David Greenhalgh

A stochastic differential equation SIS epidemic model

In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals

Convergence Criteria for Genetic Algorithms

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Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity.

. We then prove that this SDE has a unique global positive solution

Some results on optimal control applied to epidemics

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Some bounds on estimates for reproductive ratios derived from the age-specific force of infection

and establish conditions for extinction and persistence of

Vaccination campaigns for common childhood diseases

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Analytical threshold and stability results on age-structured epidemic models with vaccination

. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases.

When a predator avoids infected prey: a model-based theoretical study

In this paper we discuss convergence properties for genetic algorithms. By looking at the effect of mutation on convergence, we show that by running the genetic algorithm for a sufficiently long time we can guarantee convergence to a global optimum with any specified level of confidence. We obtain an upper bound for the number of iterations necessary to ensure this, which improves previous results. Our upper bound decreases as the population size increases. We produce examples to show that in some cases this upper bound is asymptotically optimal for large population sizes. The final section discusses implications of these results for optimal coding of genetic algorithms.

Awareness programs control infectious disease - Multiple delay induced mathematical model

Many classical mathematical models for animal infections assume that all infected animals transmit the infection at the same rate, all are equally susceptible, and the course of the infection is the same in all animals. However for some infections there is evidence that seropositives may still transmit the infection, albeit at a lower rate. Animals can also experience more than one episode of the infection although those who have already experienced it have a partial immune resistance. Animals who experience a second or subsequent period of infection may not necessarily exhibit clinical symptoms. The main example discussed is bovine respiratory syncytial virus (BRSV) amongst cattle. We consider simple models with vaccination and homogeneous and proportional mixing between seropositives and seronegatives. We derive an expression for the basic reproduction number, R(o), and perform an equilibrium and stability analysis. We find that it may be possible for there to be two endemic equilibria (one stable and one unstable) for R(o)<1 and in this case at R(o)=1 there is a backwards bifurcation of an unstable endemic equilibrium from the infection-free equilibrium. Then the implications for control strategies are considered. Finally applications to Aujeskys disease (pseudorabies virus) in pigs are discussed.

Some threshold and stability results for epidemic models with a density-dependent death rate

Abstract This paper deals with a mathematical model for controlling an epidemic by the removal and isolation of infected people. The objective is taken to be to maximize the expected number of people removed at some terminal time. Some simple results are found for a deterministic model with a homogeneously mixing population by using the maximum principle. It is found that the optimal policy with the above objective function is to wait until a switching time and then attempt to remove as many infected people as possible. Next a stochastic model is discussed, and under certain assumptions similarresults are obtained. For the stochastic homogeneous mixing case the relationship between the switching times, the starting state of the epidemic, and the terminal time is explored.