Chapter 4
Classical SIR Model and the Exact
Solution of Differential Equations
Differential equations, initial conditions, exact and approximate solutions for the
classical SIR model are presented. Simple formulas for probability of meeting an
infected person and calculations of the effective and basic reproduction numbers are
given.
The SIR model for an infectious disease can be written as follows [53–56]:
dS
dt
¼ ÀaSI
ð4:1Þ
dI
dt
¼ aSI À qI
ð4:2Þ
dR
dt
¼ qI
ð4:3Þ
Here t is time; the number of susceptible persons is S (persons who are sensitive to
the pathogen and not protected); the number of infected is I (persons who are sick
and spread the infection; please do not confuse with the number of still ill person,
so known active cases); the number of removed is R (persons who do not spread
the infection anymore, this number is the sum of isolated, recovered, dead people
and persons who leaved the region); a and q are constants measured in day
½
À1 if
time is measured in days. In some books and papers, the constant a ¼ aN pop (N pop
is the number of persons in a country or in a region, i.e., the volume of population)
is used instead of a. If the value N pop is known, the use of a or a is equivalent.
The parameter a is called the infection rate, since according to (4.1) it shows
how quick the susceptible persons become infected. Large values of this parameter
correspond to severe epidemics with many victims. This parameter accumulates
many characteristics. First it shows how strong (virulent) is the pathogen and what
is the way of its spreading. For airborne droplets transmission (typical for
© 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_4
23
Classical SIR Model and the Exact
Solution of Differential Equations
Differential equations, initial conditions, exact and approximate solutions for the
classical SIR model are presented. Simple formulas for probability of meeting an
infected person and calculations of the effective and basic reproduction numbers are
given.
The SIR model for an infectious disease can be written as follows [53–56]:
dS
dt
¼ ÀaSI
ð4:1Þ
dI
dt
¼ aSI À qI
ð4:2Þ
dR
dt
¼ qI
ð4:3Þ
Here t is time; the number of susceptible persons is S (persons who are sensitive to
the pathogen and not protected); the number of infected is I (persons who are sick
and spread the infection; please do not confuse with the number of still ill person,
so known active cases); the number of removed is R (persons who do not spread
the infection anymore, this number is the sum of isolated, recovered, dead people
and persons who leaved the region); a and q are constants measured in day
½
À1 if
time is measured in days. In some books and papers, the constant a ¼ aN pop (N pop
is the number of persons in a country or in a region, i.e., the volume of population)
is used instead of a. If the value N pop is known, the use of a or a is equivalent.
The parameter a is called the infection rate, since according to (4.1) it shows
how quick the susceptible persons become infected. Large values of this parameter
correspond to severe epidemics with many victims. This parameter accumulates
many characteristics. First it shows how strong (virulent) is the pathogen and what
is the way of its spreading. For airborne droplets transmission (typical for
© 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_4
23