DOI: https://doi.org/10.21203/rs.3.rs-2066503/v1
In this work, an SEIR epidemic model with time-dependent transmission rate parameters for the multiple waves of COVID-19 infection was investigated. It is assumed that the transmission rate is determined by the superposition of the Gaussian pulses. The interaction of these dynamics is represented by recursive equations. Analysis of the overall dynamics of disease spread is determined by the effective reproduction number Re(t) produced throughout the infection period. As a result, the development of the epidemic over time has been successfully studied and the phenomenon of multiple waves of COVID-19 infection in the world has been explained.
The use of epidemic models makes it possible to simulate the dynamics of disease transmission to detect emerging outbreaks and assess public health interventions1–5. A standard method, SEIR (susceptible-exposed-infectious-removed), was developed to describe the epidemic dynamics, which takes into account the time-dependent coefficients and offers the possibility to analyze the epidemic dynamics6–8. The method was applied to the 2019 coronavirus (COVID-19) pandemic caused by the severe acute respiratory syndrome Coronavirus-2 named SARS-CoV-23,4,9–11. In addition, this method allows us to evaluate trends and forecasts in the number of cases of infection in order to assess possible influencing factors such as the availability of vaccines12,13 or restrictive measures by central authorities14,15 may affect the outbreak of the epidemic. The SEIR model with time-varying parameter specifications has been used in several studies known in the literature16–20. However, they cannot consider external influences, such as actions that contain the spread of an infection that may occur at different times during the development of the infection itself or possible changes in the health condition of the infected individual due to pharmacological development. The approach discussed here introduces time-dependent model parameters. Specifically, we assumed the time-dependent infection rate, given the amount of contact between people during the spread of the disease using the Gaussian function. This assumption aims to provide a reliable approach in the analysis of the evolution of the epidemic as a whole so that policymakers can take appropriate initiatives to reduce transmission and selective actions taking into account the specific characteristics of each region. As an implementation and test model, we apply it to the situation of the spread of Covid-19 in the world by using data from sources21 with the quality of the observed data and its validation is an important conclusion in assessing the reliability of the results of this model. Additional information from the results of our model is to know the overall dynamics of the spread of the Covid-19 disease in the world which is determined by the effective reproduction number produced during the entire infection period. So that the evolution of the epidemic can be monitored and used as important information for policymakers and the general public to take action to anticipate the next wave of Covid-19 cases.
The analysis is based on data collected from the website (https://www.worldometers.info/about/) run by an international team of developers, researchers, and volunteers to collect various statistical data from around the world real to represent -time. For the COVID-19 pandemic, they manually analyze, validate and collect real-time data from thousands of sources around the world. By definition, the total number of cases is the cumulative total number of reported cases that were found to be clinically confirmed (criterion by country). Active Cases = (Total Cases) - (Total Deaths) - (Recovered). This is data on the number of people currently infected with the virus. We will implement our model using COVID-19 data in the world with a reported population of 7,953,091,060 on May 22, 2022.
We started by introducing the SEIR model, which is one of the most widely used extensions of the standard SIR model, the epidemiological model based on Ordinary Differential Equations (ODE). In general, the SEIR models classify the population into four classes: Susceptible (S), Exposed (E), Infected (I), and Recovered or Removed (R). The model describes how different parts of the population change over time. The total population is N = S + E + I + R. The interaction of these dynamics is represented by recursive formulation22,23:
These four equations are called Euler formulas and can be employed to generate a set of values that reflect what is happening with the epidemic over a period of time. Where r is the number of contact per unit time, \(\)is the probability of disease transmission per contact which is transmission rate, \(\)is a per-capita rate of progression to infectious state which is the loss of latency rate, and\(\) is the per-capita recovery rate. This formulation assumes full population composition rather than population dynamics (i.e. natural births and deaths are considered negligible during epidemic periods)13. In order to calculate anything from these formulas we need to have explicit values for β, σ, γ, S, E, I, and R. In the standard SEIR model the parameter β, σ, and γ is a constant that does not depend on time. However, the characteristics of the epidemic show us that these parameters can change and vary over time. Note that in some models for diseases where contacts are not well defined (e.g. influenza and COVID-19), \(r\)(the number of contacts per unit time multiplied by the probability of disease transmission per contact) are combined into one parameter (often also referred to as the constant of \(\), the number of adequate contacts per unit of time)24,25. For this reason, we evaluate the SEIR model by proposing to express the number of suitable contacts per unit of time \(\) is expressed as a function of time, the Gaussian pulse26, as defined:
Where \({r}_{n}\) and \({d}_{n}\) are amplitude (number of contacts) and half of the pulse duration respectively. So that, equations (1) and (2) are simplified as:
\(\) \({S}_{t+1}={S}_{t}-\left(\frac{{\beta }_{t}{S}_{t}{I}_{t}}{N}\right)dt \left(6\right)\)
In fact, the average number of new infections caused by a single infected person at the time of the partially susceptible population can be determined by the effective reproduction number Re generated throughout the infection period, which varies proportionally to the population of the susceptible population over time be changed:
It describes whether the infection population is increasing or not. It increases when Re > 1; decreases when Re < 1. When Re = 1, the disease is in equilibrium19,27−30.
Until now, the COVID-19 pandemic is still ongoing in almost all countries. As of June 12, 2022, 540,470,607 confirmed cases of COVID-19 have been reported to WHO, including 6,331,332 deaths31. Although the spread of Covid-19 in the world has decreased relatively in 2022, more new cases are surging and could potentially cause the spread of infections in the next few waves. To examine and analyze the overall dynamics of the spread of Covid-19 infection in the world, we used recent data observed from January 22, 2020, to May 22, 2022. The best parameters from the simulation results using the multiple wave infection models are given in Table 1. These values were obtained by estimating the best results from the simulations compared to the daily global observed COVID-19 data in terms of the coefficient of determination or correlation index (R2) obtained by the non-linear regression method32,33 which are detailed in the supplementary file. These parameters are used to study the dynamics of the effective reproduction number of Covid-19 infections over more than a year.
| THE WORLD | n1 | n2 | n3 | n4 | n5 | n6 | n7 | n8 | n9 | n10 | n11 | n12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| N (22 May 2022) | 7,953,091,060 | ||||||||||||
| I0 (initial number of infected) | 970 | ||||||||||||
| \(\) (per days) | 0.0333 | ||||||||||||
| \(\) (per days) | 0.0495 | ||||||||||||
| rn (number contact per day) | 0.97 | 0.27 | 0.22 | 0.19 | 0.06 | 0.05 | 0.04 | 0.03 | 0.03 | 0.10 | 0.01 | 0.01 | |
| dn (days) | 14 | 13 | 14 | 21 | 84 | 75 | 41 | 24 | 179 | 21 | 10 | 10 | |
| to (day) | 0 | 11 | 43 | 68 | 154 | 308 | 451 | 561 | 673 | 720 | 785 | 851 | |
| R2 (coef. of determination) | 0.9525 |
From Table 1 it can be seen that the twelve Gaussian pulse models are matched to the world case data for COVID-19 with a correlation index of R2 of 0.9525. Although the accuracy of the final wave prediction depends on the available data and information, a good approach to the transmission rate plays an important role in our model, which can be used to make good predictions based on sufficient data. Considering that the COVID-19 pandemic is still a long and fluctuating process, we successfully developed the multi-wave model to simulate the spread of infection due to an infectious disease. Figure 1 shows the latest case of COVID-19 from the world with our multi-wave model using twelve Gaussian pulses as the transmission rate from January 22, 2020, to May 22, 2022.
These results explain that the approximation of the transmission rate function, extended to twelve Gaussian pulses, shows good accuracy of the model over the observed data.
The effective reproduction number (Re), which is an important parameter in the overall dynamics of the spread of COVID-19 infection in the world, can also be seen in Fig. 2 (green line). From this, it can be clearly deduced that there have been four waves of infection of the COVID-19 disease worldwide, the peaks of which occur in December 2020, April 2021, August 2021, and February 2022 when the value of Re decreased and reached an equilibrium equal to one. Assuming that the transmission rate parameter uses Gaussian pulses, we find that this model provides a good and realistic estimate of the effective reproduction number by providing information on the number of secondary infections expected from infected individuals over time.
In this article, we proposed a new transmission rate model using Gaussian pulses based on the SEIR epidemic model. It can explain the phenomenon of multiple waves of infection and also successfully analyze the overall dynamics of the spread of COVID-19 infection in the world. The simulation results show that the effective reproduction number (Re), an important parameter in the spread of infectious diseases, is also well determined.
AUTHOR CONTRIBUTIONS
S.S. formulated the problem, performed the numerical calculations, and wrote the manuscript. D.H. checked the equations and edited the manuscript.
CONFLICT OF INTERESTS
The authors declare that they have no competing interests.
ACKNOWLEDGEMENT
This project is supported by Hibah Riset Data Pustaka dan Daring Universitas Padjadjaran Tahun Anggaran 2021 with contract number: 1735/UN6.3.1/LT/2021
DATA AVAILABILITY
The datasets generated and/or analysed during the current study are available in the https://www.worldometers.info/coronavirus/ and its supplementary file.
ADDITIONAL INFORMATION
We clarify and confirm that this study did not involve humans.