The SEIR Model

This package solves the SEIR model for the spread of a virus. In the example, the parameters of Wang et al. (2020) are used for the case of Wuhan, China. The model is available in Matlab on Peter Forsyth webpage https://cs.uwaterloo.ca/~paforsyt/SEIR.html. In fact, the R package is based on the work of Peter. You are invited to read his page for more details about the model. The system of differencial equations is: \[\begin{eqnarray*} \dot{S}(t) &=& -\beta(t) \frac{S(t)I(t)}{N}\\ \dot{E}(t) &=& \beta(t) \frac{S(t)I(t)}{N} - \sigma E(t)\\ \dot{I}(t) &=& \sigma E(t) - \gamma I(t) +c\frac{R(t)I(t)}{N}\\ \dot{R}(t) &=& \gamma I(t)- c \frac{R(t)I(t)}{N} \end{eqnarray*}\] where \(\beta(t)=\gamma RZero(t)\) and \(RZero(t)\) is the number of individuals infected by an infectious person, \(\gamma\) is 1 over the recovery period (in days), and \(\sigma\) is 1 over the latent period (days). It is a function of time because the transmission if function of the social distancing policy in place.

The R package

The package is available on R-Forge (https://r-forge.r-project.org/) and can be installed by typing the following command in R:

install.packages("SEIR", repos="http://R-Forge.R-project.org")

The main function is solveSEIR. It creates an object of class ``seir’’. The main arguments are (See Peter’s Matlab code for the source):

In this document, we use the default parameter used by Peter Forsyth. The important factor is \(RZero(t)\), which is determined by the arguments tyee an r0. The default type is ``LIN’’ and the default r0 is:

matrix(c(0, 20, 70, 
    84, 90, 3, 2.6, 1.9, 1, 0.5), ncol = 2)
##      [,1] [,2]
## [1,]    0  3.0
## [2,]   20  2.6
## [3,]   70  1.9
## [4,]   84  1.0
## [5,]   90  0.5

It means that the value of RZero(t) changes at t=20, 70, 84 and 90 (days). The type argument determines how it changes. By default it looks like the following:

Therefore, \(RZero(t)\) is a linear interpolation between the different values. The other option is to have a constant \(RZero(t)\) between periods as in the following graph.

The first \(RZero(t)\) is the default option. If we run the function with all default values we obtain (the print method summarizes the result):

Sol <- solveSEIR()
Sol
## SEIR model for the spread of a virus
## ************************************
## Initial condition
##  Total Cases (thousands): 0.8400
##  Active Cases (thousands): 0.8400
## Condition after 180 Days
##  Total Cases (thousands): 121.4800
##  Active Cases (thousands): 4.1200

The object contains the following elements:

names(Sol)
## [1] "y"     "t"     "h"     "RZero" "Pop"

The element y is an \(nT\times4\) matrix, where \(nT\) is the number of grid points (the closest integer of T/h), with the \(i^{th}\) row being the solution \(\{S(t_i), E(t_i), I(t_i), R(t_i)\}\). The element t is the time grid and \(RZero\) is an \(nT\) vector with the \(i^{th}\) element being \(RZero(t_i)\). The “[” method can be used to get the solution at different points in time:

Sol[c(10,50,100,150)]  
##            S         E         I           R
## 10  10998599  659.1349  1097.600    484.1839
## 50  10977389 4795.2626  9218.391   9437.0239
## 100 10903111 4199.9912 24144.378  69384.5890
## 150 10883650 1193.8311  7305.064 108691.5642

It is possible to use those elements to plot the solution, but the easiest ways is to use the plot method. The plot method gives you options to plot any of the curve:

plot(Sol, "acase")
plot(Sol, "tcase")

plot(Sol, "both")
plot(Sol, "rzero")

You can also plot them one by one:

plot(Sol, "E")
plot(Sol, "I")

Second Wave

As a last example, we can reproduce the second wave case produced by Peter Forsyth on his website. Suppose RZero stays at 0.5 for 30 days and then the government start relaxing the social distancing restrictions. The following is what he proposes:

zbreaks <- matrix(c(0,  20,  70,  84,  90, 120, 240, 360,
                    3, 2.6,  1.9, 1,  0.5, 0.5, 1.75, 0.5), ncol = 2)
Sol <- solveSEIR(T=450, r0=zbreaks)
plot(Sol, "rzero")

The new solution is:

plot(Sol, "acase")
plot(Sol, "both")