I ¼ m i ln S À S þ N i À R i À m i lnðN i À I i À R i Þ
ð 9:8Þ
It follows from (9.8) that function I has a maximum at S ¼ m i and tends to zero at
infinity. The corresponding number of susceptible persons at infinity S i1 [ 0 can
be calculated from a nonlinear equation
S i1 ¼ ðN i À I i À R i Þe
S i1 ÀN i ÀR i
m i
ð9:9Þ
Formula (9.9) follows from (9.8) at I = 0.
As in Chap. 4, we have solved (9.1)–(9.3) by introducing the function
VðtÞ ¼ IðtÞ þ RðtÞ;
ð9:10Þ
corresponding to the number of victims or cumulative confirmed number of cases.
For many epidemics (including COVID-19 pandemic), we cannot observe dependencies SðtÞ; IðtÞ and RðtÞ but observations of the accumulated number of cases V j
corresponding to the moments of time t j provide information for direct assessments
of the dependence VðtÞ.
It follows from (9.2) and (9.3) that:
dV
dt
¼ a i SI
ð9:11Þ
Equations (9.4), (9.8) and (9.11) yield:
dV
dt
¼ a i ðN i À VÞ m i lnðN i À VÞ þ V À R i À m i lnðN i À R i À I i Þ
½
Š
ð 9:12Þ
Integration of (9.12) provides an analytical solution for the set of Eqs. (9.1)–
(9.3):
F
Ã
i ðV; N i ; I i ; R i ; m i Þ ¼ a i ðt À t
Ã
i Þ;
ð9:13Þ
F
Ã
i ¼
Z V
R i þ I i
dU
ðN i À UÞ m i lnðN i À UÞ þ U À R i À m i lnðN i À R i À I i Þ
½
Š
:
ð9:14Þ
Thus, for every set of parameters N i ; I i ; R i ; m i ; a i and a fixed value of V, integral
(9.14) can be calculated and a corresponding moment of time can be determined
from (9.13). Then, functions I(t) and R(t) can be easily calculated with the use of
formulas (9.8) and:
S ¼ N i À V; R ¼ V À I
ð9:15Þ
9 General SIR Model and Its Exact Solution
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