Stochastic modeling of pneumonia transmission dynamics and implications for public health control

This study investigates the transmission dynamics of pneumonia using deterministic and stochastic SCIR compartmental models. The main objective is to examine how environmental randomness, contact-rate fluctuations, and individual-level interactions influence pneumonia persistence, extinction, and public health control outcomes.

The population was divided into four epidemiological classes: susceptible, carrier, infected, and recovered individuals. A deterministic model based on ordinary differential equations was first formulated, and stochastic perturbations were then introduced through Itô stochastic differential equations to represent uncertainty in disease transmission. Theoretical analysis was conducted to establish positivity, boundedness, global existence of solutions, and extinction-persistence conditions. A stochastic reproduction threshold was derived to quantify how environmental noise modifies the deterministic threshold. Since the stochastic model has no closed-form solution, numerical simulations were performed in MATLAB R2023a using Euler-Maruyama, stochastic Euler, stochastic Runge-Kutta, and stochastic non-standard finite difference methods.

The stochastic model produced positive, bounded, and globally defined solutions. The derived stochastic threshold showed that environmental noise reduces the effective transmission potential of pneumonia and can lead to disease extinction even when the deterministic reproduction number predicts persistence. Sensitivity analysis indicated that increasing noise intensity lowers the effective reproduction threshold and promotes extinction, whereas higher transmission rates support disease persistence. Numerical simulations confirmed the stability of the pneumonia-free and pneumonia-present equilibria under the selected parameter settings. The stochastic non-standard finite difference method preserved positivity, boundedness, and convergence more effectively than Euler-Maruyama, stochastic Euler, and stochastic Runge-Kutta methods, particularly for larger step sizes.

The stochastic SCIR framework provides a more realistic representation of pneumonia transmission under uncertain environmental and demographic conditions. The results show that stochastic fluctuations can qualitatively alter disease dynamics by reducing effective transmission and inducing extinction even when deterministic analysis predicts persistence. These findings highlight the importance of incorporating stochastic effects into pneumonia modeling and public health decision-making. The stochastic non-standard finite difference method offers a stable and biologically meaningful tool for simulating stochastic epidemic models.