Global Dynamics of Secondary DENV Infection with Diffusion
During the past eras, many mathematicians have paid their attentions to model the dynamics of dengue virus (DENV) infection but without taking into account the mobility of the cells and DENV particles. In this study, we develop and investigate a partial differential equations (PDEs) model that describes the dynamics of secondary DENV infection taking into account the spatial mobility of DENV particles and cells. The model includes five nonlinear PDEs describing the interaction among the target cells, DENV-infected cells, DENV particles, heterologous antibodies, and homologous antibodies. In the beginning, the well-posedness of solutions, including the existence of global solutions and the boundedness, is justified. We derive three threshold parameters which govern the existence and stability of the four equilibria of the model. We study the global stability of all equilibria based on the construction of suitable Lyapunov functions and usage of Lyapunov–LaSalle’s invariance principle (LLIP). Last, numerical simulations are carried out in order to verify the validity of our theoretical results.
Mathematical models and their analysis have been proven to be an efficient and significant approach to understand the within-host dynamics of viral infections such as dengue virus (DENV), human immunodeficiency virus (HIV), hepatitis C and B virus (HCV/HBV), human T lymphotropic virus type I (HTLV-I), and recently, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). DENV is the casual of dengue fever which is one of the morbidity and mortality diseases. It can transmit to humans via Aedes aegypti and Aedes albopictus mosquitoes. Annually, about 50–100 million infected individuals by DENV are reported worldwide. The most epidemic regions are sub-Saharan Africa and Southeast Asia . Several symptoms of dengue that can appear on the infected individual are high fever, vomiting, nausea, joint pains, headache, and pain behind the eyes . DENV aims and infects the following types of cells: monocytes, dendritic cells, hepatocytes, macrophages, and mast cells [3–6]. There are four serologically various dengue viruses DENV (1–4) that can infect the human . When a DENV enters the human body first time, the immune response is enhanced . Cytotoxic T lymphocytes (CTLs) and antibody immune responses are two main components of the immune system against viruses. CTLs destroy the DENV-infected cells, while antibodies kill DENV particles and clear it from the body.
During the recent years, several mathematical models have been developed which describe within-host DENV primary infection [9–17]. These models are based on the virus dynamics model introduced by Nowak and Bangham . The World Health Organization (WHO)  has reported that an infected individual by one serotype will have lifelong immunity against that serotype but only temporary and partial cross-immunity to the other three serotypes. Mathematical models of DENV dynamics pertaining to secondary infection with another serotype have been developed in [20–25]. Gujarati and Ambika  have formulated the following DENV infection model:where is the time, and , , , , , and are the concentrations of the target cells, DENV-infected cells, DENV particles, B cells, heterologous antibody previously formed on primary infection, and homologous antibody against the new virus serotype of the secondary infection, respectively. The parameter represents the creation rate of the target cells. The DENV particles infect the target cells at rate . The DENV-infected cells produce viruses at rate . The B cells are created at constant rate and proliferated at rate . The death rates of the compartments , , , , , and are given by , , , , , and , respectively. The two types of antibodies and are generated from the B cells at rates and and neutralize the DENV at rates and . The terms and represent the rates at which antibody virus complex affects the antibody growth. is a correlation factor that quantifies the similarity between the individual serotypes. We observe that the global stability of the models presented in [20–25] is not well studied.
All of the DENV infection models in the abovementioned works are given by ordinary or delay differential equations under the assumption that the cells and DENV particles are well mixed. Spatial structure plays an important role in understanding the dynamical behavior of viral infection within a host. In recent years, spatial dependence has been incorporated into mathematical models of several viral infections such as hepatitis B virus (HBV) , hepatitis C virus (HCV) [27, 28], human immunodeficiency virus (HIV) [29–31], and human T lymphotropic virus type I (HTLV-I) . To the best of our knowledge, the DENV infection model with diffusion has not been studied before. Therefore, the aim of the present study is to focus on the dynamical behavior of DENV infection with diffusion. Following the work of Hattaf , our proposed model takes into account the spatial mobility of all compartments.
2. Mathematical DENV Dynamics Model
We develop a DENV infection model with secondary infection and diffusion aswhere is the position. The heterologous and homologous antibodies are activated at rates and , respectively. Here, is the Laplacian operator and is the diffusion coefficient, where . The spatial domain (where ) is bounded and connected; moreover, its boundary is smooth.
The initial conditions are given bywhere , , are the continuous functions. In addition, we take the following homogeneous Neumann boundary conditions:where is the outward normal derivative on the boundary . These boundary conditions indicate that cells and viruses cannot cross the isolated boundary .
3. Well-Posedness of Solutions
Proof. We denote , the set of all bounded and uniformly continuous functions from to , with norm . We define the positive cone which induces a partial order on . This shows that the space is a Banach lattice [35, 36].
For any initial data , we define byIt is clear that is locally Lipschitz on . We can rewrite systems (2)–(6) with initial conditions (7) and boundary conditions (8) as the following abstract functional differential equation:where and . One can show thatHence, for any , systems (2)–(6) with (7)-(8) has a unique nonnegative mild solution defined on , where is the maximal existence time interval on which the solution exists [35–37]. In addition, this solution also is a classical solution for the given problem.
We defineThen, using systems (2)–(6), we obtainSince , then we getwhere . Thus, satisfiesLet be a solution of the following ODE:This gives that . On the basis of comparison principle , we obtain . Hence,which implies that , , , , and are bounded on . The standard theory for semilinear parabolic systems implies that . This shows that solution is defined for all , , and also is unique and nonnegative.
Theorem 2. There exist three threshold parameters , , and with and , such that(i)If , then the system has a single equilibrium (ii)If and , then the system contains only two equilibria and (iii)If and , then the system contains three equilibria , , and (iv)If and , then the system contains three equilibria , , and (v)If and , then the system contains four equilibria , , , and
Proof. To calculate the equilibria of systems (2)–(6), we letThen, solving the system of algebraic equations (18)–(22), we get four equilibria such as the following:(i)Infection-free equilibrium , where (ii)Persistent DENV infection equilibrium with ineffective antibodies is , where(iii)Persistent DENV infection equilibrium with only effective heterologous antibody is , where(iv)Persistent DENV infection equilibrium with only effective homologous antibody is , where Where Clearly, and (v)Clearly from (iii) and (iv), if and , then , , , and all existHere, represents the basic infection reproduction number, represents the heterologous antibody immune response activation number, and is the homologous antibody immune response activation number.
5. Global Stability
In this section, we investigate the global asymptotic stability of all equilibria by the Lyapunov method. The construction of Lyapunov functions are based on the works presented in [40–44]. To prove Theorems 1–4, we need to define a function and the arithmetic-geometric mean inequality:which implies
Neumann boundary conditions (8) and divergence theorem imply thatfor . Thus, we obtain
For convenience, we drop the input notation, i.e., .
Consider a function and define
Let be the largest invariant subset of
Theorem 3. Let , then is globally asymptotically stable (GAS).
Proof. Define asClearly, for all and . We calculate along the solutions of model (2)–(6) asCollecting terms of equation (34), we obtainConsequently, we calculate as follows:Using equality (30), equation (36) is reduced to the following form:Therefore, , for all , and with equality holding when . The solutions of models (2)–(6) converge to . The elements of satisfy , and then, . Equation (6) reduces toThis yields . Hence, , and by applying LLIP [45–47], we get that is GAS.
Theorem 4. If and , then is GAS.
Proof. Define asCalculate asThe equilibrium conditions of imply thatWe get andCalculate the time derivative of and use equality (30) to getSince and , then utilizing inequality (28), we obtain for all . Furthermore, at . The solutions of (2)–(6) tend to . It follows that is GAS by using LLIP.
Theorem 5. Let and then is GAS.
Proof. Define asCalculate asCollecting terms of equation (45), we getApplying the equilibrium conditions,we getCalculate the time derivative of and use equality (30) to getSince , then using inequality (28), we obtain for all . Furthermore, at . The solutions of (2)–(6) tend to . It follows that is GAS by using LLIP.
Theorem 6. Let and , then is GAS.
Proof. Define asCalculating along the solutions of models (2)–(6), we getCollecting terms, we getApplying the equilibrium conditions of ,we getCalculating the time derivative of and using equality (30), we obtainSince , then using inequality (28), we obtain , for all . Furthermore, at . The solutions of (2)–(6) tend to . It follows that is GAS by using LLIP.
6. Numerical Simulations
In this section, we numerically illustrate the global stability of equilibria by choosing the domain as with a step size 0.02. The step size for time is given by 0.1. Following the works presented in [31, 48–50], we consider the following initial conditions for systems (2)–(6):