Modeling a SI epidemic with stochastic transmission: hyperbolic incidence rate

Area of Science:

• Mathematical epidemiology
• Stochastic modeling
• Nonlinear dynamics

Background:

• The study builds upon the Roberts and Saha (1999) susceptible-infectious (SI) epidemic model.
• It incorporates a novel hyperbolic type nonlinear incidence rate, previously unused in such models.

Purpose of the Study:

• To analyze a new stochastic SI epidemic model with a hyperbolic incidence rate.
• To investigate the asymptotic behavior of the deterministic and stochastic versions of the model.
• To explore the impact of environmental stochasticity on disease transmission dynamics.

Main Methods:

• Development of an ordinary differential equation for the deterministic model, analogous to the double Allee effect equation.
• Analysis of the asymptotic behavior of the deterministic model as time approaches infinity.
• Derivation and analysis of a stochastic differential equation (SDE) to model environmental variations in disease transmission.
• Application of the Fokker-Planck equation to find the invariant measure of the SDE.

Main Results:

• The limit of the deterministic model solution as time tends to infinity was determined.
• The existence and uniqueness of a solution for the derived stochastic differential equation were proven.
• An explicit expression for the invariant measure was found and its properties were studied.
• Simulations were used to compare the long-term behaviors of the deterministic and stochastic models.

Conclusions:

• The novel hyperbolic incidence rate provides a unique approach to modeling epidemic dynamics.
• The stochastic model captures the influence of environmental fluctuations on disease spread.
• The findings offer a deeper understanding of epidemic behavior under varying conditions.