We construct a vaccine inventory model in which the demand is assumed to follow a modest SIV-type epidemic model. Since the continuous form of the epidemic model is not easily solvable, we apply the forward Euler method to construct a discrete form of the model, which is used to generate the time-dependent values of the vaccine’s demand at integer time steps over a specified epidemic duration. The vaccine inventory model expresses the total cost incurred by a company that monopolistically fulfils the vaccine demand over the epidemic duration as a function of the inter-order time interval, taking into account the vaccine’s order, purchase, and holding costs. We conduct numerical simulations to apply our epidemic and inventory models in two qualitatively different scenarios: a disease-free scenario characterised by a low vaccination coefficient and an endemic scenario characterised by a high vaccination coefficient. A sensitivity analysis reveals that the disease’s basic reproduction number depends most sensitively on the population’s birth rate and the disease’s infection coefficient in both scenarios, the disease’s epidemic peak depends most sensitively on the vaccination coefficient in both scenarios, and the company’s minimum total cost depends most sensitively on the vaccination coefficient in the endemic scenario and on the vaccine’s per-unit purchase price in the disease-free scenario.