Stability of a Discrete-Time Pathogen Infection Model with Adaptive Immune Response
This paper studies the global stability of a discrete-time pathogen dynamic model with both cell-mediated and antibody immune responses. Both latently and actively infected cells are incorporated into the model. We discretize the continuous-time model by using the nonstandard finite difference (NSFD) method. We establish that NSFD preserves the nonnegativity and boundedness of the solutions of the model. We derive four threshold parameters which govern the existence and stability of the steady states. We establish by using the Lyapunov method, the global stability of the five steady states of the model. We illustrate our theoretical results by using numerical simulations.
Studying and analysing the inward host dynamics of pathogens that infect the human body such as the human immunodeficiency virus (HIV), hepatitis C virus (HCV), hepatitis B virus (HBV), and human T-cell leukemia virus (HTLV) have received great attention from scientists and researchers (see, e.g., [1–12]). The dynamical behavior of pathogens can be understood by using mathematical modeling. The adaptive immune system plays an important role in controlling pathogenic infections. B cells and cytotoxic T lymphocytes (CTLs) are the main types of lymphocytes. B cells produce antibodies (antibody immune response) to neutralize the pathogens. CTLs (cell-mediated immune response) attack and kill cells infected by pathogen. In the literature, the impact of antibody immune response on pathogenic infections has been studied in several mathematical models (see, e.g., [13–20]). Further, the effect of cell-mediated immune response on pathogen dynamics has been modeled in several works (see, e.g., [21–27]). Wodarz  has combined both CTL and antibody immune responses in the following pathogen dynamic model:where , , , , and are the concentrations of healthy cells, infected cells, free pathogens, antibodies, and CTLs at time , respectively; the healthy cells are generated, dead, and become infected at rates , , and ; the term represents the death rate of infected cells; the infected cells are killed by CTLs at a rate ; the free pathogens are produced from infected cells at a rate , die at a rate , and neutralized by the antibodies at a rate ; the antibodies and CTLs are proliferated at rates and , respectively, while they die at rates and , respectively.
In models (1)–(5), it is assumed that cells generate mature pathogens directly after they are infected. However, it has been reported in  that a number of intracellular processes are needed to produce new mature pathogens. It has been shown in [30–32] that latent HIV reservoirs serve as a major barrier in HIV treatment. Pathogen dynamic models with latently infected cells and adaptive immunity have been studied in [33–35].
In the previous mentioned works, the pathogen dynamics are modeled by systems of nonlinear differential equations which are difficult to be solved analytically. Therefore, we can use an approximate solution which is obtained by discretizing the continuous-time model. Nonstandard finite difference (NSFD) has become one of the famous discretization methods which has been utilized in solving different types of differential equations . NSFD can preserve the basic and global properties of several types of nonlinear systems . NSFD has been used to discretize different pathogen dynamic models (see, e.g., [37–40]). The effect of the cell-mediated immune response has been considered of discrete-time pathogen dynamic models in [41, 42]. In addition, Elaiw and Alshaikh [43–45] have studied discrete-time pathogen infection models with antibodies.
The present paper aims to study the impact of both antibody and cell-mediated immune responses on the pathogen dynamic model. The model takes into account two types of infected cells, latently infected cells and actively infected cells. We use the NSFD method to discretize the continuous-time model. We show that the model is well posed. Further, we construct Lyapunov functions to prove the global stability of the steady states.
2. The Model
We consider the following continuous pathogen infection model with cell-mediated immune response, antibody immune response, and latency:where is the concentration of latently infected cells at time ; a fraction of healthy cells becomes actively infected cells, and the remaining fraction will become latently infected cells; the terms and represent the activation and death rates of the latently infected cells, respectively.
We apply the NSFD method  on systems (6)–(11), and we obtainwhere is the time step size and are the approximations of the solution of systems (6)–(11) at the discrete-time points , . The denominator function is chosen such that [46, 47]. Let us consider the following form of :
We define the setwhere , , , , and .
Proof. From equations (12)–(17), we obtainWe show that exists uniquely and is positive. If , then from equations (21) and (22) and the initial condition (19), we get and . From equations (23) and (26), we getThen,whereSince and , then , and hence equation (27) admits a unique positive root, . It follows from equation (23) thatNext, we consider . It follows from equations (24) and (25) thatThen, we getwhereSince and , then , and hence there exists a unique positive root of equation (32), .
From equation (24), we getTherefore, the solution exists uniquely and is nonnegative.
For , repeat the above process; we can show that exists uniquely and is positive. Utilizing the mathematical induction, for all , we know that exists uniquely and is positive.
To investigate the boundedness of solutions, we defineUsing equations (12)–(17), we getThen,Hence,Using lemma 2.2 in , we haveThen, , , , , , , and . Therefore, the solution converges to as .
2.2. Steady States
Systems (12)–(17) have five steady states:(i)Pathogen-free steady state , where .(ii)Chronic pathogen steady state without immune response , where in which is the basic reproduction number given as Thus, exists if .(iii)Chronic pathogen steady state with only antibody immune response , where in which Clearly, . If , then exists.(iv)Chronic pathogen steady state with only cell-mediated immune response , where in which Therefore, exists if .(v)Chronic pathogen steady state with both antibody and cell-mediated immune responses , where in which Thus, exists when and .
2.3. Global Stability Analysis
In this section, we study the global stability of the steady states by using the Lyapunov method. We define a function aswhich yields
Proof. Define a function aswhereThe solution of equation (51) is given byClearly, for all and at . We calculate asUsing inequality (49), we getFrom equations (12)–(17), we haveSince, , then for all , we have . Hence, the sequence is monotonically decreasing. Since , then , and hence . This yields , , and . For the case , we have , , , and . From equations (14) and (15), we obtain and . If , we have , , and . From equations (12), (14), and (15), we obtain , , and . Hence, the pathogen-free steady state is GAS.
Theorem 2. If and , then is GAS.
Proof. Define asWe have for all , and moreover . Computing asUsing inequality (49), we getFrom equations (12)–(17), we haveFrom the steady state conditions of , we haveand we getSince and , then is monotonically decreasing. We have , and then and hence , which implies , , , , and . We have four cases:(i) and , and then from equation (14), Applying the steady state condition (60), we obtain . In addition, from equation (15), we get From equation (60), we get .(ii), , and . From equation (63), we get .(iii), , and . From equation (62), we get .(iv), , , and . It follows that if and , then , , , , and . Then, is GAS.