Stability Analysis of Some Mathematical Models of Epidemics

Abstract

The spread of communicable diseases is often described mathematically by compartmental models. Many epidemiological models have a Disease Free Equilibrium (DFE) at which the population remains in the absence of disease. The classical Susceptible Infected Removed (SIR) models are very essential as conceptual models like as predator-prey and competing species models in ecology. Some Susceptible Infected (SI) and Susceptible Infected Susceptible (SIS) type models have been considered in this study. There are two major types of control strategies available to limit the spread of infectious diseases, viz. pharmaceutical interventions (drugs, vaccines, etc.), and non-pharmaceutical interventions (social distancing, quarantine, etc.). Vaccination is important for the elimination of infectious diseases as an effective preventive strategy. Vaccination of susceptible individual has been introduced through Susceptible Infected Removed Susceptible (SIRS) models. Effective vaccines have been used successfully to control smallpox, polio and measles. Some models have been presented in this study for the transmission dynamics of infectious diseases to analyze the stability of various equilibrium points mathematically. Some Susceptible Vaccinated Infected Susceptible (SVIS) and Susceptible Vaccinated Infected (SVI) models have been introduced in this study by including a new compartment ‘V’ for vaccinated individual in SIS and SI type models respectively. The above models have various kinds of parameters. Mainly the stability is analyzed by bifurcation curves in the SVIS models. The basic reproductive number (R0) can be calculated due to DFE in the SVI model. Some controlling methods have been given by changing the parameters in the SVI model through R0.