Differential equations epidemic models

Here we are introduced to the mathematical background to the theory of epidemics. This allows us to predict if an epidemic can occur. The regular recurrence of epidemics, and the similar shapes of consecutive epidemics of a disease can be modelled through mathematical equations. If this way of thinking about infectious diseases in terms of models seems somewhat unfamiliar, one should only consider all the results of similar models that we meet daily in other circumstances. One of the most complex examples is given by the weather forecasts, which are made using supercomputers, where large systems of equations are linked to huge libraries of previous meteorological patterns. Although mathematical models presents an air of exactitude, no model will be better than the assumptions on which they were built. It is sometimes difficult to see the exact assumptions underlying the model, but in any good publication they should be stated clearly. The models are useful not only for predictions, but they can aid in understanding complex contexts, recognising which factors are the most important determinants of the dynamic of the epidemics in the population and which we should study better and try to measure more exactly.The models also let us evaluate control policies and also are good tools for teaching infectious disease epidemiology.

For building such simple models, the population is divided into distinct classes according to the health of its members. A typical subdivision consists in susceptible (S), currently infected and infectious (I), and removed class (R) of individuals who can no longer contract the disease because they have recovered with immunity, have been placed in isolation, or have died. If the disease confers a temporary immunity to its victims, individuals can also move from the third class to the first. Time scales of epidemics can vary greatly from weeks to years. Vital dynamics of a population (the normal rates of birth and mortalities in the absence of disease) can have a large influence on the course of an outbreak. Whether or not immunity is conferred on individuals can also have an important impact. Many models using the general approach with variations on the assumptions have been studied.

The first model was proposed by Kermack and MacKendrick in 1927 to explain the rapid raise and fall of cases observed frequently in epidemics such as the great plague in London (1665-1666), the cholera epidemic in London (1865) and the plague in Bombay (1906).The model assumes:

1. the population is fixed, so that no one enters and no one leaves or dies
2. the incubation period is zero, and
3. the duration of infectivity is just as long as the clinical disease

The illness is brought into the group by someone who broke the first rule temporally, and contracted it outside.

Before the first case is infected, S is obviously equal to one, since everyone is susceptible , and I and R are both equal to zero. As the epidemic spreads, S will decrease and R will increase. Intuitively, I should first increase, and then decrease.

We will set up three equations to show how these three categories in the population will change over time. In doing this, we will use the time derivative, this means that X is presently changing. If there is a minus sign in front of the derivative, this means that X is presently decreasing, otherwise it is increasing.

At any time during the epidemic, the three equations will be:
 

The first equation shows that the number of susceptible is decreasing. The actual rate of decrease is, where:
represents the proportion of contacts between susceptible and infectious individuals if all possible contacts take place completely at random;

c is the average number of contacts that has any person in the population, and

b is the risk of transmission in each of these contacts.

In the second equation, the number of infected increases at the same rate that the susceptible become infected, but after time D, an infected person becomes immune and is taken out of the infectious group.

In the third equation, we show that the people become immune at the same rate they leave the infectious group.

JAVA demonstration of a mathematical epidemologic model.
 

References

Kermack , W. O., McKendrick, A.G.(1927) Contributions to the mathematical theory of epidemics (Part I) Proc. Roy. Soc., A, 115: 700-21.