Optimal immunisation policies for epidemics

Abstract

Policies of immunising the susceptibles in the general stochastic epidemic are considered. With costs assigned to the infection and immunisation of susceptibles, an optimal policy is found which at any stage minimises the expected future cost. Optimal policies are also found for the general deterministic epidemic and for an epidemic with carriers model. The three policies are compared and numerical examples are provided. An optimal policy is described in the situation when both isolation of infectives and immunisation of susceptibles are allowed.

Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis

“…The parameters used are β = 73, γ = 36.5, u max = 100, r I = 1 and r V = 1.5 (see formula (3) for the meaning of the parameters r I and r V and Section 2.2 for u max ). The solid curve represents the epidemic evolution when there is no vaccination (which is the state of the art solution, see [2,23,27]) and the dashed curve plots the epidemic evolution when there is some partial vaccination. The cost for the first trajectory is 0.51 and for the second is 0.49.…”

“…Consider the epidemic in Figure 1 (see caption for the detail of the parameters) where the abscissa represents the number of the susceptible in the population, and the ordinate the proportion of infected people. In the literature several proposals for the best vaccination strategy are presented (see for example [2,10,23,27]); however previous works operated under specific assumptions on the value function (see below) and consequently did not always selected the best vaccination policy.…”

“…Initially X 1 (0) + X 2 (0) + X 3 (0) = 1 and X 4 (0) = 0 (but X 4 need not be continuous in 0). See figure 2 for a graphical view of system (1). Note that (1) implies X 1 (t)+X 2 (t)+ X 3 (t) + X 4 (t) = 1, ∀t ≥ 0.…”

“…Historically one of the first to consider this problem, Abakuks explores two alternatives: in [2] a restrictive class of vaccination policies which allows at any time to immunize either all or none of the susceptible (therefore optimal policy immunizes either at once or never); in [1] the author considers policies which at any time during the course of the epidemic allow to immunize any number of the susceptible.…”

“…Otherwise the optimal vaccination policy is to vaccinate: find numerically (using (31)) the quantity ∆ such that Remark 4. In all situations the algorithm above solves at most once the evolution equation (1).…”