Spatial profiles of a reaction-diffusion epidemic model with nonlinear incidence mechanism and varying total population

Abstract This paper considers a susceptible-infected-susceptible epidemic reaction-diffusion model with no-flux boundary conditions and varying total population. The interaction of the susceptible and infected people is described by the nonlinear transmission mechanism of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and q &gt; 0. In Peng et al ( SIAM J. Math. Anal. (arXiv:2411.00582)), we have studied a model with a constant total population. In the current paper, we extend our analysis to a model with a varying total population, incorporating birth and death rates. We investigate the asymptotic profiles of the endemic equilibrium when the dispersal rates of susceptible and/or infected individuals are small. Our work is motivated by disease control strategies that limit population movement. To illustrate the main findings, we conduct numerical simulations and provide a discussion of the theoretical results from the view of disease control. We will also compare the results for the models with constant or varying total population.