Remove all objects from workspace.

r remove (list = objects() ) Load add-on packages - deSolve - contains lsoda function - differential equation solver.

library (deSolve)
## 
## Attaching package: 'deSolve'
## 
## The following object is masked from 'package:graphics':
## 
##     matplot

Function to compute derivatives of the differential equations.

seir_model = function (current_timepoint, state_values, parameters)
{
  # create state variables (local variables)
  S = state_values [1]        # susceptibles
  E = state_values [2]        # exposed
  I = state_values [3]        # infectious
  R = state_values [4]        # recovered
  
  with ( 
    as.list (parameters),     # variable names within parameters can be used 
         {
           # compute derivatives
           dS = (-beta * S * I)
           dE = (beta * S * I) - (delta * E)
           dI = (delta * E) - (gamma * I)
           dR = (gamma * I)
           
           # combine results
           results = c (dS, dE, dI, dR)
           list (results)
         }
    )
}

Parameters

transmission_rate = 0.7                 # transmission rate
infectious_period = 4.5                 # infectious period
latent_period = 1.9                    # latent period

Compute values of beta (tranmission rate), gamma (recovery rate), and delta (exposion rate)

beta_value = transmission_rate
gamma_value = 1 / infectious_period
delta_value = 1 / latent_period

Compute Ro - Reproductive number.

Ro = beta_value / gamma_value

Disease dynamics parameters.

parameter_list = c (beta = beta_value, gamma = gamma_value, delta = delta_value)

Initial values for sub-populations.

V = 0        # exposed hosts
X = 9999        # susceptible hosts
Y = 1           # infectious hosts
Z = 0           # recovered hosts

Compute total population.

N = V + X + Y + Z

Initial state values for the differential equations.

initial_values = c (S = X/N, E = V/N, I = Y/N, R = Z/N)

Output timepoints.

timepoints = seq (0, 50, by=1)

Simulate the SEIR epidemic.

output = lsoda (initial_values, timepoints, seir_model, parameter_list)

Plot dynamics of Susceptibles sub-population.

plot (S ~ time, data = output, type='b', col = 'blue')

Plot dynamics of Exposed sub-population.

plot (E ~ time, data = output, type='b', col = 'purple')

Plot dynamics of Infectious sub-population.

plot (I ~ time, data = output, type='b', col = 'red')

Plot dynamics of Recovered sub-population.

plot (R ~ time, data = output, type='b', col = 'green')

Plot dynamics of Susceptibles, Exposed, Infectious and Recovered sub-populations in the same plot.

# susceptible hosts over time
plot (S ~ time, data = output, type='b', ylim = c(0,1), col = 'blue', ylab = 'S, E, I, R', main = 'SEIR epidemic') 

# remain on same frame
par (new = TRUE)    
 
# exposed hosts over time
plot (S ~ time, data = output, type='b', ylim = c(0,1), col = 'purple', ylab = '', axes = FALSE) 

# remain on same frame
par (new = TRUE)  

# infectious hosts over time
plot (I ~ time, data = output, type='b', ylim = c(0,1), col = 'red', ylab = '', axes = FALSE) 

# remain on same frame
par (new = TRUE)  

# recovered hosts over time
plot (R ~ time, data = output, type='b', ylim = c(0,1), col = 'green', ylab = '', axes = FALSE)

The Susceptible Exposed Infectious Recovered epidemiological model computes and illustrates the relationship between the numbers of susceptile (blue), exposed (purple), infectious (red), and recovered hosts over time (green). The SEIR model shows the evidence of a latent period before an individual becomes infectious. Indiviuals are exposed before they become infectious. Over time, the recovered population increses while all others decrease.