Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases
In this chapter, we propose and analyze a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). The first model with cellular immunity, the sec
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Abstract In this chapter, we propose and analyze a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). The first model with cellular immunity, the second with humoral immunity and the third with cellular and humoral immune responses. In the three proposed models, the disease transmission process is modeled by a general incidence function which includes several forms existing in the literature. In addition, the global analysis of the proposed models is rigorously investigated. Furthermore, biological findings of our analytical results are presented. Moreover, mathematical virus models and results presented in many previous studies are extended and generalized. Keywords Virus dynamics · Immunity · Diffusion · Lyapunov functional · Global stability
1 Introduction Recently, many mathematical models used partial differential equations (PDEs) have been developed to better describe the dynamics of infectious diseases caused by viruses such as HIV and HBV. In 2007, Wang and Wang [1] proposed a PDE model to describe the HBV infection. In 2008, Wang et al. [2] modeled the intracellular time delay between infection of a cell and production of new virus particles by incorporating a discrete time delay into [1]. In 2011, Brauner et al. [3] adapted the model [1] to HIV infection. The bilinear incidence rate in three above models K. Hattaf (B) Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco e-mail: [email protected] K. Hattaf · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O Box 7955, Sidi Othman, Casablanca, Morocco © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_16
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was replaced by saturated infection rate in [4], by standard incidence function in [5], by Beddington-DeAnglis functional response in [6] and by a specific functional response in [7]. In 2015, we generalized all the above models by proposing a PDE model with general incidence function [8]. This general incidence function includes the above incidence rates and many other types existing in the literature such as the Crowley-Martin functional response [9], the incidence function was used by Zhuo [10], and the Hattaf-Yousfi functional response introduced in [11] and used in [12, 13]. On the other hand, the adaptive immunity plays a crucial role in the control of viral infection. In fact, cytotoxic T lymphocytes (CTL) cells attack the infected cells, while the B cells produce antibodies to neutralize the viral particles. The first immune response exerted by CTL cells is called the cellular immunity. However, the second immune response mediated by antibodies is called the humoral immunity. The aim of this work is to model
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