Nonlinear Mathematical Modeling of Infectious Disease Progression in Biological Systems

Abstract

The dynamics of infectious disease development within a population of interacting organisms are investigated using a mathematical modeling framework. The epidemic process is described by a system of nonlinear ordinary differential equations that account for susceptible, asymptomatically infected, symptomatic, and unreported infected individuals. The model incorporates time-dependent parameters and delay effects to capture realistic features of disease transmission and progression. An analytical approach based on the method of averaging of functional corrections is proposed and applied to obtain approximate solutions of the governing system. First- and second-order approximations are derived, enabling a detailed analysis of the temporal evolution of the epidemic process. The accuracy of the analytical results is confirmed through comparison with numerical simulations, showing strong agreement between the two approaches. The proposed model allows for the investigation of how variations in epidemiological parameters influence the speed and pattern of disease spread. In particular, transmission, recovery, and progression rates significantly affect the amplitude and duration of the epidemic outbreak. The results also highlight the important role of asymptomatic and unreported infections in sustaining disease transmission within the population.