A model for the spread of an SIS epidemic in a human population

Abstract

The aim of this work is to understand how infectious diseases spread through human populations. Attention is given to those diseases which follow the Susceptible–Infective–Susceptible (SIS) pattern. When modelling diseases spread in a human population, it is important to consider the social and spatial structure of the population. Humans usually live in groups such as work places, households, towns and cities. However, an individual’s membership of a particular group is not fixed. Rather, it changes over time. This structure determines two paths for a disease to spread through the population. Disease is spread between individuals in the same group by contact between infected and susceptible individuals, and is spread from one group to another by the migration of infected individuals. This type of population structure can be modelled by a metapopulation network. I develop a continuous–time Markov chain (CTMC) model that describes the spread of an SIS epidemic in a metapopulation network. I establish an ordinary differential equations (ODE) and a Gaussian diffusion analogue of the stochastic process by applying, respectively, the theory of differential equation approximations for Markov chains, and the theory of density dependent Markov chains. I use the ODE model to derive analytic expressions for various epidemiological quantities of interest. In particular, I obtain expressions for two threshold quantities; the basic reproduction number, and a quantity called T0 which is greater than the basic reproduction number. If the basic reproduction number is above 1, then the disease persists and if the basic reproduction number is below 1, then the disease–free equilibrium (DFE) is locally attractive. However, if T0 is less than or equal to 1, then the DFE is globally attractive. Using the theory of cooperative differential equations and the theory of asymptotically autonomous differential equations, I show the existence and global stability of a unique endemic equilibrium (EE) and the global stability of the DFE in terms of the basic reproduction number, provided that the migration rates of susceptible and infected individuals are equal. Numerical examples indicate that a unique stable EE exists when the condition on the migration rates is relaxed. The approximating Gaussian diffusion shows that the distribution of the population at the endemic level has an approximate multivariate normal distribution whose mean is centered at the endemic equilibrium of the ODE model. The results of this study can serve as a basic framework on how to formulate and analyse a more realistic stochastic model for the spread of an SIS epidemic in a metapopulation which accounts for births, deaths, age, risk, and level of infectivities. Assuming that the model presented here accurately describes the spread of an SIS epidemic in a metapopulation, another question which I address is how to control the spread of the disease. Since most control strategies such as vaccination, treatment and public awareness require a high cost for their implementation, I aim to provide a strategy whose cost is minimal and which only requires control of the migration pattern. Using convex optimisation theory, I obtain an exact analytic expression for the optimal migration pattern for susceptible individuals which minimises the basic reproduction number and the initial growth rate of the epidemic, provided that the migration rate of infected individuals follow a specific pattern. It turns out that the optimal migration pattern for susceptible individuals can be satisfied if the migration rates between any two patches (or groups) are symmetric. The control strategy obtained here can be applied to reduce the early growth rate of a disease in conjunction with or in the absence of another prevention measure.