Mathematical Analysis and Optimal Control of Listeriosis Transmission in a Coupled Human-Food-Bacteria System

Source abstract

Listeriosis is a foodborne disease caused by the bacterium \textit{Listeria monocytogenes}, posing a significant public health threat, particularly to immunocompromised individuals, pregnant women, and newborns. This study develops and analyzes a risk-structured mathematical model for listeriosis transmission dynamics that incorporates high-risk and low-risk susceptible human populations, food contamination, and environmental bacterial reservoirs. The model, formulated as a system of eight nonlinear ordinary differential equations, captures the interactions between human infection, food contamination, and bacterial growth. Three equilibrium points are derived, including disease-free, bacteria-free, and endemic equilibrium points. The contaminated food threshold ($R_L$) and food safety index ($R_C$) are established as key epidemiological thresholds. Local stability analyses using the Routh-Hurwitz criterion demonstrate that the disease-free and bacteria-free equilibria are locally asymptotically stable whenever $R_L < 1$, while the endemic equilibrium is stable when $R_C < 1$. Global stability of the disease-free equilibrium is established using the Castillo-Chavez method. The model is extended to incorporate three time-dependent control measures: food safety practices ($u_1$), public health education ($u_2$), and treatment of infected individuals ($u_3$). An optimal control problem is formulated with the objective of minimizing infected individuals, contaminated food products, and bacterial presence while minimizing intervention costs. Pontryagin's Maximum Principle is applied to characterize the optimal control strategies. Numerical simulations using forward-backward sweep methods demonstrate that all control strategies significantly reduce disease burden, (combining all three controls) proving most effective in reducing contaminated food by 88.5\% and exposed and infectious populations by 90\%. The findings provide valuable insights for public health policymakers in designing evidence-based, cost-effective intervention programs for listeriosis control in Nigeria and similar resource-limited settings.