DYNAMIC OF AN SIR EPIDEMIC MODEL WITH A SATURATED TREATMENTRATE UNDER STOCHASTIC INFLUENCE

Abstract

Adding a saturating treatment logistic growth rate under both deterministic and stochastic frameworks is suggested as an addition to the conventional SIR (Susceptible-Infected-Recovered) epidemic model in this research. Treatment efficacy is constrained in real-world situations, especially when the number of infected persons is considerable, contrary to the idealized linear treatment rate assumed by the classic SIR model. We propose a continuously differentiable treatment function to account for this, which characterizes the sluggish reaction of medical therapy when the healthcare system is overloaded. In this model, the saturation effect is represented by a function that shows how the treatment rate grows at first, but then approaches a maximum limit when resources are limited. To account for the dynamic between susceptible and infected people, the model uses a bilinear incidence rate. To mimic the declining rewards of medical treatments as the illness spreads, the availability of therapy modifies this rate, reaching saturation at higher infection levels. Derivation of the fundamental reproduction number (R₀) provides a crucial starting point for comprehending the epidemic's spread. The likelihood of the illness spreading increases when R₀ > 1, and it decreases with decreasing values, suggesting that the sickness will ultimately die out. First, for the deterministic model, we study the local and global stability of the disease-free and endemic equilibrium points. By examining the effects of changing the treatment function and other model parameters on epidemic control, the stability analysis sheds light on this topic.

We include the inherent uncertainties and random fluctuations in real-world epidemic dynamics into the stochastic model. Variations in treatment efficacy, healthcare capacity, and external variables like environmental changes or governmental initiatives are all examples of what might cause these oscillations. We demonstrate that, subject to certain constraints relating to the strength of the stochastic perturbations, the endemic equilibrium is stable on a global scale. By shedding light on how the system acts when faced with ambiguity, our study demonstrates that the epidemic might stabilise into an endemic state, even in the face of random oscillations. At last, numerical examples are given to back up the analytical results. By simulating the effects of treatment saturation and stochastic disease dynamics on epidemic development, the simulations reveal the model's practical consequences. More accurate forecasts of epidemic outcomes, especially in resource-limited situations, are produced by combining treatment saturation with stochastic impacts, as shown by the numerical findings. In conclusion, our research offers a more refined paradigm for simulating epidemic breakouts in the face of constrained healthcare resources and random disruptions. When it comes to dealing with large-scale epidemics in real-world situations, where healthcare facilities are often overloaded, the results have significant implications for public health strategy.