On a reaction–diffusion system modelling infectious diseases without lifetime immunity

• Published in: European journal of applied mathematics Vol. 33; no. 5; pp. 803 - 827
• Main Author: YIN, HONG-MING
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.10.2022
• Subjects: Applied mathematics; Asymptotic properties; Bacteria; Biological models (mathematics); Cholera; Coronaviruses; COVID-19; Elliptic functions; Energy methods; Epidemics; Immunity; Infectious diseases; Influenza; Mathematical models; Medical research; Pandemics; Partial differential equations; Viruses
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792521000231

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.