An epidemic-inventory model with delivery and consumption delays for controlling non-immunizing diseases

Infectious diseases that do not provide lasting immunity remain a global concern; however, timely logistical interventions can help mitigate their impact through effective management of medicine inventories. This study introduces a coupled Susceptible-Infected-Susceptible (SIS) epidemic-inventory dynamic model to investigate the effect of logistics on disease dynamics. The medicine inventory is governed by a controllable replenishment rate, while the disease transmission rate decreases with the level of available stock, creating a biologically grounded feedback loop between resources and infection dynamics. The model incorporates delivery and consumption delays to represent, respectively, the lag between ordering stock and its arrival, as well as the lag between purchase of the medicine and its use. Using Pontryagin’s Maximum Principle, we derive optimal replenishment policies that minimize total costs over a finite planning horizon. Both delayed and non-delayed settings are analyzed for high- and low-income populations. Numerical simulations using gonorrhea data reveal that optimal control trajectories typically involve a single regime in the non-delayed setting, and two regimes in the delayed setting. Sensitivity analysis indicates that, in low-income settings, limited access to preventive measures (the medicine stock modeled in this study) offsets the benefits of higher replenishment capacity, suggesting that improvements in distribution and accessibility should precede enhanced delivery efforts. Finally, longer delays reduce optimal replenishment rates and the number of infected individuals. The proposed model can be readily applied to other non-immunizing diseases and provides actionable insights to support healthcare decision-making.