Modelo SIR

In past classes we saw that a system of ODEs to describe the spread of an epidemic is given, in its simplest form, by the SI model (susceptible S and infected I):where the parameter β is a measure of how fast the epidemic spreads. For this system we find that the exact solution (taking as initial condition I(0) = 1 ) is given by:In this class and the next we will consider models closer to what might be a real situation. Let us now consider that an infected individual may eventually recover and this leads to think of a third population compartment that we will call R of recovered individuals. Let us assume that the growth rate of the population of recovered dR/dt is proportional to the number of infected I and this implies a modification in equation [2] and add an extra equation, resulting in the following 3 equations:Where in this case μ represents the rate of recovery of individuals. Here the new term μ / is subtracted in equation [6] because the recovered individuals imply a reduction in the growth of the infected population, while equation [7] is the proportionality relationship between the population growth R and the number of infected I.This system is already interesting enough that with it we can explain some of the things that are happening today regarding the spread of SARS-COV-2. So the first thing we will do is to solve it numerically with RK4 just as we did with the Lorenz system in the last class. For this we can use much of the notebook from the last class. For example the function rk4vec() remains identical and we only need to modify the class that contains the equations of the dynamic system, for which case we will have the following:With this class already defined it is easy to run a simulation of SIR in a similar way as we did with the Lorenz system and also make a visualization:In this code cell we consider a situation where the recovery rate μ is greater than the rate of spread of the epidemic β and we take an initial condition with a population of susceptibles of 1000, a single contagion and zero people recovered. The result is seen in the following graph:The solution of this system, under the conditions and parameters previously indicated, shows that:The curve of susceptibles always decreases with time, the same as with the SI system.The curve of infected has a bell shape where there is a maximum point that indicates the most critical moment of contagion in the population. It is important to anticipate this peak because it can exceed the capacity of the health system if the number of simultaneous infectees is very large.The recovered curve always increases over time.These three conditions reflect a situation where the epidemic eventually fades over time and we say that the population has reached what is called Herd immunity or herd immunity, which is a state where individuals have already recovered because the vast majority have managed to acquire immunity against the virus that caused the epidemic.When governments say that we must flatten the curve so as not to overload the health system, they are referring to the infected curve that we have just shown previously, we know that the most popular measure to control this has been through quarantine and social distancing and this can be modeled in a certain way through the constant β spread of the epidemic. That is to say these social distancing policies can be quantified with a smaller value of β, let's see a comparison of how a smaller value of β affects the maximum peak of infected.And with this final graph we evidence that a reduction in the rate of spread has the effect that the curve becomes wider but so also the maximum peak of contagions is reduced, we place as an example a horizontal line that represents the theoretical capacity of a health system just to show that the reduction in the rate of spread translates into that flattening of the curve that eventually implies that a health system is not saturated in consideration.To conclude this class, it is important to note that the SIR system is just another approximation to a real epidemic situation. These equations do not pretend to predict with complete certainty a given situation like the one we are currently experiencing with Covid-19, but the intuition we can develop with them allows us to understand fundamental aspects of the dynamics of these systems and thus understand the effect of social distancing policies. In the next class we will see other interesting details of this system and an additional improvement that we will leave as the final project of the course.

Taller de Aplicación de Modelos Numéricos

Modelo SIR