Resource Allocation for Epidemic Control Across Multiple Sub-populations.

The number of pathogenic threats to plant, animal and human health is increasing. Controlling the spread of such threats is costly and often resources are limited. A key challenge facing decision makers is how to allocate resources to control the different threats in order to achieve the least amount of damage from the collective impact. In this paper we consider the allocation of limited resources across n independent target populations to treat pathogens whose spread is modelled using the susceptible-infected-susceptible model. Using mathematical analysis of the systems dynamics, we show that for effective disease control, with a limited budget, treatment should be focused on a subset of populations, rather than attempting to treat all populations less intensively. The choice of populations to treat can be approximated by a knapsack-type problem. We show that the knapsack closely approximates the exact optimum and greatly outperforms a number of simpler strategies. A key advantage of the knapsack approximation is that it provides insight into the way in which the economic and epidemiological dynamics affect the optimal allocation of resources. In particular using the knapsack approximation to apportion control takes into account two important aspects of the dynamics: the indirect interaction between the populations due to the shared pool of limited resources and the dependence on the initial conditions.

Complex Dynamical Behaviour in an Epidemic Model with Control.

We analyse the dynamical behaviour of a simple, widely used model that integrates epidemiological dynamics with disease control and economic constraint on the control resources. We consider both the deterministic model and its stochastic counterpart. Despite its simplicity, the model exhibits mathematically rich dynamics, including multiple stable fixed points and stable limit cycles arising from global bifurcations. We show that the existence of the limit cycles in the deterministic model has important consequences in modelling the range of potential effects the control can have. The stochastic effects further interact with the deterministic dynamical structure by facilitating transitions between different attractors of the system. The interaction is important for the predictive power of the model and in using the model to optimize allocation when resources for control are limited. We conclude that when studying models with constrained control, special care should be given to the dynamical behaviour of the system and its interplay with stochastic effects.

Trade-off between disease resistance and crop yield: a landscape-scale mathematical modelling perspective.

The deployment of crop varieties that are partially resistant to plant pathogens is an important method of disease control. However, a trade-off may occur between the benefits of planting the resistant variety and a yield penalty, whereby the standard susceptible variety outyields the resistant one in the absence of disease. This presents a dilemma: deploying the resistant variety is advisable only if the disease occurs and is sufficient for the resistant variety to outyield the infected standard variety. Additionally, planting the resistant variety carries with it a further advantage in that the resistant variety reduces the probability of disease invading. Therefore, viewed from the perspective of a grower community, there is likely to be an optimal trade-off and thus an optimal cropping density for the resistant variety. We introduce a simple stochastic, epidemiological model to investigate the trade-off and the consequences for crop yield. Focusing on susceptible-infected-removed epidemic dynamics, we use the final size equation to calculate the surviving host population in order to analyse the yield, an approach suitable for rapid epidemics in agricultural crops. We identify a single compound parameter, which we call the efficacy of resistance and which incorporates the changes in susceptibility, infectivity and durability of the resistant variety. We use the compound parameter to inform policy plots that can be used to identify the optimal strategy for given parameter values when an outbreak is certain. When the outbreak is uncertain, we show that for some parameter values planting the resistant variety is optimal even when it would not be during the outbreak. This is because the resistant variety reduces the probability of an outbreak occurring.