, $d_{i1}$, $d_{v1}$, $d_{c1}$, $\sigma_{11}$, $\sigma_{21}$, $\sigma_{31}$, $\sigma_{41}$, $\sigma_{51}$, $\eta_{11}$, $\eta_{21}$, and $\eta_{31}$ satisfy $|\Lambda_1| < 1$, $|d_{s1}| < 1$, $|d_{l1}| < 1$, $|d_{i1}| < 1$, $|d_{v1}| < 1$, $|d_{c1}| < 1$, $|\sigma_{11}| < 1$, $|\sigma_{21}| < 1$, $|\sigma_{31}| < 1$, $|\sigma_{41}| < 1$, $|\sigma_{51}| < 1$, $|\eta_{11}| < 1$, $|\eta_{21}| < 1$, and $|\eta_{31}| < 1$. All fixed constants $\Lambda_0$, $m_{s0}$, $d_{l0}$, $d_{p0}$, $d_{v0}$, $d_{c0}$, $\sigma_{10}$, $\sigma_{20}$, $\sigma_{30}$, $\eta_{10}$, $\sigma_{40}$, $\sigma_{50}$, $\eta_{20}$, and $\eta_{30}$ are provided in Table 2. Due to the absence of biological data for our simulations, we have selected parameter values arbitrarily, and they do not possess any biological meaning.

Three environmental situations were considered. The first case involves all parameters being constants. The second case considers only the transmission rates $\sigma_1(t)$, $\sigma_2(t)$, $\sigma_3(t)$, $\sigma_4(t)$, and $\sigma_5(t)$ as $\omega$-periodic functions. The third situation examines the scenario where all parameters are $\omega$-periodic functions.

3.1.   Fixed parameters

In this first situation, we consider the case where all parameters are constant. Model (2.1) then takes the form

such that the positive initial condition $(S^0, L^0, P^0, V^0, C^0) = (0.01, 4, 7, 3, 6) \in \mathbb{R}_+^5\;$. Let us denote by $\mathcal{R}_0$, the basic reproduction number. It can be determined through the next-generation matrix method ^[22,23]. Let

and then

Therefore, the next-generation matrix $FV^{-1}$ is given by

Therefore, $\mathcal{R}_0$ is given by

We provide several numerical examples to validate the obtained theoretical results. The behavior of the trajectories of (3.2) with respect to time is shown in Figure 1 (right) and in $LPV$ coordinates in Figure 1 (left), which represent the main variables of the disease where $\mathcal{R}_0 > 1$. As can be seen, the solution converges to the positive steady state, reflecting the persistence of HIV. To validate global stability, we consider several initial conditions in Figure 2, and all trajectories converge to the same steady state. In Figure 3 (left), the behavior of the trajectories of (3.2) in $LPV$ coordinates and the behavior of the trajectories with respect to time (Figure 3, right) are shown for $\mathcal{R}_0 < 1$. Once again, the theoretical results are confirmed, as the solution converges to the HIV disease-free steady state $\mathcal{A}_0 = \left(\Lambda_0, 0, 0, 0, 0\right)$, confirming the extinction of HIV. To further validate the global stability of the HIV disease-free steady state $\mathcal{A}_0$, several initial conditions were considered in Figure 4, and all trajectories converge to the same disease-free steady state.

3.2.   Periodic transmission rates

In the second situation, we perform numerical tests on (2.1), where only the incidence rates ($\sigma_1(t)$, $\sigma_2(t)$, $\sigma_3(t)$), the neutralization rate ($\sigma_4(t)$), and the T-lymphocytes impairment rate ($\sigma_5(t)$) depend on time $t$, and are assumed to be $\omega$-periodic functions. The model then takes the form

such that the positive initial condition $(S^0, L^0, P^0, V^0, C^0) = (0.01, 4, 7, 3, 6)\in \mathbb{R}_+^5\;$. We used the time-averaged system to approximate $\mathcal{R}_0$. The behavior of the trajectories of (3.3) with respect to time is shown in Figure 5 (right), and in $LPV$ coordinates in Figure 5 (left), where $\mathcal{R}_0 > 1$. As can be seen, the solution converges to a periodic trajectory, confirming HIV persistence. Several initial conditions were considered in Figure 6, and all trajectories converge to the same periodic solution. In Figure 7, we display the behavior of the trajectories of (3.3) in $LPV$ coordinates (left) and with respect to time (right) for $\mathcal{R}_0 < 1$. Again, the theoretical results are confirmed, as the solution converges to the HIV disease-free steady state $\mathcal{A}_0 = \left(\Lambda_0, 0, 0, 0, 0\right)$, confirming HIV extinction. In Figure 8, several initial conditions were considered, and all trajectories converge to the same disease-free steady state.

3.3.   Totally periodic environment

In the third step, we assume that all parameters are $\omega$-periodic functions, and the system is expressed as

given an initial condition with non-negative values

Again, as in the case of model (3.3), the time-averaged system was used to calculate $\mathcal{R}_0$. The behavior of the trajectories of (3.4) with respect to time is shown in Figure 9 (right), and in $LPV$ coordinates in Figure 9 (left), where $\mathcal{R}_0 > 1$. As can be seen, the solution converges to a periodic trajectory, confirming HIV persistence. Several initial conditions were considered in Figure 10, and all trajectories converge to the same periodic trajectory. In Figure 11, we display the behavior of the trajectories of (3.4) in $LPV$ coordinates (left) and the behavior of the trajectories with respect to time (right) for $\mathcal{R}_0 < 1$. Again, the theoretical results are confirmed, as the solution converges to the HIV disease-free periodic solution $\mathcal{A}_0(t) = \left(\Lambda^*(t), 0, 0, 0, 0\right)$, confirming HIV extinction. Several initial conditions were considered in Figure 12, and all trajectories converge to the same disease-free steady state.

4.   Conclusions

This paper extends the system studied in ^[17], which models HIV transmission in blood cells by generalizing the infection, neutralization, and impairment rates. We defined the basic reproduction number $\mathcal{R}_0$ as the spectral radius of an integral operator. It is demonstrated that the HIV-free periodic solution $\mathcal{A}_0(t)$ is globally asymptotically stable when $\mathcal{R}_0 < 1$, and that HIV persists when $\mathcal{R}_0 > 1$, exhibiting asymptotic periodic behavior. We provide several numerical tests for three situations, fixed parameters, periodic transmission rates, and a fully periodic environment, all of which confirm the theoretical results, showing that the solution converges to a limit cycle.

Acknowledgments

The author is grateful to the anonymous reviewers for their valuable and constructive feedback, which helped improve the presentation of the paper.

Conflict of interest

The author declares no conflict of interest.