Computational modelling and bifurcation analysis of reaction diffusion epidemic system with modified nonlinear incidence rate

ABSTRACT

The aim of this work is to design two novel implicit and explicit finite difference (FD) schemes to solve SIR (susceptible, infected and recovered) epidemic reaction–diffusion system with modified saturated incidence rate. Since this model is based on population dynamics, therefore solution of the continuous system possesses the positivity property. The proposed finite difference schemes retain the positivity property of sub population which is an essential feature in population dynamics. Von Neumann stability analysis reveals that proposed FD schemes are unconditionally stable. It is verified with the help of Taylor's series expansion that proposed FD schemes are consistent. The proposed implicit scheme is unconditionally consistent, i.e. for $h=\tau$. On the other hand the proposed explicit scheme gives conditional consistency for $h={\tau }^3$. The proposed FD schemes are compared with two other FD schemes, i.e. forward Euler and Crank Nicolson scheme. Simulations are performed for the verification of all the attributes for the underlying FD schemes. Furthermore, stability of the reaction diffusion system is discussed by applying Routh–Hurwitz criteria. Bifurcation values of infection coefficient are also obtained from Routh–Hurwitz condition.

KEYWORDS:

• FD schemes
• SIR reaction–diffusion model with modified incidence rate
• positivity
• convergence
• bifurcation value

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K57
• 65N06
• 65P30

Acknowledgments

The authors would like to thank anonymous referees for the useful comments and suggestions which led to an improved paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Nauman Ahmed  http://orcid.org/0000-0003-1742-585X

M. Rafiq  http://orcid.org/0000-0002-2165-3479