A stochastic vector-borne epidemic model: Quasi-stationarity and extinction
Introduction
Vector borne diseases have a great impact on human health in terms of mortality. Mosquitoes are perhaps the best known disease vectors, with various species playing a role in the transmission of infections such as malaria, yellow fever, dengue fever and West Nile virus. Mathematical modeling of malaria began with Ross’s model [1]. The Ross model is deterministic and reflects the basic mechanism that both humans and mosquitoes are necessary for the transmission of infection. Despite its simplicity it has been used to establish an important threshold result and to study the effects of various methods of controlling malaria infection [2], [3], [4], [5].
Consider a population in which a vector borne disease is introduced. In the current paper we use stochastic models to answer the question: what might happen? Recurrence of epidemic outbreaks can be explained by the combined influence of epidemic and demographic forces. Stochastic models that account for these two forces in a closed population predict that the infection will eventually become extinct. The time to extinction is an important measure of the persistence of the infection.
Recently, a stochastic model for a vector borne epidemic has been suggested by Llyod et al.[3]. In their model, they have examined the impact of stochastic effects on the invasion and persistence of vector-borne infection. The disease invasion probabilities are derived using branching process methodology. In [6], Bolzoni et al. extended this model to incorporate multiple hosts.
The aim of the present paper is to study the time to extinction for a stochastic model for vector borne diseases. The mathematical problem of determining the time to extinction has proved to be surprisingly difficult (e.g Nåsell [7]) even for human-to-human transmittable disease, and vector borne diseases are more complicated. The basic reproduction number of an infection is the most important concept in mathematical epidemiology, and is important also when studying properties of the extinction time. This quantity, denoted by can for human-to-human diseases be defined as the expected number of new cases generated by one typical infectious individual in a large susceptible population. For vector borne diseases it is defined similarly, but now this number has to be computed “via” the number of infected vectors. If as is assumed in this paper, then we say that the population is above threshold. Introducing an infective to a susceptible community above threshold may, as is well known, lead to a large outbreak. Many of the models that have been employed in vector-borne settings have been deterministic [2], [8], [9], [10], [11], ignoring the possible importance of random effects. Stochastic effects can be significant during the period immediately after the introduction of the infection into a population [3]: disease invasion is often highly stochastic. Random effects are also the cause leading to extinction from an endemic setting [12]. Deterministic models are hence not of much use when aiming to derive expressions for the time to extinction, because extinction is caused by random fluctuations from the expected (or deterministic) curve.
The rest of the paper is structured as follows. Section 2 is devoted to a brief review of the deterministic Ross model and the corresponding stochastic formulation. Section 3 is devoted to the law of large numbers of the stochastic epidemic process and Section 4 to its diffusion limit which is the limits of the stochastic models. In Section 5 it is shown that in a finite population the time to extinction is exponentially distributed if the process is started in quasi-stationarity. An approximate expression for τ, the mean parameter of the exponential distribution, is derived in Section 5.3 where we approximate the quasi-stationary distribution by the stationary distribution of the limiting diffusion (e.g Nåsell [7], used the same approach for a different model). In Section 6 we studied the influence of different parameters on the expected time to extinction by using numerical illustrations as well as stochastic simulations.
Section snippets
The deterministic Ross-Macdonald model
The Ross-Macdonald model assumes that each host and vector are, at any point in time, either susceptible to the infection or have the infection and are infectious (incubation periods are hence ignored as well as immunity). The host population size is assumed to be constant and its size is denoted by NH. The number of hosts that are infectious at time t is written as IH(t), which means that there are susceptible hosts. The corresponding fractions of the host population are given by IH(t
Law of large numbers for the stochastic vector-borne epidemic
In this section we assume that the populations of hosts and vectors are large. We relabel the numbers of hosts NH and vectors NV by and . Further, we relabel by . We consider a process . We derive a law of large numbers for to show that the stochastic process converges to a deterministic process . To obtain a non-trivial limiting deterministic process, we start the proportion process
Diffusion limit of the stochastic vector-borne epidemic
In the previous section we shown that for large host and vector populations, the normed jump Markov vector process is approximately equal to the deterministic vector function when N → ∞. In the current section we work on the diffusion approximation of the process assuming that . Define the centered and scaled process by
The change in the scaled state variable X during the time interval from t to is
The quasi-stationarity distribution
In the previous section it was shown that when the host and vector population sizes are fairly large then the epidemic process may be approximated by an Ornstein-Uhlenbeck process with a specified bivariate normal distribution as its stationary distribution. This approximation can only be valid before the epidemic goes extinct, that is, before there are no infectious host and vector present (i.e ). In the present section we show that for any N the time to extinction starting in
Numerical illustrations and stochastic simulations
In the current section we compute τ numerically using τ1. First we do this for a range of values of the different parameters in order to study which parameters are most influential. We vary each parameter separately 25% up and down from the type value except for the parameter k. To avoid the situation where the parameter k is varied 25% up from the type value. The type value around which we vary the parameters is given and motivated as follows. As we deal with the time to extinction, we
Discussion
In the present paper we study the time to extinction of a super-critical host-vector epidemic and how it depends on various parameters. Since the corresponding deterministic model stays endemic forever it is of little use for the studied problem. In the paper it is shown that the time to extinction starting in quasi-stationarity is exponentially distributed. An approximate expression for the mean parameter of this exponential distribution is derived from the diffusion approximation of the
Acknowledgements
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