A diffusive dengue disease model with nonlocal delayed transmission

Abstract

In this paper, we derive a nonlocal delayed and diffusive dengue transmission model with a spatial domain being bounded as well as unbounded. We first address the well-posedness to the initial-value problem for the model. In the case of a bounded spatial domain, we establish the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number ${\mathfrak{R}}_0$. Also, a set of sufficient conditions is further obtained for the global attractivity of the endemic steady state where all the parameters are spatially independent. In the case of an unbounded spatial domain, and when the coefficients are all constants, we show that there exist traveling wave solutions of the model. Numerical simulations are performed to illustrate our main analytic results.

Introduction

Dengue fever is one of the most common arboviral diseases in tropical regions of the world, which is transmitted to humans through the bite of infected Aedes mosquitos, principally Aedes aegypti, and are therefore considered to be arbovirus (arthropod-borne viruses). Dengue is caused by a group of four antigenically distinct flavivirus serotypes: DEN-1, DEN-2, DEN-3, and DEN-4. Infection by any dengue serotype produces permanent immunity to it, but apparently only temporary cross immunity to the other serotypes [31]. Thus, a person living in an endemic area can have as many as four dengue infections during his lifetime, one with each serotype. Moreover, the mosquitoes never recover from the infection and ends with their life during their infective period [8]. The incidence of dengue has grown dramatically around the world in recent decades, mostly in urban and peri-urban areas. It is endemic in more than 110 countries in Africa, the Americas, the Eastern Mediterranean, South-east Asia and the Western Pacific. It infects 50 to 100 million people worldwide a year, leading to 50 million hospitalizations, and approximately 12,500 to 25,000 deaths a year [2], [8], [31]. Thus, dengue fever remains a threat to human beings in many places in the world.

It is the high infection rate of dengue fever and high death rate of its severe form dengue hemorrhagic fever that cause public health problems and damped the economic [2], [8], [19], [31]. Therefore, there is an essential need for more information on the temporal (or spatial) patterns of disease burden, distribution and control strategies. In an effort to understand the dynamics of the spread of the disease, many mathematical models dealing with the transmission of dengue disease were developed. For example, Esteva and Vargas [5] set up an SIR v.s. SI epidemiological model. Using the results of the theory of competitive systems and stability of periodic orbits, they established the global stability of the endemic equilibrium. In fact, they found a threshold parameter R₀ that determined the global dynamics, that is, if R₀ < 1 the disease-free equilibrium is globally stable, namely, the disease will die out; if R₀ > 1, the unique disease equilibrium is globally stable, which implies that disease will always remain endemic above the threshold value. Recently, Wang and Zhao [28] presented a time-delayed dengue transmission model, and also established a threshold dynamics in terms of the basic reproduction number R₀. It is also shown that the disease will die out if R₀ < 1 and the disease persists if R₀ > 1. Moreover, they gave a set of sufficient conditions for the global attractivity of the endemic equilibrium. Actually, the models obtained in the mentioned papers based on continuous time models in the form of ordinary differential equations (ODEs). Such models assume that the populations are well mixed, and the transmissions are instantaneous. In reality, the environment in which a population lives is often heterogeneous making it necessary to distinguish the locations. Particulary, due to the large mobility of people within a country or even worldwide, spatial effects cannot be ignored in studying the spread of endemic. In order for a model to be more realistic, such as in a spatially continuous environment, Wang and Zhao [27] first proposed a nonlocal and time-delayed reaction-diffusion model of dengue fever, and established a threshold dynamics in terms of the basic reproduction number R₀, which states that the disease-free equilibrium is asymptotically stable if R₀ < 1 and the disease is uniformly persistent if R₀ > 1.

In the epidemiology, one of the important and fundamental problems is to study the stability of the steady states, since this characterizes whether a disease will become endemic and this is a major concern for public health offices. Meanwhile, for the reaction–diffusion disease model under consideration, the traveling wave describes the disease population invading into the susceptible population from an initial disease-free steady state to the final, also endemic steady state. Results on this topic may help one predict how fast a disease invades geographically, and accordingly, take necessary measures in advance to prevent the disease, or at least, decrease possible negative consequences. The important role of the traveling wave solutions in many scientific problems has been well recognized (see [7], [20], [26], [32] and the references therein). In the past decades, much work has been done for traveling waves of various epidemic models.

In this paper, we shall follow the approaches in the standard model (see [5], [27], [28]) to incorporate special movements in spatially heterogeneous environments, and derive a nonlocal and time-delayed reaction–diffusion model of dengue fever with a spatial domain being bounded as well as unbounded. Here, we are mainly interested in the extinction and persistence of disease of the model. Moreover, we study the existence of traveling wave solutions of the model.

The organization of the current paper is as follows. In next section, we derive a nonlocal and time-delayed reaction–diffusion dengue fever model (2.3) for the cross infection between mosquitos and human individuals. In Section 3, we address the well-posedness by proving the non-negativeness and boundedness of solutions to the initial-value problem for the model, and introduce the basic reproduction number ${\mathcal{R}}_0$. In the case of a bounded spatial domain, we study the threshold dynamics for the spatially heterogeneous system (2.3)–(2.5) on the extinction and persistence of disease in terms of ${\mathcal{R}}_0$. In Section 4, in the case of an unbounded spatial domain, we establish the existence of traveling wave solutions for system (4.1) when R₀ > 1. Section 5 is devoted to numerical simulations. The conclusion and a brief discuss in Section 6 complete this paper.