Chapter 2
Early Stages of Epidemics
and Exponential Growth
The early stages of the COVID-19 epidemic outbreaks and the pandemic dynamics
in March and April 2020 will be analyzed. The statistics-based method of parameter
estimations for the exponential growth will be presented.
For the initial stage of every epidemic, the exponential growth in the number of
cases V over time t is typical:
V ¼ e
ct þ b
ð2:1Þ
where c and b are constant parameters. This fact was confirmed also for COVID-19
pandemic (see, e.g., [6–32]) and follows from the simple mathematical model,
stating that the increase in the number of cases dV during the time interval dt is
proportional to their number V:
dV ¼ cVdt
ð2:2Þ
Formula (2.1) yields the solution to the differential Eq. (2.2) and means that
corresponding points V j (taken in logarithmic scale) versus time must follow the
straight lines. This simple model does not take into account the restricted volume of
population (according to Eq. (2.1) the number of cases tends to infinity as time
increases) and the fact that some infected persons become isolated or dead (i.e., they
stop to spread the infection). To simulate the deviation from the exponential growth
(2.1) and stabilization of the number of cases at some saturation level, more
complicated approaches are necessary. We will use two of them (classical and
generalized SIR models) later.
Equation (2.1) yields the linear dependence
y ln V ¼ ct þ b
ð2:3Þ
© 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_2
7
Early Stages of Epidemics
and Exponential Growth
The early stages of the COVID-19 epidemic outbreaks and the pandemic dynamics
in March and April 2020 will be analyzed. The statistics-based method of parameter
estimations for the exponential growth will be presented.
For the initial stage of every epidemic, the exponential growth in the number of
cases V over time t is typical:
V ¼ e
ct þ b
ð2:1Þ
where c and b are constant parameters. This fact was confirmed also for COVID-19
pandemic (see, e.g., [6–32]) and follows from the simple mathematical model,
stating that the increase in the number of cases dV during the time interval dt is
proportional to their number V:
dV ¼ cVdt
ð2:2Þ
Formula (2.1) yields the solution to the differential Eq. (2.2) and means that
corresponding points V j (taken in logarithmic scale) versus time must follow the
straight lines. This simple model does not take into account the restricted volume of
population (according to Eq. (2.1) the number of cases tends to infinity as time
increases) and the fact that some infected persons become isolated or dead (i.e., they
stop to spread the infection). To simulate the deviation from the exponential growth
(2.1) and stabilization of the number of cases at some saturation level, more
complicated approaches are necessary. We will use two of them (classical and
generalized SIR models) later.
Equation (2.1) yields the linear dependence
y ln V ¼ ct þ b
ð2:3Þ
© 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_2
7