The logistics of a pandemic

There's much been said about the need for vaccination to reduce the spread of Covid through the population. Professor Doherty has just released a very influential and convincing report on the sorts of vaccination levels we need to achieve and it seems to have pushed our Federal Govt into some action, at last.

However, I haven't seen anyone explain how modelling a pandemic or any other type of population dynamics is done. So I thought I'd give it a go. Warning, it involves maths, but it's pretty straightforward.

There is a very well known mathematical formulation called the logistic function or logistic equation. It's taught in first year calculus in its most basic form. It's a very simple differential equation:dN/dt=rN(K-N)/K. The rate of change of the population of virus (N) at any particular time(t) is defined by a function of the rate of maximum population growth (r), the population of virus at that time (N) and the carrying capacity (K). When dN/dt is positive, the population of virus is growing, when it's negative, that population is falling.

The rate of maximum population growth, for a virus, is when there is a new freely-mixing population of hosts that hasn't been exposed to the virus. This is pretty obvious. The carrying capacity is a little more complex. Farmers understand that any piece of land has a maximum stocking level, that varies with seasons and weather patterns. If that's exceeded, some stock will not have enough food to survive. That's its carrying capacity. For a virus, the carrying capacity is affected by various factors too, including how dense the host population is, how freely they mix together, any measures taken to prevent the virus from moving between hosts and the number of people within a population who have an immune response that reduces the virus's ability to infect them and potentially turn them into a vector for the virus to infect others.

This equation is the basis of all of the epidemiology. We know that the virus requires contact between infected hosts and new, uninfected people. We know what mechanisms it uses to spread between hosts. We know roughly what proportion of people who are exposed will become infected. And we know that if we give people's immune system a boost with a vaccine we can reduce the number of hosts.

We also know that if no measures are taken, the virus will rapidly infect some significant proportion of the population (the natural carrying capacity in this case) and then, as the virus infects more and more people, the rate of change in new virus cases will go from positive (growing) to negative (falling) due to acquired immunity reducing its ability to infect new hosts. Eventually, it will no longer be an epidemic disease, but will become endemic, with a small number of hosts who don't spread it much. This has often been referred to as "herd immunity", once again drawing on the experience of farmers in combatting highly infectious diseases in their herds and flocks. However, viruses don't remain unchanged, they mutate, creating new variants that previously trained immune systems are unable to fight and so the cycle repeats. We know that Corona viruses generally are prone to rapid mutation and we've seen Covid-19 mutate several times, with the current Delta variant much more effective at infecting new hosts (the rate of maximum population growth - r in the equation above - is higher), so the rate of change in the population of the virus is bigger and more positive.

So, the advice is to:

• socially distance: that reduces the density of the population and limits the availability of new hosts.
• wear a mask: that reduces the ability of the virus to move between hosts
• get tested and socially isolate if positive: that actively reduces the carrying capacity.
• get vaccinated: that increases the number of people who have an immune system that can recognise and fight the virus and once again reduces carrying capacity

I hope this helps to understand a little of the maths behind the management of the epidemic. It's not guesswork. The logistic equation can take much more complex forms and I'm certain that the modelling being used by decision makers does just that, but the basic form of the equation is beautifully simple and clear.

No magic required.