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 continuoustime 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 T 0 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 T 0 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. iThe 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 imp...

Modeling Information Propagation in General V2V-enabled Transportation Networks

“…A recent study investigated the delay to forward messages in a two-dimensional road and introduced an algorithm for selecting the path with the lowest expected delay (He et al, 2017). Others have studied the metapopulation spreading process in which a group of individuals is viewed as a node and individuals move along link connecting nodes (Levins, 1969;Hanski et al, 2004;Sattenspiel and Dietz, 1995;Keeling and Rohani, 2002;Gallos and Argyrakis, 2004;Watts et al, 2005;Shausan, 2015;Sani et al, 2007).…”