Global Properties of a General Latent Pathogen Dynamics Model with Delayed Pathogenic and Cellular Infections
This paper studies the global dynamics of a general pathogenic infection model with two ways of infections. The effect of antibody immune response is analyzed. We incorporate three discrete time delays and both latently infected cells and actively infected cells. The infection rate and production and clearance/death rates of the cells and pathogens are given by general functions. We determine two threshold parameters to investigate the global stability of three equilibria. We use Lyapunov method to establish the global stability. We support our theoretical results by numerical simulations.
The study of mathematical modeling and analysis of pathogen infection is helpful for understanding within-host pathogen dynamics; therefore, it has attracted the attention of many researchers during the last decades [1–19]. In , Nowak and Bangham proposed a simple model for the interaction between a replicating virus and host cells. In the same work, they proposed a second model that improves the first model by including the cytotoxic T lymphocytes (CTL) immune response against infected cells. Several extensions of  without immunity are given in [2–16] and with CTL immune response in [18, 19]. Antibodies which are produced from the B cells play a prominent role in pathogen dynamics. The function of antibodies is to attack and kill pathogens. Based on the first simple model introduced in  and the fact that antibody immune response is more effective than CTL in some infections , Murase et al.  proposed a mathematical model that describes the interactions between the target cells, the pathogens, and the antibody immune response. This model is given as follows:where , , , and are the concentrations of susceptible cells, actively infected cells, pathogens and antibodies, respectively. In the literature, several modifications have been done on model (1)-(4) (see, e.g., [22–29]).
In model (1)-(4), the infection rate is given by bilinear incidence which may be not reasonable to describe the dynamics of the pathogen during the several stages of infection . Pathogen dynamics models with antibodies and general incidence rate have been studied in several works (see, e.g., [24–26]), but with only pathogen-to-cell transmission. In , the authors considered only cell-to-cell transmission in order to model the interaction of the healthy and infected cells. Both ways of transmissions have been considered in [31–37], but without humoral immunity. In , Lin et al.  presented a mathematical model with humoral immunity and both ways of transmissions that are modeled by two bilinear functions.
Although antiviral drug therapies can significantly limit the level of viruses in the blood, there is still a low viral load due to ongoing latently infected cells reactivation. These cells play the role of viral reservoirs and are considered a major hurdle to virus clearance in patients under treatment. Variant models have been developed to study the dynamics of pathogens in the presence of latent reservoirs (see, e.g., [24, 39–43]). Recently, Elaiw et al.  have studied the global stability of a general pathogen dynamics model with delay and with both cellular and pathogenic infections by assuming that all the infected cells are active. In , a class of latently infected cells has been added in pathogen dynamics with two routes of infections but with specific forms of the incidence rate and production and clearance/death rates of the pathogens and cells.
The main objective of this study is to develop a general pathogenic infection model with two ways of infections, antibody immune response, three discrete time delays, and both latently infected cells and actively infected cells. This model is described by the following nonlinear system:where the fractions and with are the proportions of infection that lead to latency and actively, respectively. The functions , , , , , , , and are continuously differentiable and satisfy the following hypotheses:
(H1) (i) there exists such that for ,
(ii) for all ,
(iii) there exists and such that for all
(H2) (i) for all and , ,
(ii) and for all , , for all ,
(iii) there are , and such that , , , and for all
(H3) and for all
1.1. Properties of Solution
Proof. System (5)-(9) can be written as , whereandwhereIt follows thatAny solution of system (5)-(9) with the initial conditions (11) satisfies for all . Hence, is positively invariant for system (5)-(9).
From (5), we obtain . It follows that . Let . Then where It follows that , where , and then and Moreover, let Thenwhere Then, , where This yields and , where . Then, , , , , and are ultimately bounded.
Lemma 2. Let (H1)-(H3) be satisfied; then there exist two bifurcation numbers and such that
(i) there exists only one equilibrium if and ,
(ii) there exist only two equilibria and if , and
(iii) there exist three equilibria , and if and .
Proof. Let be any equilibrium satisfyingFrom (24), we have either or . First, we consider ; then using (H2), we obtain . From (20)-(23), we obtainwhereHypothesis (H2) implies that the following functions:are strictly increasing and continuous.
Substituting (27) into (20), we getLetAccording to (H1) and (H2), we have and
Substituting in (27), we get , , and . This yields pathogen-free equilibrium Now from (29), we getMoreover, from (25), we haveTherefore, (30) becomesHypothesis (H2) implies thatFrom (H1), we have Since and , hence, ifthen and there exists such that If inequality (34) is satisfied, then from (27) and (H1)-(H2), we haveIt follows that an infected equilibrium without antibodies exists when condition (34) is satisfied. Therefore, we can define the basic reproduction number for system (5)-(9) asLet , whereThe other possibility of (24) is andLet in (20)-(22) and define function asBased on (H1) and (H2), we have and . Hence, there exists such that . Thus, and . Now, from (23), we haveFrom (H2), we have if , then . Now we can definewhich represents the antibody immune response activation number. We have ; then if , there exists an infected equilibrium with antibodies .
1.3. Global Properties
It should be noted that finding an appropriate Lyapunov functional for higher-order nonlinear delay differential equations is a more difficult task. For the construction of Lyapunov functional of fifth-order nonlinear differential equations with multiple deviating arguments, we refer the reader to the work . Here, we study the global stability of equilibria of system (5)-(9) by means of direct Lyapunov method. We first define a function and use the notation = .
Proof. Define aswhere is defined by (26). It is seen that for all , while at the pathogen-free equilibrium , we get Calculating along system (5)-(9), we obtain Collecting terms of (43), we get From (H3), we have It follows that We haveTherefore, we obtainFrom (H1) and (H2), we haveTherefore, if , then for all . Moreover, when , , and . The solutions of system (5)-(9) tend to the largest invariant subset of . For each solution in satisfies and from (8), we getThen, . Similarly, from (7), we get . It follows that . From LaSalle’s invariance principle, is GAS when .
Remark 4. From (H1)-(H3), we have Thus,
Obviously, the equilibrium does not exist if , and if . When , we have the following theorem.
Proof. Let be given as follows:We have for all and . Calculating , we obtainSimplifying (54), we get The components of satisfy We get