Study of a SIR Model with Vaccination and Treatment under an Imprecise Environment

Abstract

Presently, compartmental epidemic models are widely employed to analyze the dynamic behavior of infectious diseases and to propose effective preventive and control measures. However, many studies have been done on fixed or precisely defined parameter values, which may not accurately represent real-world scenarios. In real situations, model parameters often vary due to factors such as changes in human immunity, the genomes of disease pathogens, climatic fluctuations, situation-dependent treatment, and varying vaccination rates. Such a changing nature of parameter values is known as impreciseness in parameter values. To study the effect of the changing nature of parameter values on disease dynamics, we have developed and studied a  $SIR$ (Susceptible-Infected-Recovered class) model incorporating both vaccination and treatment controls within an environment of interval imprecision. To address parameter uncertainty within intervals, we have proposed a linear parametric approach for modeling interval-valued parameters. We have rigorously investigated the positivity and boundedness of model solutions and determined the basic reproduction number $R_0(s)$ using the next-generation matrix approach, where $s$ represents the interval parameter. Furthermore, we have analyzed the local stability, global stability, and bifurcation of equilibrium points of the developed system. Our findings reveal that when values of $s$ are low, indicating low levels of disease transmission rate, the result is a nearly disease-free system where neither vaccination nor treatment control is necessary. Additionally, we have demonstrated that relying solely on either vaccination or treatment is insufficient to control the epidemic. Through our analysis, we have identified the necessity for a comprehensive control policy for the disease.