The 1918 Spanish influenza pandemic claimed around fifty million lives worldwide. Interventions were introduced to reduce the spread of the virus, but these were not based on quantitative assessments of the likely effects of different control strategies. One hundred years later, mathematical modelling is routinely used for forecasting and to help plan interventions during outbreaks in populations of humans, animals and plants.

In two recent linked theme issues of the journal Philosophical Transactions of the Royal Society B, Oxford Mathematician Dr Robin Thompson and Dr Ellen Brooks-Pollock (University of Bristol) commissioned articles about modelling epidemics in humans, animals and plants. While there are differences between pathogens in these different types of host population, there are also a number of similarities between the questions that modelling is used to address.

As an example, when a pathogen first arrives in a host population, mathematical models can be used to assess whether or not initial cases will fade out, or whether they will lead on to a major epidemic. This analysis can in fact be performed using only a quadratic equation!

Probability of a major epidemic

To estimate the risk of a major epidemic when a pathogen first arrives in a population, we denote by $q_I$ the probability that a major epidemic does not occur, starting from $I$ infected individuals. The probability of a major epidemic when the pathogen first arrives in the population (i.e. starting from one infected individual) is therefore $1 - q_1$.

Conditioning on whether the first individual infects another host, or instead recovers, leads to the equation \begin{eqnarray*} q_1 = \text{P} {\rm (infection)} q_2 + \text{P} {\rm (recovery)} q_0. \end{eqnarray*} The variable $q_0$ represents the probability that a major epidemic does not occur starting from $0$ infected hosts. If there are no infected hosts, then a major epidemic will certainly not occur, so $q_0 = 1$. A major epidemic failing to occur from two infected hosts requires no major epidemic from either infected host (and for each of these hosts, the probability of no major epidemic is $q_1$). If infection lineages from the two infected individuals are independent of each other, then $q_2 = {q_1}^2$.

As a result, the probability of a major epidemic starting from one infected host is $1 - q_1$, where $q_1$ is the minimal solution of the quadratic equation \begin{eqnarray*} q_1= \text{P} {\rm (infection)} {q_1}^2 + \text{P} {\rm (recovery)}. \end{eqnarray*} Calculations like these are increasingly used for epidemic forecasting. For example, the probability of a flare up of Ebola in Nigeria was estimated in this way during the 2014-16 Ebola epidemic in West Africa. 

Mathematical models will continue to play important roles for forecasting and guiding interventions during epidemics. To find out more, check out this review article or contact Robin.

(The image above is a visualisation of air traffic routes over Eurasia, which could be used to inform models of global pathogen transmission. Credit: Globaïa, 2011).

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