Chapter 9
General SIR Model and Its Exact
Solution
In order to simulate different epidemic waves, we will propose a generalized SIR
model which assumes constant value of its parameters for every separate wave and
present an analytical solution of corresponding equations.
In previous chapters, we have demonstrated that dynamics of COVID-19 pandemic can be very different. Changes in quarantine conditions, human behavior,
pathogen activity, weather, etc., can cause changes in the course of the pandemic,
namely the so-called epidemic waves. The mathematical simulation of these waves
needs development of some criteria of the wave detection presented in previous
chapter. Here, we will modify the SIR model to make it applicable for simulations
of different epidemic waves. Some results have been already published in [89, 94].
We suppose that the SIR model parameters are constant for every epidemic
wave, i.e., for the time periods: t
Ã
i
t t
Ã
i þ 1 ; i ¼ 1; 2; 3; . . .. Then for every wave,
we can use the equations, similar to (4.1)–(4.3):
dS
dt
¼ Àa i SI;
ð9:1Þ
dI
dt
¼ a i SI À q i I;
ð9:2Þ
dR
dt
¼ q i I:
ð9:3Þ
As before, S is the number of susceptible persons (who are sensitive to the
pathogen and not protected), I is the number of infected persons (who are sick and
spread the infection; please do not confuse with the number of still ill persons, so
known active cases) and R is the number of removed persons (who no longer
spread the infection; this number is the sum of isolated, recovered, dead and
infected people who left the region). Parameters a i and q i are supposed to be
constant for every epidemic wave.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
I. Nesteruk, COVID-19 Pandemic Dynamics,
https://doi.org/10.1007/978-981-33-6416-5_9
127
General SIR Model and Its Exact
Solution
In order to simulate different epidemic waves, we will propose a generalized SIR
model which assumes constant value of its parameters for every separate wave and
present an analytical solution of corresponding equations.
In previous chapters, we have demonstrated that dynamics of COVID-19 pandemic can be very different. Changes in quarantine conditions, human behavior,
pathogen activity, weather, etc., can cause changes in the course of the pandemic,
namely the so-called epidemic waves. The mathematical simulation of these waves
needs development of some criteria of the wave detection presented in previous
chapter. Here, we will modify the SIR model to make it applicable for simulations
of different epidemic waves. Some results have been already published in [89, 94].
We suppose that the SIR model parameters are constant for every epidemic
wave, i.e., for the time periods: t
Ã
i
t t
Ã
i þ 1 ; i ¼ 1; 2; 3; . . .. Then for every wave,
we can use the equations, similar to (4.1)–(4.3):
dS
dt
¼ Àa i SI;
ð9:1Þ
dI
dt
¼ a i SI À q i I;
ð9:2Þ
dR
dt
¼ q i I:
ð9:3Þ
As before, S is the number of susceptible persons (who are sensitive to the
pathogen and not protected), I is the number of infected persons (who are sick and
spread the infection; please do not confuse with the number of still ill persons, so
known active cases) and R is the number of removed persons (who no longer
spread the infection; this number is the sum of isolated, recovered, dead and
infected people who left the region). Parameters a i and q i are supposed to be
constant for every epidemic wave.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
I. Nesteruk, COVID-19 Pandemic Dynamics,
https://doi.org/10.1007/978-981-33-6416-5_9
127