MODELING THE TRANSMISSION OF MEASLES DISEASE USING A STOCHASTIC-DELAY DYNAMICAL SYSTEM

Abstract

Measles is a highly contagious viral disease that spreads through coughing, sneezing, and direct contact with an infected person's secretions. It can also be transmitted from mother to fetus during pregnancy and can lead to serious complications, including blindness, ear infections, pneumonia, encephalitis, and even death. To protect against measles, it is recommended to receive the measles, mumps, and rubella (MMR) vaccine in two doses. In this study, we developed a deterministic mathematical model to analyze the transmission dynamics of measles, incorporating vertical transmission and protective interventions. The model was subsequently extended to include delay, stochastic, and stochastic-delay frameworks, accounting for the typical 7 to 18-day incubation period and various environmental factors influencing disease spread. Our qualitative analysis shows that the solutions of state variables remain positive and bounded within the invariant region. We computed the disease-free equilibrium point, the endemic equilibrium point, and the basic reproductive number (𝑅0). Notably, the basic reproductive number for the stochastic-delay model was found to be the smallest, reflecting the effects of time delay and stochastic factors. We analyzed the stability of the disease-free and endemic equilibrium points, determining that the disease-free equilibrium point is locally asymptotically stable when R₀ is less than one. The global stability of this point was confirmed using the Lyapunov function principle. Numerical simulations, conducted using MATLAB software, corroborated our analytical results. Based on our findings, we recommend that minimizing contact between infected individuals and contaminated air, along with administering two doses of the MMR vaccine particularly to newborns and pregnant women can significantly mitigate the spread of the measles virus. We urge public health stakeholders to take proactive measures to prevent the spread of this preventable disease