LDE model
As early as 1845, Verhust proposed the LDE model, which is an ordinary differential equation (ODE) based on Malthus’ quantification of total biological growth to characterize the self-growth of disease in a population [
16,
31]. In recent years the model has been widely used in the analysis of epidemiological characteristics of infectious diseases and the study of early warning mechanisms of infectious diseases [
32]. Its main feature is the fitting of data to determine the particular specific time of the development of infectious diseases, with the following equation:
$$\frac{dn}{dt}= kn\left(1-\frac{n}{N}\right)$$
(1)
Where dn/dt is the rate of change of the cumulative number of infectious disease cases
n at time
t,
k is the correlation coefficient and
N is the upper limit of cumulative infectious disease cases. The general solution of eq. (
1) is as follows:
$$n=\frac{N}{1+{e}^{- kt-c}}$$
(2)
This equation includes three parameters
k,
N and
c. The meanings of
k and
N are the same as in eq. (
1) and directly determine the trend of the cumulative number of cases
n with
t. The
c is a constant calculated by integration during the solution of eq. (
1) and is important when solving for the three inflection points of the logistic curve. The first order derivative of eq. (
2) is expressed in terms of time
t. The eq. (
3) is as follows:
$$\frac{dn}{dt}=\frac{Nk{e}^{- kt-c}}{1+{e}^{- kt-c}}$$
(3)
The equation expresses the curve of new cases over time. If we take the derivative of eq. (
3), which is the second order derivative of eq. (
2), we can obtain an equation for the curve of the rate of increase or decrease in the number of new cases. The rate of change in the number of new cases is zero at the peak of the epidemic, so let the second order derivative of eq. (
2) be equal to zero and solving for the inflection point from increase to decrease of the number of new cases i.e., solving for the value of t at the peak of the epidemic, where
\(t=-\frac{c}{k}\). The second-order derivative of eq. (
3), which is the third-order derivative of eq. (
2), gives the equation for the “acceleration” curve of the increase and decrease in new cases, and if this “acceleration” is equal to 0, the “acceleration” of new cases can be obtained. If this “acceleration” is equal to 0, the inflection point of the change in the “acceleration” of new cases can be obtained, as shown in eq. (
4):
$$t=\frac{-c\pm 1.317}{k}$$
(4)
These two inflection points divide the process of infectious disease epidemic development into a gradual increase, a rapid increase and a slow increase, and the horizontal coordinate of the first inflection point corresponding to the gradual increase to the rapid increase is
\({t}_1=\frac{-c-1.317}{k}\) [
20]. The horizontal coordinate corresponding to the second inflection point from the fast to the slow growth period is
\({t}_2=\frac{-c+1.317}{k}\) [
20].
GLDE model
The GLDE model is improved to introduce the shape parameter
λ into the LDE model, thus improving the model warning accuracy with the following differential equation:
$$\frac{dn}{dt}=\frac{kn}{\lambda}\left[1-{\left(\frac{n}{N}\right)}^{\lambda}\right]$$
(5)
Where
\(\frac{dn}{dt}\) is also the rate of change of cumulative infectious disease cases
n at time
t, the significance of the
k and
N parameters is consistent with the significance of the parameters in the LDE model above. Then the general solution of eq. (
5) is as follows:
$$n=\frac{N}{{\left(1+{e}^{- kt+c}\right)}^{\frac{1}{\lambda }}}$$
(6)
The equation includes four parameters,
k,
N,
c and
λ, where
k and
N have the same meaning as in eq. (
5) and directly determine the trend of the cumulative number of cases
n with
t.
c is a constant resulting from the integration of eq. (
5), which is important when solving for the 3 inflection points of the generalized logistic curve.
λ is the shape parameter that determines the location of the distribution of the generalized logistic curve. When λ is greater than 0 and less than 1, the distribution is skewed to the left. When λ is greater than 1, the distribution is skewed to the right, and when λ is equal to 1, it is symmetrical, that is, the general logistics distribution. Expressing the first order derivative of eq. (
6) in terms of time
t, the eq. (
7) is as follows:
$$\frac{dn}{dt}=\frac{kn}{\lambda }{e}^{- kt-c}$$
(7)
This equation expresses the curve of new cases over time. If we take the derivative of eq. (
7), which is the second order derivative of eq. (
6), we can obtain an equation for the rate of increase or decrease in the number of new cases. The rate of change in the number of new cases is zero at the moment when the epidemic reaches its peak, so let the second order derivative of eq. (
6) be equal to zero and finding the inflection point at which there is an increase to decrease of the number of new cases, that is, the value of
T at the peak of the epidemic, by solving for
\(T=-\frac{c+\ln \lambda }{k}\). The second-order derivative of eq. (
7), which is the third-order derivative of eq. (
6), gives the equation for the “acceleration” curve of the increase and decrease in new cases, and if this “acceleration” is equal to 0, the “acceleration” of new cases can be obtained as the inflection point for the change in “acceleration” of new cases is
$$T=-\frac{c-\ln \left(\frac{3\pm \sqrt{5}}{2}\lambda \right)}{k}$$
(8)
These two inflection points divide the development process of infectious disease epidemic into progressive, rapid and slow phases. The horizontal coordinate of the first inflection point from progressive to rapid phase is: \({T}_1=-\frac{c-\ln \left(\frac{3-\sqrt{5}}{2}\lambda \right)}{k}\), and the horizontal coordinate of the second inflection point from rapid to slow phase is \({T}_2=-\frac{c-\ln \left(\frac{3+\sqrt{5}}{2}\lambda \right)}{k}.\)