Infection dynamics for fluctuating infection or removal rates regarding the number of infected and susceptible individuals

In general, the rates of infection and removal (whether through recovery or death) are nonlinear functions of the number of infected and susceptible individuals. One of the simplest models for the spread of infectious diseases is the SIR model, which categorizes individuals as susceptible, infectious, recovered or deceased. In this model, the infection rate, governing the transition from susceptible to infected individuals, is given by a linear function of both susceptible and infected populations. Similarly, the removal rate, representing the transition from infected to removed individuals, is a linear function of the number of infected individuals. While nonlinear infection and removal rates have been extensively studied in deterministic epidemiological models, analytic results for stochastic dynamics with general nonlinear rates remain limited. This work presents an analytic expression for the number of infected individuals considering nonlinear infection and removal rates. In particular, we examine how the number of infected individuals varies as cases emerge and obtain the expression accounting for the number of infected individuals at each moment. This work paves the way for new quantitative approaches to understanding infection dynamics.