Analyzing Stability and Dynamics of an Epidemic Model with Allee’s Effect and Mass Action Incidence Rates Incorporating Treatment Stategies.

Abstract This paper focuses on transforming a continuous SIR epidemic model into its discrete
counterpart using a non-standard finite difference scheme, motivated by the rich dynamic behaviors often exhibited by discrete models. The study investigates the impact of Allee’s Effect
on disease progression by analyzing the existence of equilibrium points and determining their
stability and bifurcation conditions. Augmenting the model with these factors significantly
enhances our capacity to predict the course of the epidemic. Similarly, assessing the impact
of treatment on epidemic diseases is pivotal for disease control. A thorough understanding of
factors influencing disease dynamics, including Allee’s Effect, is crucial for effective modeling
and prediction of epidemic progression. The transmission rate of the disease fluctuates based
on individual isolation or treatment rates, and incorporating Allee’s Effect adds an extra layer
of complexity to this dynamic. Analyzing mathematical models based on these and similar
factors, including Allee’s Effect, allows us to glean insights into the disease’s trajectory. This
research investigates epidemic disease models that integrate mass action involving treatment,
while also considering the influence of Allee’s Effect. In our investigation, we explore the
existence of equilibrium points and establish the stability and bifurcation conditions of these
points in the context of Allee’s Effect. We complement our analysis with numerical examples.
To compare the behavior of the epidemic disease models under consideration, including those
accounting for Allee’s Effect, we present graphs illustrating the dynamic behaviors of our
populations.