Abstract

Dengue is an epidemic disease rapidly spreading throughout many parts of the world, which is a serious public health concern. Understanding disease mechanisms through mathematical modeling is one of the most effective tools for this purpose. The aim of this manuscript is to develop and analyze a dynamical system of PDEs that describes the secondary infection caused by DENV, considering (i) the diffusion due to spatial mobility of cells and DENV particles, (ii) the interactions between multiple target cells, DENV, and antibodies of two types (heterologous and homologous). Global existence, positivity, and boundedness are proved for the system with homogeneous Neumann boundary conditions. Three threshold parameters are computed to characterize the existence and stability conditions of the model’s four steady states. Via means of Lyapunov functional, the global stability of all steady states is carried out. Our results show that the uninfected steady state is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the disappearance of the disease from the body. When the basic reproduction number is greater than unity, the disease persists in the body with an active or inactive immune antibody response. To demonstrate such theoretical results, numerical simulations are presented.

1. Introduction

There are several types of viruses that infect humans and cause many diseases, such as HIV (human immunodeficiency virus), HCV (hepatitis C virus), HBV (hepatitis B virus), HTLV-I (human T lymphotropic virus type I), HPV (Human papilloma virus), Impetigo, EVD (Ebola virus), Chicken pox virus, Monkeypox virus, Rotavirus, Poliomyelitis, Measles, DENV (dengue virus), and COVID-19.

The infection with dengue is induced by a mosquito-borne virus called dengue virus (DENV), which belongs to the family Flaviviridae. Dengue first appeared in the Philippines and Thailand during the 1950s. In 1970, only 9 countries experienced a severe dengue epidemic. Now, the disease is endemic in over 100 countries in the WHO regions of Africa, the Americas, the Eastern Mediterranean, South-East Asia, and the Western Pacific. DENV is transmitted to humans by mosquito bites. Aedes aegypti mosquitoes and Aedes albopictus mosquitoes are the common vectors of transmission [1, 2]. Dengue is most often asymptomatic (people may not even realize they are sick at all) or may have mild symptoms, but it can also cause a severe flu-like illness that often occurs with symptoms such as high fever, headache, rashes, nausea, leukopenia, joint pain, pain in the bones or muscles, severe bleeding, plasma leakage and/or organ impairment. In most cases, symptoms last 2–7 days, after the incubation period has passed for 4–10 days after mosquito bites [3, 4]. Many studies have indicated that the types of cells which can be infected by DENV are monocytes, hepatocytes, dendritic cells macrophages, and mast cells [527]. Several aspects of the immune system play a crucial role in controlling and eliminating DENV infection, including two important immune responses: cytotoxic T lymphocytes (CTL) and humoral immunity [6].

1.1. Primary and Secondary DENV Dynamics Models

Dengue virus (DENV) can cause dengue fever in four distinct but genetically related serotypes (DENV1-4). There is lifelong immunity to one serotype of dengue virus, but not cross-protection to other serotypes and an individual may become infected four times in a lifetime [2, 28, 29]. Obtaining a second infection with DENV can lead to DSS (dengue shock syndrome) or DHF (dengue hemorrhagic fever), which occurs after being infected again with a DENV serotype different from the first time [30]. In this case, the infection is called Secondary infection and the most common symptoms of DHF and DSS are thrombocytopenia, hemorrhage, and plasma leakage [27]. Recently, infection caused by DENV has been the subject of numerous mathematical studies.

Primary DENV dynamics models: During primary infections, the body produces antibodies that neutralize the specific serotype of the infection (type-specific responses). Several mathematical models that describe Primary DENV infection within hosts have been studied (see [818]). Some related works regarding the immune responses have been investigated in primary DENV infections, including: cytotoxic T lymphocytes (CTLs) responses has been introduced by [8, 11, 14], a study of both humoral and innate immune responses has been published by [16], humoral immune response [30], Models of both humoral and CTL responses [12, 13, 17, 31]. In [30], Gujarati and Ambika introduced a mathematical model which describes the primary DENV infection with humoral immunity as follows:

For time , the variable is the concentration of healthy monocytes which are created at rate , and die at rate . denotes DENV-infected monocytes which die at rate and produce DENV particles at the rate . become DENV-infected at rate . are cleared at a rate . and represent B cells and neutralizing antibodies, respectively. The term denotes the neutralization rate of the DENV by antibodies, and are the production rate, die rate and stimulation rate of B-cells, respectively. gives the rate of antibody production from the B-cells. Antibody virus complexes affect antibody production at the rate , and the antibodies die at the rate .

Secondary DENV dynamics model: During secondary infections, the body produces antibodies that neutralize two or more serotypes. Several related studies regarding secondary DENV infections are discussed in [827, 30, 32]. There have been various immune responses studied in secondary DENV infections, among them: secondary DENV infection model with humoral immunity by [20, 24, 30], both humoral and CTL immunity [22, 27], the mobility of cells and viruses has been studied by [25], Alshaikh et al. [26] developed a model of secondary DENV infection that takes into account that the DENV can infect multiple classes cell types, Alves Rubio et al. [32] studied and analyzed a mathematical model to explain antibody-mediated enhancement in heterologous of secondary infections and assess the effect of limiting cloning for plasma cells. The following model is a secondary DENV infection model with humoral immunity, which was introduced by [30]:

Here, at time , denotes the concentration of heterologous antibodies previously produced during the primary infection, and is the concentration of homologous antibody specific to the new DENV serotype of the secondary infection. Neutralization rate of the DENV by antibodies is represented by the term . die at rate . The antibodies and are generated by B-cells at rates and , respectively, where is a correlation factor used to measure the similarity of one serotype to another. Antibody growth is affected by the antibody virus complex at rates and . All other variables and parameters of the model have the same biological concept as explained in the model (1). All parameters of models (1) and (2) are positive.

It has been noteworthy that most DENV infection models in the above works did not consider the spatial structure (diffusion), as the models were presented by ordinary or delayed differential equations assuming that all components (DENV and cells) are homogeneous. While there have been other studies of various viral infections that have considered the spatial effects of various viral infections. They hypothesized that the spatial distributions of cells and virus particles are not fully mixed (see, e.g. [25, 3338]). Among the studies that introduced DENV infection, taking into account diffusion, was Elaiw and Alofi study [25], they modeled a DENV infection with diffusion, by taking into consideration the spatial mobility of cells and DENV particles. Furthermore, [25] considered the DENV can infect one target of cells, but there are several works that have indicated that the dengue virus can infect multiple types of cells (e.g., [827]).

In view of the foregoing, the DENV infection with both multiple target cells and diffusion did not a subject of mathematical studies before. Therefore, in this work, we present a model to study and analyze secondary DENV infection. Inspired by the study of Elaiw and Alofi [25] and Alshaikh et al. [26], we focus on that DENV can infect multiple target cells considering the mobility of every component of the model.

2. Model Formulation

As a result of the biological and mathematical considerations, we propose DENV infection model considering a system of multiple target cells with two different types of antibodies ( and ) and diffusion as follows:where is the position. and are the reproduction rates of heterologous and homologous antibodies. The spatial domain (where ) is connected and bounded, and the boundary of is smooth. Finally, is the Laplacian operator and denotes the diffusion coefficient of cells and DENV particles, where .

For model (3), we consider the following homogeneous Neumann boundary conditions:

And initial conditionswhere is the outward normal derivative on the boundary , , , are continuous functions. The boundary conditions (4) point out that cells and DENVs cannot pass the isolated boundary [39].

3. Biologically Realistic Domain

To ensure that the model (3) is biologically acceptable, we define a bounded domain for the compartment concentrations.

Proposition 1. Assume that for all . Then, model (3) with any initial satisfying (5) has a unique, nonnegative, and bounded solution defined on .

Proof. Let be the set of all bounded and uniformly continuous functions from to , with the norm , where is the Euclidean norm on . We define the positive cone which induces a partial order on . Therefore, the space is a Banach lattice [40, 41].
Now, for any initial datawe define byClearly, is locally Lipschitz on . With boundary conditions (4) and initial conditions (5), the system (3) can be rewritten as an abstract functional differential equation with the following formulawhereandAs a result, it can be shown thatHence, for any , system (3) with conditions (4) and (5) has a single non-negative solution defined on , where the interval is the maximal existence time period where the solution exists [4042]. Moreover, for the given problem, this solution also constitutes a classical solution.
To prove boundedness, letthenwhere . Since , we havethenThus, satisfiesLet be a solution of the following ordinary differential equation (ODE):This implies that . Based on the principle of comparison [43], we have . HenceTherefore, are bounded on .
Additionally, letthenwith . Since , and , we obtainThus, satisfieswhere . Suppose the following ODEhas a solution , this gives that . As above, we obtain . Thus,Therefore, it implies , and are bounded on .
According to semilinear parabolic system (standard theory); [44]. This proves that the solution is defined for all , and it is unique and nonnegative.

4. Steady States and Thresholds

Through this section, we deduce three threshold parameters which define the existence of all possible steady states of model (3). To calculate the steady states, we solve the following system of algebraic equations

and then, we have the following:(i)If , there is an uninfected steady state , with .(ii)If , there is an infected steady state with inactive immune antibody response , whereand satisfiesTo determine such , in case of the equilibrium , we have , thenSubstituting from equation (25) into equation (32) we getwhere . From equation (33), we can define a function as:Now, we need to show that such that . Note thatThen, if , such that . This yields that the steady state exists when . Therefore, the basic reproduction number of our model (3), is given by determines whether the infection is present in the host.(iii)If , an infected steady state with only active heterologous antibody is obtained, whereandIt follows that only when . In this case, is the threshold parameter that indicates the initiation of an immune reaction to a heterologous antibody.(iv)If , then an infected steady state with only active homologous antibody exists, whereand

Thus, only when . Here, stands for the threshold parameter that indicates the initiation of an immune reaction to a homologous antibody.

It is clear from the above that: , , and if and , , , and all exist.

Corollary 1. Suppose that the conditions , and exist. Then the model (3) has four steady states when the following conditions are attained:(a)The uninfected steady state is always exist,(b)The infected steady state with inactive immune antibody response exists if ,(c)The infected steady state with only active heterologous antibody exists if ,(d)The infected steady state with only active homologous antibody exists if .

5. Global Stability

This section investigates the global stability of the steady states by using Lyapunov’s method. We construct Lyapunov functions using the same technique in [25, 26, 38].

Before discussing theories of global stability, it is important to consider the following:(i)We specify , where , and .(ii)For simplicity, we omit the notation for inputs. i.e. , , and for all .(iii)Numann boundary conditions (4) and Divergence Theorem imply thatfor . Thus, we obtain(iv)Consider a function and define such that(v)Let be the largest invariant subset of

Theorem 1. For model (3), suppose that , then steady state is globally asymptotically stable (G.A.S).

Proof. Define a Lyapunov function as follows:Clearly, for all and . The partial derivative can be calculated along the solutions of model (3) as:After collecting the terms of equation (46), we haveTherefore, the derivative will be as follows:Using equalities (42), (48) is transformed toHence, for all . Additionally, when . The solutions of model (3) converge to . The elements of satisfy and then . Third equation of (3) reduces toThis yields . Hence, and by applying LaSalle’s invariance principle [4547] we get that is G.A.S.

Theorem 2. For model (3), suppose that and , then is G.A.S.

Proof. Define as:Calculating as:The steady state conditions at are given byUsing equation (53), equation (52) becomesSimplify (54), we getCalculating the time derivative of , we getUsing equalities (42) to getIt follows from the relation between arithmetic and geometric means thatNext we prove that when . From (53) we have , then