Analyzing COVID-19 outbreaks: A methodological approach with Gaussian pulse models

Abstract

This method introduces a methodological approach for comprehensively analyzing COVID-19 outbreaks using Gaussian pulse models to assess transmission rates. Our methodology is designed to provide an in-depth understanding of the intricate dynamics underlying the spread of COVID-19. By incorporating Gaussian pulse models into our approach, we capture temporal and spatial outbreaks' characteristics with high precision. This method provides a detailed overview of our methodological approach, underscoring its potential to revolutionize our comprehension of COVID-19 outbreak dynamics.

• •Our methodology involves the application of Gaussian pulse models to the transmission rate estimation.
• •Parameter estimation in epidemic modeling involves adjusting key factors like transmission rate, and effective reproduction number to best match observed data.
• •We use Microsoft Excel's Solver add-in with the GRG algorithm to find the best parameter values, improving the data fit.

Graphical abstract

Specifications table

Subject area:	Agricultural and Biological Sciences
More specific subject area:	Modeling in Epidemiology
Name of your method:	Gaussian Pulse Models for analyzing COVID-19 outbreaks
Name and reference of original method:	N.A.
Resource availability:	Setianto, S., Hidayat, D. Modeling the time-dependent transmission rate using gaussian pulses for analyzing the COVID-19 outbreaks in the world. Sci Rep13, 4466 (2023). https://doi.org/10.1038/s41598-023-31714-5[10]

Background

The COVID-19 pandemic, caused by the novel coronavirus SARS-CoV-2, has posed a global health crisis of unprecedented proportions. The outbreak, which originated in Wuhan, China in late 2019, has since spread to virtually every corner of the world, leading to millions of infections and fatalities [1]. Understanding the dynamics of the spread of the virus is crucial for effective public health interventions and policy formulation [2].

One of the key elements in understanding the spread of infectious diseases, such as COVID-19, is the transmission rate, which quantifies the rate at which the virus is transmitted from one individual to another. Accurate estimation of this rate is pivotal in modeling the disease's dynamics, predicting its future trajectory, and designing effective control strategies. However, estimating the transmission rate is a complex task due to various factors like changes in behavior, interventions, and evolving variants of the virus [3], [4], [5].

This method focuses on an innovative methodological approach for analyzing COVID-19 outbreak dynamics by employing Gaussian pulse models for transmission rate assessment. Gaussian pulse models have been widely used in various fields, including signal processing and physics, to describe the temporal and spatial characteristics of phenomena. In the context of disease spread, these models offer a unique way to capture the nuanced dynamics of outbreaks, considering factors like seasonality, local variations, and the impact of interventions [6], [7], [8], [9].

By applying Gaussian pulse models to the transmission rate, this method aims to provide a more accurate and granular understanding of how COVID-19 spreads over time and across different regions. The insights gained from this methodology can significantly contribute to public health research, enabling more effective strategies for disease control and prevention. This paper outlines the key components of our methodological approach and discusses its potential implications for enhancing our comprehension of COVID-19 outbreak dynamics.

Method details

SEIR model

We have enhanced the conventional SEIR (Susceptible-Exposed-Infectious-Recovered) model by incorporating a mechanism for changing the transmission rate over time, which is achieved through the use of Gaussian pulses [10]. In Fig. 1, the SEIR model divides the population into four distinct categories:

• •Susceptible (S): Individuals who are at risk of contracting the disease.
• •Exposed (E): Individuals who have been infected but are not yet capable of transmitting the disease to others.
• •Infectious (I): Individuals who are currently infected and can spread the disease to others.
• •Recovered (R): Individuals who have either recovered from the disease or have been removed from the infected group.

Fig. 1.

A schematic representation of the SEIR model illustrates the key components and relationships within the model

β represents the rate at which individuals get infected, σ indicates how quickly those who have been exposed to the disease become infectious, and γ signifies the rate at which individuals who are infectious are either recovered or removed from the population.

Gaussian pulse model

The modification involves adjusting the transmission rate within the model using Gaussian pulses [10], which introduces a time-dependent aspect to how the disease spreads and impacts the population. This allows for a more nuanced and accurate representation of disease dynamics over time. In the context of studying multiple waves of infection, the transmission rate plays a crucial role in determining how the disease spreads over time. Instead of assuming a constant transmission rate, this model considers the dynamic nature of transmission as defined:

$$\beta \left(t\right)=\sum _{n=1}^{\infty }{r}_n{e}^{-\frac{{\left(t-t_n\right)}^2}{{d_n}^2}}$$ (1)

Where $r_n$ and $d_n$ are amplitude (number of contacts) and half of the pulse duration respectively. Each pulse, characterized by its amplitude and half-duration, represents a distinct wave or peak in the infection rate. These waves can be associated with different phases of the outbreak, such as initial surges, subsequent peaks, and so on. By using this Gaussian pulse model, researchers can better capture the complex dynamics of how the disease spreads during different phases of an outbreak, as it allows for varying levels of infectiousness and durations of peak transmission throughout the course of the epidemic. This approach provides a more accurate representation of the real-world behavior of infectious diseases that often exhibit multiple waves of infection.

Simulation of the model

We show our simulations in Fig. 2 using the parameters from Table 1. In the simple scenario, we assume that the total population, N = 1000, the latency rate σ = 1/14 (per days), and recovery rate γ = 1/7 (per days). Fig. 2a offers a comprehensive insight into the phenomenon of a second-wave infection by employing the SEIR model with two Gaussian pulses to describe the ever-changing transmission rates, thereby presenting a dynamic portrayal of the epidemic's evolution over time. The figure serves as a visual representation of how the population transitions between distinct compartments that symbolize susceptibility (depicted in blue), exposure (represented in orange), infectivity (illustrated in red), and recovery or removal (indicated in green). In conjunction with Fig. 2b, it becomes evident that there are two distinct bell-shaped active case curves, formed by the combined populations of exposed and infectious individuals. These curves signify the occurrence of two distinct waves of infection within the epidemic. This dual-wave pattern could be attributed to various factors, such as changes in public health measures, population behavior, or the emergence of new virus variants. This graph provides a detailed analysis of the epidemic dynamics by depicting how the proportion of the infected population undergoes rapid growth followed by a decline. The observed pattern reflects the interplay between infection and protective interventions affecting the susceptible population. Specifically, as individuals either contract the disease or benefit from protective measures, the curve illustrates the dynamic shifts in the proportion of the infected population over time. Furthermore, the presence of two distinct sharp spikes in the total cases curve (depicted in black) signifies the occurrence of a second wave of infection. These spikes, encompassing the combined populations of exposed, infectious, and recovered/removed individuals, highlight the temporal dynamics and intensity of the epidemic's resurgence.

Fig. 2.

Simulation results of the (a) SEIR model with Gaussian pulse as transmission rate and (b) The total and active cases of the model.

Table 1.

Simulation parameters of the second wave model of infection.

Parameter		First	Second
N	1000
Io	1
tac(Active cases max)		58	134
$\gamma$ (per days)	0.14
$\sigma$ (per days)	0.07
Active cases max		227	229
Re max		6.79	2.89
tRe(Re max)		20	186
rn (number contact per day)		1.0	1.0
dn(days)		20	20
tn (day)		30	120

The comprehensive analysis underscores the SEIR model's effectiveness in capturing the intricate dynamics of multiple infection waves. It also sheds light on the population's transitions between different states over time, providing valuable insights into the factors that contribute to the occurrence of second waves of infection during the course of an epidemic. This multi-dimensional perspective enhances our understanding of the epidemic's behavior and evolution. In order to model the transmission rate, we employ two Gaussian pulses with parameters set as half the pulse duration (d_1,2 = 20 days). The peaks of the first and second pulses (t₁, t₂) are positioned at 30 and 120 days, respectively, while the number of contacts per day is standardized to r_n = 1. This configuration is chosen to represent the temporal characteristics of the transmission rate over the course of the epidemic. In more detail, the use of two Gaussian pulses allows for a nuanced representation of how the transmission rate changes over time. The half duration parameter (d_1,2) influences the width of each pulse, indicating the period during which the transmission rate remains elevated. Setting t₁ and t₂ as the peaks of the pulses establishes the specific time points when the transmission rate is at its maximum for the first and second waves. By fixing the number of contacts per day at r_n = 1, the model assumes a consistent level of contact for each individual, contributing to the simplicity of the analysis. This configuration provides a detailed understanding of the temporal dynamics of the transmission rate, offering insights into the intervals between waves, their intensity, and the overall pattern of the epidemic's progression. Details of the parameters used in this simulation are given in Table 1.

The effective reproduction number$R_e$ that produce during the entire infectious period and would change over time proportionally with the population of susceptible:

$$R_e\left(t\right)=\frac{{\beta }_t}{\gamma }\frac{S\left(t\right)}{N}$$ (2)

In Fig. 3, a comprehensive analysis reveals the profound influence of changes in the susceptible population on the Reproduction Number (R_e) curve, a key determinant in understanding epidemic dynamics [11].

Fig. 3.

Parameters value of (a) β(t), S(t)/N, γ, and (b) the effective reproduction number R_e(t).

The simulation intricately demonstrates that, contrary to expectations, a significant reduction in the susceptible population does not necessarily hinder the persistence of a second infection wave. This intriguing phenomenon is attributed to the sustained high daily contact rate, specifically set at r_n = 1. The R_e curve, representing the average number of secondary infections generated by a single infected individual, emerges as a crucial factor in driving the persistent second wave. Despite a marked decline in the pool of susceptible individuals, the R_e curve remains impactful, showcasing the nuanced dynamics of infection transmission. The elevated daily contact rate underscores a scenario where each infected person continues to have a considerable potential to transmit the infection, thus contributing to the sustained second wave. This detailed analysis not only emphasizes the intricate relationship between the susceptible population, the Re curve, and daily contact rates but also sheds light on the resilience of infection waves under certain conditions. It provides a comprehensive understanding of how these factors collectively shape the trajectory of an epidemic, offering valuable insights for public health strategies and interventions aimed at controlling and mitigating the impact of infectious diseases.

Implementation the model for multiple waves of infection of COVID-19 disease

The comprehensive analysis, rooted in existing literature and exemplified by the study conducted by Setianto and Hidayat [10], delves into the intricate application of Gaussian Pulse modeling within the framework of COVID-19 outbreaks. The process of parameter estimation for the SEIR epidemic model using daily cases data involves several key steps to effectively understand and model the spread of infectious diseases. Firstly, it is essential to define the SEIR model by formulating equations that describe the compartments representing Susceptible (S), Exposed (E), Infected (I), and Recovered (R) individuals. Following this, rigorous data collection of daily reported cases is undertaken to provide the necessary information for parameter estimation. Once the data is collected, the next step involves defining the objective function. In this context, the R-squared (R²) value is utilized as a measure of fit between observed and predicted cases. This allows for the quantification of how well the model captures the dynamics of the epidemic. With the objective function established, the optimization problem is set up, specifying any constraints on parameters to ensure realistic values are obtained. The optimization procedure is then executed, typically employing Excel Solver's GRG algorithm [12,13]. This iterative process adjusts the parameters of the model to maximize the R² value, thereby improving the model's predictive accuracy. Model validation is a crucial step, where the accuracy of the model is assessed by comparing its predictions with observed data. This step ensures that the model accurately captures the dynamics of the epidemic and can be relied upon for decision-making. Finally, the validated model, particularly focusing on metrics such as the effective reproduction number (R_e), is utilized for analysis. Public health officials and policymakers can leverage the insights provided by the model to monitor the spread of the disease and inform strategic interventions, such as vaccination campaigns and social distancing measures. This sophisticated modeling technique involves a nuanced modification of the standard SEIR model, incorporating time-dependent transmission rates through the mathematical expression of a sum of Gaussian pulses characterized by amplitude and half-duration parameters. This adjustment process, known as parameter estimation, is often achieved using optimization techniques like maximum likelihood estimation, least squares regression, or Bayesian inference [14]. As results, the effective reproduction number (R_e) reveals distinctive patterns globally and in Indonesia. The R_e curve illustrates six global waves with peaks in January 2021, May 2021, September 2021, March 2022, August 2022, and December 2022. Conversely, Indonesia experienced five infection peaks in February 2021, July 2021, February 2022, August 2022, and November 2022 [10]. These fluctuations suggest varying transmission intensities and timing, influenced by factors like interventions, vaccination rates, and potential emergence of new variants. Policymakers can leverage this analysis to tailor effective strategies for future wave mitigation and virus control. The analysis scrutinizes the temporal dynamics of infection waves by visually examining peaks and troughs in infectious population curves. Sensitivity analyses are applied to the amplitude and half-duration parameters of the Gaussian pulses, offering insights into how variations in these parameters influence the transmission rate and subsequent population dynamics. Furthermore, this comprehensive analysis extends beyond mathematical modeling, considering the practical implications of the findings for public health policy. Recommendations, informed by the model outcomes, may include adjustments to intervention strategies, vaccination campaigns, and measures to control infection waves. The transparency and rigor of the analysis are fortified by the acknowledgment of potential limitations inherent in the modeling approach. The study emphasizes the need for careful interpretation and contextualization of findings, recognizing the complexity and uncertainty associated with modeling infectious disease dynamics.

Conclusion

In conclusion, this method presents a comprehensive exploration of Gaussian Pulse Models in the context of COVID-19 outbreaks. The incorporation of time-dependent transmission rates, achieved by modifying the standard SEIR model, offers a nuanced and detailed insight into the dynamic nature of infection spread. By doing so, it not only advances our understanding of COVID-19 outbreaks but also contributes to the broader field of epidemiological modeling, paving the way for more refined and insightful analyses in the future.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

• Data will be made available on request.