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Two Phases Sir Model

Calculated Results
Peak day49
Infected at Peak Day2351
Phase 1 Basic Reproduction Number4.000
Phase 2 Basic Reproduction Number2.000

The Three compartments of the SIR Model

SIR model consists of three compartments:

  • S for the number of Susceptible.
  • I for the number of Infectious.
  • R for the number of recovered that includes both immune and death.
Each compartment is considered as function of time t in number of days, denoted as S(t), I(t) and R(t). The model will project the value of these three functions on given time t. The model is a reasonable prediction for infectious diseases where recovery confers lasting resistance, such as Coronavirus, measles, mumps and rubella.

Transition Rates in SIR model

The SIR model will need two inputs that represent the two transition rates:

  • Contact Rate: is a transition rate from susceptible to infectious that is defined as the average number of contacts per person times the probability of disease transmission in a contact between a susceptible and an infectious subject.
  • Recover Rate: is a transition rate from infectious to recovered that is defined as the reciprocal the duration in number of days for the infection duration.
The two phases SIR model will break the model into two phases to reflect the fact that the government will step in to control the epidemic when the virus proved to be highly contagious. The phase one contact rate will normally be much higher than phase 2 rate due to the control and isolation of infected subjects. Therefore, using the two phase model will result better projection than one single contact rate model.

Basic Reproduction Number

Basic reproduction number or basic reproduction ratio is defined as the ratio of the contact rate to the recover rate. This number reflects the number of contacts by an infected individual with others before the infected has recovered. If this number is greater than 1, the infection will spread in a population. The higher the number is, the harder to control the epidemic. For example, SARS has a basic reproduction number in the range of 2-5.

Wuhan Coronavirus

new corona virus, COVID-19, is highly contagious and caused world wide pandemic. The basic reproduction number for COVID-19 is estimated in the range of 2.13 to 3.11. The average number of days of the infection is about 14. Using that number we can input a recover rate at 0.07. Using 2.62 as the basic reproduction number, then we can input the contact rate as 0.182. Then you can input different values of the total population into the model to view the peak and number of infectious by day. By trying different population values, you can get an rough estimation of the total population infected.

Using the two phases SIR model, we can set the first 45 days as the uncontrolled period. Adding the 10 days of incubation period we can input 55 days for phase 1. In this pahse, there is no actions taken to control the infection. Therefore, the contact rate is much higher. After phase 1, the Chinese government took strong actions to control the spreading of the virus and that reduced the contact rate significantly. In this model, we input 0.28 for phase 1, 0.14 for phase 2 as the contact rate to reflect that fact. The model tells us that 75000 total infection is a reasonable projection of the total population and peak at day 49, at the peak day, the net infected number is 2351.