Basic Concepts

Introduction

It is often desirable to be able to predict the spread and path of diseases through a given population. Such knowledge can then be used to help public health officials decide on appropriate response procedures. Other information about the disease may be useful as well, such as the total length in time of the epidemic, the rate of recurrence, the equilibrium state of infectivity, or the effect of the initial infectant on further propagation. In such situations, mathematical modeling can be a useful tool.

Deterministic vs Stochastic

There are two basically different types of models for epidemiology, deterministic models and stochastic models. Deterministic models assume all contact rates between each infectious and susceptible person are equal, and that a new infection will result from each of these contacts at some given rate. Stochastic models, on the other hand, make use of probabilistic factors to determine the rate of infection. Which of these types of models is best suited to a given situation depends largely on the size of the population, as well as its desired degree of precision.

Deterministic

As one might expect from their relative straightforwardness, deterministic models have the longer history of the two groups. They have been in use from the 19th century onwards. An important advance in the development of deterministic models came from Kermack and McKendrick’s study of mouse epidemics. Their work in this field led to the establishment of the Threshold Theorem, which states that “the introduction of infectious cases into a community of susceptibles would not give rise to an epidemic outbreak if the density of the susceptibles were below a certain critical value.” (Ref 1) These are the types of models that we will be focusing on. For a large population, or when little precision is necessary, the difference between the performance of a deterministic and a stochastic model is insignificant. A deterministic model is therefore probably preferable, due to its relative simplicity.

Stochastic

Stochastic models are the younger of the groups. McKendrick came up with the first epidemiological stochastic model in 1926. They did not come into use for another 20 years, however. In terms of application, stochastic models become preferable at small populations, particularly as small as a household, when enough precision is required to outweigh the added complexity of the model.

A Model of childhood disease (chicken pox)

One of the simplest disease models that one can study to some depth is a childhood disease like chicken pox. With a disease like chicken pox, all recovered people become immune to the disease (with insignificant exceptions). Also, the latency between being infected and showing symptoms is very small. One can therefore assume that all people who recover are effectively removed from the population, and that there is no exposed group (which is present in most other disease models), who have been infected but have not yet become infectious.

The other assumptions made for this model are that all susceptibles are equally susceptible, all infectious people are equally infectious, and that the population remains fixed (by balancing birth with death and emigration).

The actual model

The total population (n) is divided into three groups: susceptibles (S), infectious (I), and recovered (R), each of which is a fraction of the total population, such that at all times,

Let rb be the average birth rate (and thefore the average death/emigration rate), and bc the contact rate, which is the average fraction of susceptibles to contract the disease from a single infective.
Then the rate of change of the number of susceptibles in the population is given by

that is the birth rate minus the number of susceptibles who leave the population by death or emigration minus the number of susceptibles who become infected.
Now, let cr be the recovery rate, the average number of infected people to recover per unit time, or 1/c is the mean infectious period. The rate of change of the number of infected people is then

the number of people who become infected minus the number who leave due to death or emigration minus the number who recover.
Finally, the recovered populations rate of change is given by

the number of people who recover from infection minus the number who leave the population due to death or emigration.

Since the equations for S and I are independent of R, the system of equations can be simplified to not include R explicitly, since it can be expressed in terms of S and I:
and

Furthermore, the solutions for these two equations must be nonlinear, due to the presence of the product SI. Finding exact solutions to nonlinear differential equations is particularly difficult. This can be accomplished in two different ways. The first is to use a numerical method, such as Runge-Kutta to approximate the solution. Otherwise, we can linearize the system of equations and find the solutions around the critical points to find an approximate solution. One way to do this is by the nullcline approach.
We set S’ and I’ to zero, and solve the resulting system, obtaining

This is then the critical point where the equilibrium is defined. When shown in the SI-plane, this point gives the solution to the linear portion of the system around which the nonlinear portion oscillates. Since the system is stable, it should converge to this point as
This value of S is also the threshold defined by the Threshold Theorem. In order for I to be positive (nonzero) at equilibrium, this value must be less than one. If I is not positive, then there are no infected people at equilibrium and there is no epidemic. Then, for a small enough ratio of r to c, decreasing the contact rate can have a significant effect, and the severity of the epidemic can be reduced or even eliminated by the implementation of public health measures such as isolation of infectives or vaccination of susceptibles.
This solution suggests that there should be recurring outbreaks of diminishing severity after the initial outbreak. This pattern is not quite what is observed, though. In fact, it is found that the severity of epidemics can depend on a number of other factors, including seasonal changes. The model can then be refined by making b a function of time rather than a constant, to portray a higher susceptibility during winter months than during the summer. By adding this seasonality function, apparently random fluctuations that look chaotic are observed. (Ref 2) The system, though perhaps aperiodic, is stable though, so not actually chaotic.