# The Mathematics of Epidemiology

The answer is that they come into contact with an infected person, and the infection spreads.

So two different factors are involved.

One is the rate at which a susceptible person comes into contact with an infected person, and the other is the odds of an infection spreading, upon contact.

The absolute most basic version of the model does not include births and deaths, so the arrows would be removed from the above depiction.

Generally the equations are adjusted for the total size of the population, which simplifies the equations a lot.

The calculations are as follows.

The fraction of susceptible people is decreased as new infections arise at a rate that’s a product of some constant (representing the odds of contact and transmission) and the product of both susceptible and infected.

We can see that if there are very few infections or very few susceptible people the rate of infection is low.

It’s when there are a fair amount of both infected and susceptible people that the pathogen spreads its most rapidly.

2.

The infected population increases with as new infections form, but declines as individuals recover, at some rate.

3.

The recovered individuals end up in the third compartment.

We can see from these three equations that there are a number of ways to reduce the impact of an infection.

One is to ensure that there are few susceptible people.

But we can also reduce the chances of a susceptible people coming into contact with an infected individual.

And our natural aversion to disease helps.

That’s one concern about asymptomatic infections that are still contagious: it can vastly increase the contact rate between individuals.

We can see from these three equations that there are a number of ways to reduce the impact of an infection.

One is to ensure that there are few susceptible people.

But we can also reduce the chances of a susceptible people coming into contact with an infected individual.

And our natural aversion to disease helps.

That’s one concern about asymptomatic infections that are still contagious: it can vastly increase the contact rate between individuals.

More Complicated ModelsIt’s possible to add more and more compartments to the basic SIR model, and there are many variations that exist, including ones that take into account vaccination and waning immunity.

But as more and more compartments are added, the difference/differential equations become more and more complicated and eventually it becomes very difficult to draw any conclusions at all.

That’s where we need another approach.

And for that, we need graph theory.

Graph TheoryI’ve had many people engage me in discussion on topics of epidemiology, without even understanding the basics of graph theory.

When people think of the word “graph” they usually think of a plot of data on a Cartesian system: an x axis and y axis plot of data.

But there’s another meaning to the word “graph” in mathematics.

A graph is a collection of vertices, which can represent a lot of different things, including people, computers, geographic locations, and so on, along with a collection of edges which describe how those vertices are connected to one another.

Graphs allow us to study the actual distribution of people and how they interact with one another.

And so graph theory is important because we can use it to construct models of epidemiological dynamics that don’t assume homogeneity.

Here’s an example fictional social network graph.

We can see that there are tons of edges/nodes along with a lot of edges/connections, but there also seems to be two distinct groups that are only connected by a few nodes.

This situation would arise in cases of two fairly isolated communities that only interact once in a while, perhaps through diplomatic relationships or trade.

Fictional social network (CC-BY 3.

0)Probability Theory & Stochastic ProcessesProbability theory and stochastic processes are areas of study that allow us to understand the real world better.

And it really helps us understand what R0 means.

For instance, an R0 above one means that an epidemic is likely to occur, while an R0 less than one means that the epidemic will tend to die out.

It’s possible that a pathogen that isn’t all that virulent, and with an R0 less than one, will still result in large scale infections.

But it tends not to happen.

And it’s the long term tendencies in epidemics that we want to control with vaccines.

It’s also important to understand that the basic reproduction number is not a rate.

It is the estimated average number of infections generated upon an initial infection introduced into the population.

How multiple infections influence the rate of infection is more complicated, as indicated by the SIR model.

The proportion of infected to susceptible individuals actually changes the instantaneous rate of infection.

Computer SimulationWhile not exactly part of the mathematical discussion, computers simulation of epidemics allows us to get a lot of information from very complicated models.

With modern computers, we can produce very robust simulations, with lots of individuals that have different properties, including varying degrees of immunity, different social behavior, etc.

We can run dozens, and even hundreds or thousands of simulations, and start to get an idea of what kind of situations would arise in the real world, in ways that we could never achieve simply by looking at highly simplified models.

Further ReadingIs the Antivax Movement Being Used as a Scapegoat?Vaccination is important, but the medical community is too quick to blame the antivax movement for recent measles…medium.

comFurther Discussion on the Measles OutbreakWhat’s really going on here?medium.

com.