Non-immunizing infectious diseases cannot be eradicated and therefore generate persistent health and intervention costs over time. Steering such diseases toward a stable endemic equilibrium is a natural long-run policy objective. This paper contributes by formulating an integrated infinite-horizon SIS-inventory control model to capture the spread of non-immunizing infectious diseases. Inventory is governed by a controllable replenishment rate that influences the recovery rate, directly linking logistical capacity to epidemiological outcomes. We derive regime-dependent trajectories implied by the Pontryagin Maximum Principle and establish conditions for the existence, reachability, and saddle-type local stability of steady states under three tractable pure replenishment regimes: zero, moderate, and maximum. We find that zero replenishment fails to support convergence to the disease-free equilibrium and leads to persistence of infection, whereas maximum replenishment yields a unique endemic steady state. In contrast, moderate replenishment may generate multiple equilibria. Threshold effects emerge, showing that insufficient replenishment can undermine system stabilization. Numerical experiments illustrate transition dynamics, convergence speeds, and sensitivity to key parameters, yielding practical insights for long-run epidemic management.
Dmitry Tsadikovich (Sun,) studied this question.