Application of logistic differential equation models for early warning of infectious diseases in Jilin Province - BMC Public Health

Study design

This study was conducted in accordance with the route of determining and constructing the LDE and GLDE models for the fitting and comparison of the effects of infectious diseases, and the estimation of warning times and comparison of differences between the two models. The GLDE is constructed by first introducing the shape parameter λ into the LDE. As LDE models were suitable for early warning of seasonal or cyclical diseases, acute infectious diseases with seasonal or cyclical characteristics were selected according to the weekly data collected for the prevalence and incidence of the disease. For the selected diseases, the epidemic cycle was segmented and the actual number of incidences (in weeks) was fitted using the two models respectively, and the goodness-of-fit test was performed on the data from the LDE and GLDE models. The parameters obtained in the fit were used to estimate the epidemic acceleration weeks (EAW) and the recommended warning weeks (RWW), and to compare the differences in warning durations estimated by the two models. The research design methodology involved is shown in Fig. 1 below.

Data collection and criteria for inclusion and exclusion

In this study, the data on diseases were obtained from the China Information System for Disease Control and Prevention (CISDCP). Data were collected for 22 infectious diseases in Jilin Province from January 1, 2005 to December 31, 2019, where the data information included the date of disease onset. Diagnosis of all diseases followed the diagnostic criteria for infectious diseases developed by the National Health Commission of the People's Republic of China. Demographic data were obtained from the Jilin Provincial Statistical Yearbook, including the total population, birth rate and death rate of Jilin Province for each year. The LDE models for early warning of the onset of chronic infectious diseases did not have practical application, and the LDE models were suitable for the early warning of seasonal and periodic infectious diseases. Therefore, in this study, the 22 diseases collected were classified into acute infectious diseases (HFMD, Mumps, Shigellosis, Scarlet fever, HFRS, Influenza, Rubella, Measles, Hepatitis A, Acute hemorrhagic conjunctivitis, Pertussis, Meningococcal meningitis, Typhoid and paratyphoid, Malaria) and chronic infectious diseases (Tuberculosis, Hepatitis B, Hepatitis C, Syphilis, Brucellosis, Gonorrhea, Hepatitis E, AIDs) according to their onset progression rate [28,29,30], and the acute infectious diseases with seasonal or cyclical characteristics were selected to be included in the fitting and early warning of the LDE models.

Model building

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 ${T}_2=-\frac{c-\ln \left(\frac{3+\sqrt{5}}{2}\lambda \right)}{k}.$

Simulation method and statistical analyses

In this study, diseases were fitted in segments according to the epidemiological cycle of the disease, based on the fluctuation of the disease epidemic curve. This was done using Berkeley Madonna 8.3.18 (developed by Robert Macey and George Oster of the University of California at Berkeley. Copyright©1993–2001 Robert I. Macey & George F. Oster) for modelling and the system of equations was solved using Runge–Kutta method of order four to find the best-fit curve and parameters. SPSS 21.0 (IBM Corp, Armonk, USA) was used to determine the goodness of fit of the model fit curve. The index for determining the goodness of fit was the root mean square (RMS) of the simulated and actual data [33, 34], and the larger the R², the better the fit between the actual and simulated data and the test was P = 0.005.

Establishing the timing of the warning

Equation (4) and Eq. (8) were used to calculate the two inflection points at which the speed of disease changes from slow to fast and from fast to slow in each epidemic cycle, namely the EAW and the warning removed weeks (WRW). As it takes time to implement health decisions and interventions and to produce the corresponding prevention and control effects, leaving the epidemic to develop until the "epidemic acceleration time" would result in a lag. Therefore, the mean and standard deviation (s) of the EAW for each epidemic cycle of the diseases were calculated. It is possible to consider an early warning time of 1–2 standard deviations ahead of the epidemic acceleration time, namely the RWW.

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