# Epidemics

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Dr. Angela McLean accepted the invitation on 10 July 2007 (self-imposed deadline: 10 October 2007).

This article will briefly cover: observed patterns of epidemics of infectious diseases and the underlying processes thought to generate those patterns.

## The SIR Model

Figure 1: Animation showing the damped oscillations of the SIR model with births and deaths. Note that the susceptibles and infected populations are 90 degrees out of phase, with the infecteds lagging the susceptible.[Parameters $$R_0=17, \gamma=1/13, \mu=1/(70 \times 365)$$

## Macroparasites

Macroparasites are, as the name suggests, much larger parasitic organisms such as helminths, flukes or other worms. The mathematical modelling of this type of infection requires several extra components to be included in the standard model. Firstly, worm burden (number of worms within a host) must be modelled as is important for both transmission and severity of the disease. Secondly, transmission is more complex, many macroparasites have free-living stages outside the host and hosts can be infected multiple times which will increase the burden. Finally, worm burden shows a highly over-dispersed distribution with a few individuals having incredibly high numbers of parasites. We will not discuss these elements in any more detail, but note that all of these factors have parallels in the study of micro-parasites.

## Stochastic Dynamics

Stochastic dynamics refers to the situation where noise is incorporated into the basic dynamics. Two main approaches have been explored, one is to add suitably scaled noise to the standard differential equations, the other is to adopt an individual-level, event-based approach. We discuss both of these below:

### Stochastic Differential Equations

Consider the following modification to the standard differential equation for the level of infection within the population: $\frac{dI}{dt} \; = \; \beta S I \, - \, \gamma I \, + \, f(S,I)\xi$ where $$\xi(t)$$ is a source of Gaussian noise (see Stochastic Dynamical Systems for more information about this type of system). The function $$f$$ determines how the noise scales with the population dynamics, several common forms are:

• $$f$$ is a constant. This is the simplest form of noise that can be investigated, which can lead to unrealistic negative population values.
• $$f(S,I) = \sqrt{\beta S I + \gamma I}\ .$$ This form of noise mimics event-driven (demographic) stochasticity, and is derived from the fact that events form a Poisson process. The variance of the noise for each event (infection and recovery) is equal to the mean rate, and the variance of the two noise terms add together.
• $$f(S,I) = \sqrt{ v_{\beta} (S I)^2 + v_{\gamma} (I)^2} \ .$$ This form of noise corresponds to external parameter noise, due to factors external to the model such as temperature or humidity. Here, $$v_{\beta}$$ and $$v_{\gamma}$$ measure the variance in the transmission rate ($$\beta$$) and recovery rate ($$\gamma$$).
• $$f(S,I) = \sqrt{k_1 (\beta S)^2 I + k_2 (\beta I)^2 S + k_3 (\gamma)^2 I}\ .$$ This final formulation is due to heterogeneities in the parameters associated with individuals, and the variation in the mean values. The parameters $$k_1\ ,$$ $$k_2$$ and $$k_3$$ measure the heterogeneity in infectivity, susceptibility and recovery rates.

Which of these forms of noise is used (or even a combination of them is possible) is largely dependent on the problem being modelled, and the expected source of noise.

### Event-based Stochasticity

In recent years event-based (demographic) stochasticity has been used increasingly by applied researchers. This dominance of event-based stochasticity over stochastic differential equations can be attributed to one main factor: event-based models respect the individual nature of the population, such that the population is composed on an integer number of susceptible and infected individuals. For the simple SIR model two events can occur:

• Infection This occurs with a probabilistic rate $$\beta X Y/N$$ and leads to integer changes in the population variables $$X \to X - 1$$ and $$Y \to Y + 1\ .$$
• Recovery This occurs with a probabilistic rate $$\gamma Y$$ and again leads to integer changes in the population variable $$Y \to Y - 1$$ and $$Z \to Z + 1\ .$$

We can therefore calculate the rate that any even occurs is simply the sum of all the individual rates$\beta X Y / N + \gamma Y\ .$ Making the standard assumption that events are Poisson, the time to the next event (whatever it might be) is then: $\delta t = \frac{ - \log(RAND_1) }{\beta X Y / N + \gamma Y} \ .$ where $$RAND_1$$ is a randomly number, uniformly distributed between 0 and 1. This calculation tells us the time to the next event, but not which event it is; for the simple SIR with two events this can be done with relative ease. Picking a second random number, $$RAND_2\ ,$$ (again uniformly distributed between 0 and 1), then the event is infection if $RAND_2 < \frac{\beta X Y / N}{\beta X Y / N + \gamma Y}$ in which case $$Y$$ is increased by one and $$X$$ is decreased by one. Otherwise we assume the event is recovery and we decrease $$Z$$ by one and increase $$Z$$ by one. This process can now be repeated for event after event, increasing the time each time. As such this provides a fast and robust means of simulating the dynamics of infection in a population and accounts for the chance nature of transmission and recovery.

### Implications of Stochasticity

Including stochasticity in models has several effects not evident in deterministic models:

1) The most obvious is that variability enters the simulations. Therefore where the deterministic models may predict an equilibrium prevalence of infections, stochastic models will display variation in both the number of suceptible and infected individuals -- what is more, there is generally a negative covariance between infected and susceptible populations. This variablity means that multiple stochastic simulations are generally required, and there results must be treated statistically.

2) The second element is that population size is important, we can no-longer simply rescale the parameters. It is generally found that large population sizes experience relatively less stochasticity and therefore are closer to the deterministic ideal.

3) Given that the deterministic approach to equilibrium is oscillatory, stochasticity can often excite oscillations. These oscillations, which can be substantial in small populations, can be distinguished from seasonally forced oscillations by the fact that they are not locked to any regular (multi-)annual cycle.

4) Finally, given that transmission and recovery are stochastic processes, there is always the chance that the infection will die-out -- even though it is deterministically stable. This phenomena has been well documented for measles, where populations below 300,000 suffer regular stochastic extinctions (followed by re-introduction of infection from elsewhere) while in populations above 500,000 measles appears to persist.

It is still an open challenge to understand how the stochastic nature of epidemics can be utilised in the more effecient control of infections.