Oxford Mathematician explains SIR Travelling Wave disease model for COVID-19 (Coronavirus)

The SIR model is one of the simplest ways to understand the spread of a disease such as COVID-19 (Coronavirus) through a population. Allowing the movement of populations makes the model slightly more realistic and results in ‘Travelling Wave’ solutions.

In this video, Oxford University Mathematician Dr Tom Crawford explains how including population migration modifies the original SIR model. He then goes on to use the results of the model to answer two important questions:

  1. How fast will the disease spread?
  2. How severe will the epidemic be?

The answers to these questions are discussed in the context of the current COVID-19 (Coronavirus) outbreak. The model tells us that to reduce the impact of the disease we need to lower the ‘contact ratio’ as much as possible – which is exactly what current social distancing measures are designed to do.

Watch the first video on the basic SIR model here.


  1. Hello Tom, I am a student and I have study one of the epidemic model, when I found the speed of the travelling wave solution, I plot c with different parameter . when I plot c vs. beta(transmission rate) I got that with very very small value of beta the value of c(speed of the infectious) increase how I can explain the reason biologically. Thank you a lot


    • The wave speed c represents the speed at which the ‘wave of the epidemic’ passes through the population. If the transmission rate is very small, then the outbreak will die very quickly (since very few infections are being passed on) and so the wave speed will be very fast (because the epidemic ends very quickly).


  2. I appreciate your response, I have the same notice for the turning rate of the infected. For very very small value of turning rate of the infected the speed c increase. Is it with the same reason? Thank you Tom.


  3. Hello Tom, I am afraid to say that , I am still confused. When I plot c vs. alpha(recovery rate) , the value of c decrease with increasing the rate of recovery rate. Is it because when the population recover, it will be possible to be infected a gain and that’s why the wave speed will decrease(because the epidemic persist). Could you please clarify to me if I misunderstand? Also the speed of infected population and wave speed have positive relation, why?


  4. With all models you need to make some deductions for yourself as to how the mathematical results relate to the real-world. Basically, we convert the real-world problem into a series of equations, which we then solve/understand mathematically, and then interpret those results back in the context of the original problem. I honestly don’t know enough about disease transmission from a biological perspective to be able to answer all of your questions. If the maths is correct and it tells you a certain relationship exists, then the next step would be to talk to a biologist (or to do some research yourself) to see if that relationship makes sense. Mathematical modelling works best with an inter-disciplinary approach!


  5. Hello Tom, I have question regarding the travelling wave. I got two steady states with three variable , they are stable in one region and on other region one stable and the second one unstable . I want to draw the travelling wave for the steady (N,0,0) and (a,0,b) where first component is susceptible, second one is infective and third one is recovery. From N to a (I did) for 0 to b (I did) for some specific values. But how I can draw the third one where in both steady states are in free-disease(I mean 0 to 0). Thank you for your nice explanation.


  6. I’m a little unsure as to what’s going on here as in the model we only solve for S and I and so the phase plane is 2D. The dynamics of a 3D phase plane are very different as trajectories are no longer trapped which leads to all kinds of different (and much more complex) behaviours.


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