SI Model

Remove all objects from workspace.

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.

si_model = function (current_timepoint, state_values, parameters)
{  
  # create state variables (local variables)
  S = state_values [1]        # susceptibles
  I = state_values [2]        # infectious

  with (
    as.list (parameters), # variable names within parameters can be used 
         { # compute derivatives
           dS = (-beta * S * I)
           dI = ( beta * S * I) 
     
           
           # combine results
           results = c (dS, dI)
           list (results)
         } 
        )
}

Parameters

contact_rate = 10                 # number of contacts per day
transmission_probability = 0.07   # transmission probability
infectious_period = 5             # infectious period

Compute values of beta (tranmission rate) and gamma (recovery rate).

beta_value = contact_rate * transmission_probability
gamma_value = 1 / infectious_period

Compute Ro - Reproductive number.

Ro = beta_value / gamma_value

Disease dynamics parameters.

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

Initial values for sub-populations.

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

Compute total population.

N = X + Y + Z

Initial state values for the differential equations.

initial_values = c (S = X/N, I = Y/N)

Output timepoints.

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

Simulate the SI epidemic.

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

Plot dynamics of Susceptibles sub-population.

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

Plot dynamics of Infectious sub-population.

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

Plot dynamics of Susceptibles, 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, I', main = 'SI epidemic') 
text(30,0.10, "S", col = 'blue')

# 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) 
text(30,0.90, "I", col = 'red')

# remain on same frame
par (new = TRUE)  

Description: This infectious disease model has two (2) compartments: susceptible (S) and infected(I). The disease dynamic is moving from susceptible to infected stage. This disease is long last infection in host and host does not develop any immunity to be in recovery stage. An example of this disease dynamic is HIV. Susceptible host infected with HIV virus will not develop any immunity, it is long last infection in that host.