Hybrid deterministic-stochastic modeling of typhoid fever using jump diffusion for epidemic control and decision support

Abstract

A hybrid deterministic-stochastic (jump-diffusion) model for typhoid fever dynamics, addressing the limitations of classical deterministic models in capturing random environmental fluctuations and sudden outbreak events is pre sented. The model extends a deterministic seven-compartment framework by introducing both Brownian motion and discrete (Poisson jump) stochastic perturbations in the infected, treated, and recovered compartments. Ana lytical results establish the positivity and well-posedness of solutions, derive explicit expressions for disease-free and endemic equilibria, conditions for reducible nonlinear jump-endemic model. The basic reproduction numbers for both deterministic (R0) and stochastic (Rjp) systems were established. Sensitivity analysis using normalized indices and Partial Rank Correlation Coefficient (PRCC) methods identifies the symptomatic transmission rate (β1), vacci nation rate (ν), and recovery/treatment rates (γ1,δ) as the most influential parameters affecting R0. PRCC results show β1 and α have the strongest positive impact on R0, while γ1 and ν have strong negative effects, highlighting the importance of vaccination and treatment in disease control. Numerical simulations demonstrate that increasing ν or γ1 can reduce R0 below unity, leading to disease eradication, while higher β1 or positive jump intensities can sustain outbreaks even under control measures. The stochastic reproduction number Rjp is shown to decrease with stronger diffusion but increase with higher jump intensity, emphasizing the need for adaptive, non-constant inter ventions. The jump-diffusion framework thus provides a robust tool for epidemic decision support under uncertainty.