As mentioned in the chap. 4, parameters a i show how quick the susceptible
persons become infected (see (9.1)). Large values of this parameter correspond to
severe epidemics with many victims. These parameters accumulate many characteristics. First, they show how strong (virulent) is the pathogen and what is the way
of its spreading. Parameters a i accumulate also the frequency of contacts and the
way of contacting. In order to decrease the values of a i , we have to minimize the
number of our contacts and change our contacting habits. For example, we have to
avoid the public places, use masks there, and minimize or cancel traveling. We have
to change our contact habits: to avoid handshakes and kisses. First, all these simple
things are very useful to protect yourself. In addition, if most people follow these
recommendations, we have chance to diminish the values of parameters a i and
reduce the negative effects of the pandemic.
The parameters q i characterize the patient removal rates, since Eq. (9.3)
demonstrates the increase rate of R. The inverse values 1=q i are the estimations for
time of spreading infection s i during ith epidemic wave. So, we are interested in
increasing the values of parameters q i and decreasing 1=q i . People and public
authorities should work on this and organize immediate isolation of suspicious
cases.
Since the derivative dðS þ I þ RÞ=dt is equal to zero (it follows from summarizing Eqs. (9.1)–(9.3)), the sum
N i ¼ S þ I þ R
ð9:4Þ
must be constant for every wave and is not the volume of population as mentioned
in Chap. 4.
To determine the initial conditions for the set of Eqs. (9.1)–(9.3), let us suppose
that at the beginning of every epidemic wave t
Ã
i :
Iðt
Ã
i Þ ¼ I i ; Rðt
Ã
i Þ ¼ R i ; Sðt
Ã
i Þ ¼ N i À I i À R i
ð9:5Þ
In particular, when the first wave of the epidemic starts with one infected person,
the initial conditions (9.5) can be written as follows:
Iðt
Ã
1 Þ ¼ 1; Rðt
Ã
1 Þ ¼ 0; Sðt
Ã
1 Þ ¼ N 1 À 1:
ð9:6Þ
Initial conditions (9.6) were used in Chap. 6 to simulate the first epidemic waves
in different countries.
It follows from (9.1) and (9.2) that
dI
dS
¼
m i
S
À 1; m i ¼
q i
a i
ð9:7Þ
Integration of (9.7) with the initial conditions (9.5) yields:
128
9 General SIR Model and Its Exact Solution
persons become infected (see (9.1)). Large values of this parameter correspond to
severe epidemics with many victims. These parameters accumulate many characteristics. First, they show how strong (virulent) is the pathogen and what is the way
of its spreading. Parameters a i accumulate also the frequency of contacts and the
way of contacting. In order to decrease the values of a i , we have to minimize the
number of our contacts and change our contacting habits. For example, we have to
avoid the public places, use masks there, and minimize or cancel traveling. We have
to change our contact habits: to avoid handshakes and kisses. First, all these simple
things are very useful to protect yourself. In addition, if most people follow these
recommendations, we have chance to diminish the values of parameters a i and
reduce the negative effects of the pandemic.
The parameters q i characterize the patient removal rates, since Eq. (9.3)
demonstrates the increase rate of R. The inverse values 1=q i are the estimations for
time of spreading infection s i during ith epidemic wave. So, we are interested in
increasing the values of parameters q i and decreasing 1=q i . People and public
authorities should work on this and organize immediate isolation of suspicious
cases.
Since the derivative dðS þ I þ RÞ=dt is equal to zero (it follows from summarizing Eqs. (9.1)–(9.3)), the sum
N i ¼ S þ I þ R
ð9:4Þ
must be constant for every wave and is not the volume of population as mentioned
in Chap. 4.
To determine the initial conditions for the set of Eqs. (9.1)–(9.3), let us suppose
that at the beginning of every epidemic wave t
Ã
i :
Iðt
Ã
i Þ ¼ I i ; Rðt
Ã
i Þ ¼ R i ; Sðt
Ã
i Þ ¼ N i À I i À R i
ð9:5Þ
In particular, when the first wave of the epidemic starts with one infected person,
the initial conditions (9.5) can be written as follows:
Iðt
Ã
1 Þ ¼ 1; Rðt
Ã
1 Þ ¼ 0; Sðt
Ã
1 Þ ¼ N 1 À 1:
ð9:6Þ
Initial conditions (9.6) were used in Chap. 6 to simulate the first epidemic waves
in different countries.
It follows from (9.1) and (9.2) that
dI
dS
¼
m i
S
À 1; m i ¼
q i
a i
ð9:7Þ
Integration of (9.7) with the initial conditions (9.5) yields:
128
9 General SIR Model and Its Exact Solution