SIR Model

In order to describe the dynamics of the contagion in a population, the SIR (Sus-ceptible, Infected, Recovered) model has been widely used (Kermack and McK-endrick 1927), or SEIR, which introduces the Exposed. In the current health crisis, these models are being used to model the COVID-19 pandemic. This document uses this SIR model expressed in its differential form to describe the dynamics of infections in Chile, which will begin with an S(0) level of initial susceptibility, i.e. those who are not immune to the virus. Then, from an initial infected,the virus will spread at a certain rate generating a level of active infected in each period t (It). In this way the infected can remain ”infective” for 1/gamma days.

Where Beta corresponds to the daily transmission rate and gamma to the period of infection, Ro=beta/gamma and the basic number of reproduction, i.e. the number of people infected by an individual when the whole population is susceptible and Rt=Ro*St-1/N is the effective number of reproduction, similar to Ro, this corresponds to the number of people infected by an individual when a St-1/N fraction of people are susceptible.

Data

The model uses data from Covid-19 in Chile published by the Ministry of Science. Specifically we focus on the Active Infected and the recovered

Methodology

The parameters of the SIR model presented in equation (1) are estimated by maximum likelihood, modelling the errors using a Poisson distribution.

The pomp object in R is the following:

pomp(
  data=subset(mdata,select=c("Infectados","dia")),
  times="dia",t0=0,
  skeleton=vectorfield(
    Csnippet("
      DS = -beta*S*I/N;
      DI = beta*S*I/N-gamma*I;
      DR = gamma*I;")),
  rinit=Csnippet("
      S = S_0;
      I = I_0;
      R = N-S_0-I_0;"),
  paramnames=c("beta","gamma","S_0","I_0", "R_0", "N"),
  statenames=c("S","I", "R")) ->  covidA1

Fitting SIR using Maximum likelihood

Normal errors

First, we estimated by normal errors distribution.

## NULL

Poisson errors

Estimating one parameter: beta

Estimating two parameters: beta y p

We can get an idea about uncertainty and, in particular, get confidence intervals by comparing the probabilities we get with different values of the parameter. The probability profile for each estimate is built, for this the value that maximizes the likelihood value is outlined over the other parameters.

Confidence interval

In the same way, it should be taken into account that the estimate obtained by the MLE for the number of active infected is a probability distribution, so we take several simulations to model the mean, and the 5 and 95 percentiles.

For different gamma values:

Bootstrapping

Because beta is a transmission rate, which varies according to the different measures implemented, for example, quarantines, and the social distancing that is imposed by the government or voluntary for fear of contagion, determines variations in this parameter over time. Therefore, estimates are made in moving windows. That is, the procedure described above is applied to a subset of data equal in length to the window, which varies from 14 to 28 days.

Parameter Estimation on a Stochastic SIR Model

Alejandra Cruces

06-07-2020