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. *e 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.


Introduction
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. e most epidemic regions are sub-Saharan Africa and Southeast Asia [1]. Several symptoms of dengue that can appear on the infected individual are high fever, vomiting, nausea, joint pains, headache, and pain behind the eyes [2]. DENV aims and infects the following types of cells: monocytes, dendritic cells, hepatocytes, macrophages, and mast cells [3][4][5][6]. ere are four serologically various dengue viruses DENV (1)(2)(3)(4) that can infect the human [7]. When a DENV enters the human body first time, the immune response is enhanced [8]. 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][10][11][12][13][14][15][16][17]. ese models are based on the virus dynamics model introduced by Nowak and Bangham [18]. e World Health Organization (WHO) [19] 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][21][22][23][24][25]. Gujarati and Ambika [20] have formulated the following DENV infection model: where t > 0 is the time, and K(t), L(t), M(t), B(t), N(t), and P(t) 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. e parameter δ represents the creation rate of the target cells. e DENV particles infect the target cells at rate μ KM.
e DENV-infected cells produce viruses at rate τL. e B cells are created at constant rate β and proliferated at rate φBM. e death rates of the compartments K, L, M, B, N, and P are given by ξK, ϱL, ηM, cB, α 1 N, and α 2 P, respectively. e two types of antibodies N and P are generated from the B cells at rates ϑϵB and ϵB and neutralize the DENV at rates ϖ 1 MN and ϖ 2 MP. e terms ϰ 1 NM and ϰ 1 PM represent the rates at which antibody virus complex affects the antibody growth. ϑ ∈ (0, 1) is a correlation factor that quantifies the similarity between the individual serotypes. We observe that the global stability of the models presented in [20][21][22][23][24][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) [26], hepatitis C virus (HCV) [27,28], human immunodeficiency virus (HIV) [29][30][31], and human T lymphotropic virus type I (HTLV-I) [32]. To the best of our knowledge, the DENV infection model with diffusion has not been studied before. erefore, the aim of the present study is to focus on the dynamical behavior of DENV infection with diffusion. Following the work of Hattaf [33], our proposed model takes into account the spatial mobility of all compartments.

Mathematical DENV Dynamics Model
We develop a DENV infection model with secondary infection and diffusion as where u ∈ Γ is the position. e heterologous and homologous antibodies are activated at rates λ 1 MN and λ 2 MP, respectively. Here, Δ is the Laplacian operator and d x is the diffusion coefficient, where x ∈ K, L, M, N, P { }. e spatial domain Γ ⊂ R m (where m ≥ 1) is bounded and connected; moreover, its boundary zΓ is smooth. e initial conditions are given by where G ℓ (u) ≥ 0, ℓ � 1, . . . , 5, are the continuous functions. In addition, we take the following homogeneous Neumann boundary conditions: where z/z B �→ is the outward normal derivative on the boundary zΓ. ese boundary conditions indicate that cells and viruses cannot cross the isolated boundary [34].
It is clear that H is locally Lipschitz on X + . We can rewrite systems (2)-(6) with initial conditions (7) and boundary conditions (8) as the following abstract functional differential equation: Hence, for any G ∈ X + , systems (2)-(6) with (7)-(8) has a unique nonnegative mild solution (K(u, t), L(u, t), is the maximal existence time interval on which the solution exists [35][36][37]. In addition, this solution also is a classical solution for the given problem.
We define en, using systems (2)-(6), we obtain Since Journal of Mathematics Let Ψ(t) be a solution of the following ODE: which implies that K(u, t), L(u, t), M(u, t), N(u, t), and P(u, t) are bounded on Γ × [0, T m ). e standard theory for semilinear parabolic systems implies that T m � +∞ [39]. is shows that solution (K(u, t), L(u, t), M(u, t), N(u, t), P(u, t)) is defined for all u ∈ Γ, t > 0, and also is unique and nonnegative.

Global Stability
In this section, we investigate the global asymptotic stability of all equilibria by the Lyapunov method. e construction of Lyapunov functions are based on the works presented in [40][41][42][43][44]. To prove eorems 1-4, we need to define a function g(s) � s − 1 − ln s and the arithmetic-geometric mean inequality: which implies Neumann boundary conditions (8) and divergence theorem imply that For convenience, we drop the input notation, i.e., Consider a function Π ℓ (K, L, M, N, P) and define Let Y ℓ ′ be the largest invariant subset of Clearly, Π 0 (K, L, M, N, P) > 0 for all (K, L, M, N, P) > 0 and Π 0 (K 0 , 0, 0, 0, 0) � 0. We calculate zΠ 0 /zt along the solutions of model (2)-(6) as

Numerical Simulations
In this section, we numerically illustrate the global stability of equilibria by choosing the domain Γ as Γ � [0, 2] with a step size 0.02. e step size for time is given by 0.1. Following the works presented in [31,[48][49][50], we consider the following initial conditions for systems (2)  e initial values are arbitrarily chosen as the global stability of the equilibria presented in eorems 3-6 guarantees the convergence regardless of the selected initial conditions.

Conclusion
A dynamical model to capture the behavior of secondary DENV infection was studied. e model incorporated the spatial mobility of DENV particles and cells. e model was given by five PDEs to describe the interaction between five compartments, target cells, DENV-infected cells, DENV particles, heterologous antibodies, and homologous antibodies. We first established that the model is biologically relevant by showing that the key variables of the model are nonnegative and bounded. We found that the model has four equilibria Ω i , i � 1, 2, 3, 4, and their existence and global stability are governed by three threshold parameters, R i , i � 1, 2, 3. We performed the global stability analysis of the four equilibria by constructing suitable Lyapunov functions. We conducted some numerical simulations and found that the numerical results are fully aligned with the theoretical results. We summarize the obtained results in the following: (i) e infection-free equilibrium Ω 0 always exists, and it is GAS if R 0 ≤ 1. is case corresponds to the healthy state where the DENV particles are cleared. (ii) e persistent DENV infection equilibrium with ineffective antibodies Ω 1 exists if R 0 > 1, and it is GAS if R 1 ≤ 1 and R 2 ≤ 1. At this point, the DENV infection exists while the immune response is not active. (iii) e persistent DENV infection equilibrium with only effective heterologous antibody Ω 2 exists if R 1 > 1, and it is GAS if R 2 ≤ R 1 . is case leads to the situation where the DENV infection is chronic, while only heterologous antibody immune response is working. (iv) e persistent DENV infection equilibrium with only effective homologous antibody Ω 3 exists if R 2 > 1, and it is GAS if R 1 ≤ R 2 . At this point, the DENV infection is chronic with only an active homologous antibody immune response.
Our model can be extended to take into account the time delays during the DENV infection as   Here, it is assumed that a DENV contacts a target cell at time t − ψ 1 , and the cell becomes infected at time t. Moreover, an infected cell at t − ψ 2 produces new infectious DENV particles at time t. e factors e − k i ψ i , i � 1, 2, represent the survival rates of the cells during the delay periods.

Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.