Historical Background

Mathematical modelling of infectious diseases has a long history; see for example Bailey (1975). The first approach to epidemic modelling is generally taken to be a paper by Daniel Bernoulli on the prevention of an infectious disease, namely small-

pox, by inoculation. The analysis which was performed then by Bernoulli can be found in Daley and Gani (1999, Sec. 1.1). Nevertheless, as Bailey (1975) points out, it was another hundred years before the physical basis for the cause of the infectious disease became well-established. One of the earliest studies of epidemic modelling was introduced in a paper by Hamer (1906). The author, assumed that the probability of a new infection in the next discrete time step is proportional to the product of the number of susceptibles and the number of infectives. A few years later, Ross (1916, 1917a,b) translated this “mass action principle” or “homo- geneous mixing” to the continuous time setup. The first complete mathematical model for the spread of an infectious disease which received attention in the lit- erature was a deterministic one, introduced by Kermack and McKendrick (1927). We shall briefly describe the main features of this model in Section 2.1.2.1.

2.1.2.1 Deterministic Models

First, consider a closed population (i.e. there are neither births nor deaths nor immigration) of size _N +a and assume that at time t = 0 there are a initially infected individuals. Such an assumption of a closed population is reasonable for epidemics which occur in a time relative to the change in the population. At any given point time each individual i is in one of the three states: i) Susceptible ii) Infected iii) Removed.

The only transitions which we allow, are the following: from susceptible to infected and from infected to removed. Therefore an individual is calledsusceptible if they do not have the disease but are susceptible to infection,infectedif they have got the disease and able to infect other (susceptibles). We assume that at the end of their infectious period, they become removed either by death or immunity, i.e. cannot infect any other susceptibles. In general they do not take part in the epidemic any longer.

S

I

R

Figure 2.1: The three transition states of an individual.

uals respectively at time t_≥0. It is sufficient for describing the epidemic to keep track of (Xt, Yt, Zt) since for alltthe following equality holds: Xt+Yt+Zt =N+a. The model is then defined by the following set of differential equations:

dXt dt = −βXtYt dYt dt = βXtYt −γYt dZt dt = γYt (2.1)

with initial state (X0, Y0, Z0) = (x0, y0,0). The factorβXtYt is a crucial non−term indicating that infections occur at high rate only when there are many susceptibles and infectives. It follows from the above equation that dX/dZ =−(β/γ)X. So,

Xt =x0exp{−θZt}

and hence

Yt =N −Zt−Xt =N −Zt−x0exp{−θZt}

where θ =β/γ. Kermack and McKendrick (1927) showed that the number of in- fectivesYt is increasing unlessx0 >1/θ. That is, there will be a growing epidemic. This observation is known as the threshold result, i.e. different behavior of the epidemic will occur depending on whether x0 > 1/θ or not. Another important

observation is that as t _{→ ∞} then Zt → Z∞ < N where Z∞ is the solution of

Z = _{N −} x0exp_{−θZ_}. In other words, this a very important property which states that not everyone becomes infected. Summarizing, we should note that many of the epidemic models used today have this general epidemic model as

their basis.

2.1.2.2 Stochastic Models

Stochastic epidemic models were also being developed early in the 20th _century along deterministic ones. McKendrick (1926) was the first to propose a stochastic version of the general epidemic model. However, at that time, there was more interest in discrete₋time models and this model did not receive much attention. A model which attracted more attention that time was the chain-binomial model proposed of Reed and Frost in lectures in 1928 (Wilson and Burke, 1942, 1943). In the standard Reed−Frost model, given the numbers Xt, Yt of the susceptibles and infectives at time t,Yt+1 has a binomial distribution with indexXt and mean

Xt(1−p)Yt and Yt+1 =Xt−Xt+1. In consequence, individuals are assumed to be

infective for a single time unit and in that time they can make an infectious contact, independently and with probability p, with any member of the population who is susceptible. This means that the number of potentially infectious contacts scales with the population size. Since the Reed−Frost model is only usually applied to small populations, this is not a problem. However, there have been various modifications to the Reed₋Frost model which refer to the number of contacts and the probability that a susceptible escapes the infection by a single infective (see for example, Dietz and Schenzle, 1985).

The stochastic models began to draw more attention and be analyzed more ex- tensively in the late 1940’s. Then, Bartlett (1949) studied the stochastic version of the model introduced by Kermack and McKendrick (1927) and since then, the amount of effort put into modelling infectious disease has blown out.

2.1.2.3 Deterministic or Stochastic?

Disease spread is an inherently stochastic phenomenon and there are a number of arguments why a stochastic model should be preferable to a deterministic one.

Real life epidemics, can either go extinct with a small number of individuals who became infected during the outbreak, or end up with a significant proportion of the population having contracted the disease. It is therefore, only stochastic models that can capture this behavior and the probability of each event occurred. Moreover, stochastic models allow us to intuitively define them since they can naturally capture the infection process between different individuals.

Isham (2005) claims that the general view in the past seems to have been that a deterministic model gives an average behaviour of a corresponding stochastic system at least asymptotically and that for large populations using a stochastic model, which is more difficult to analyse than a deterministic one, there is little to be gained. However, it is now widely accepted that both deterministic and stochastic models have their strengths and can accommodate good understanding of the underlying process (see for example Isham, 2005). We should note that it is often the case to observe a disease outbreak with an atypical behavior even in the case for large populations. Nevertheless, even if they show an average behaviour care needs to be taken when we are interested in prediction (Isham, 1991, 1993). Isham (2005) also indicates that one of the most noticeable changes of the last fifteen years has been the increased acceptance by biologists of the important role that mathematical modelling has to play in providing solutions of many of their most difficult problems. Moreover, they noticed that such models need to incorporate intrinsic stochasticity in many ways.

Similarly, the stochastic effects become more important when we are interested in determining effective control strategies or answering questions regarding recurrence and extinction of infections. It is known (see for example, Isham, 2005) that with a deterministic epidemic model with open population (i.e. allow for births, deaths, or/and immigration) if in the beginning of the epidemic R0 > 1, the infection never completely dies out. In contrast, a stochastic epidemic model may fade out completely when it reaches a state where there is a single infective and moreover it is in any case to die out eventually unless there is an external source of infection

(Isham, 2005). Taking into account the above arguments, in this thesis, we will only focus on stochastic epidemic models.