Unveiling SIR Model Parameters: Empirical Parameter Approach for Explicit Estimation and Confidence Interval Construction

RESEARCH ARTICLE • OPEN ACCESS Unveiling SIR Model Parameters: Empirical Parameter Approach for Explicit Estimation and Confidence Interval Construction Nanang Susyanto and Jayrold P. Arcede Volume 5, Issue 1, Pages 54–62, June 2024 Received 26 June 2024, Revised 11 July 2024, Accepted 13 July 2024, Published Online 18 July 2024 To Cite this Article : N. Susyanto and J. P. Arcede,“Unveiling SIR Model Parameters: Empirical Parameter Approach for Explicit Estimation and Confidence Interval Construction”, Jambura J. Biomath, vol. 5, no. 1, pp. 54–62, 2024, https://doi.org/10.37905/jjbm.v5i1.26287 © 2024 by author(s) JOURNAL INFO • JAMBURA JOURNAL OF BIOMATHEMATICS u Ǽ Х Ŷ  ʢ ǽ Ÿ ƒ ̰  Ͳ Homepage Journal Abbreviation Frequency Publication Language DOI Online ISSN Editor-in-Chief Publisher Country OAI Address Google Scholar ID Email : : : : : : : : : : : : http://ejurnal.ung.ac.id/index.php/JJBM/index Jambura J. Biomath. Biannual (June and December) English (preferable), Indonesia https://doi.org/10.37905/jjbm 2723-0317 Hasan S. Panigoro Department of Mathematics, Universitas Negeri Gorontalo Indonesia http://ejurnal.ung.ac.id/index.php/jjbm/oai XzYgeKQAAAAJ JAMBURA JOURNAL • FIND OUR OTHER JOURNALS Jambura Journal of Mathematics Jambura Journal of Mathematics Education Jambura Journal of Probability and Statistics EULER : Jurnal Ilmiah Matematika, Sains, dan Teknologi Jambura Journal of Biomathematics, Volume 5, Issue 1, Pages 54–62, June 2024 https://doi.org/10.37905/jjbm.v5i1.26287 Research Article Check for updates Unveiling SIR Model Parameters: Empirical Parameter Approach for Explicit Estimation and Confidence Interval Construction Nanang Susyanto1,∗  and Jayrold P. Arcede2  1 2 Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia Department of Mathematics, Caraga State University, Butuan City, Philippines ARTICLE HISTORY ABSTRACT. We propose a simple parameter estimation method for the Susceptible-Infectious-Recovered (SIR) model. Received 26 June 2024 Revised 11 July 2024 Accepted 13 July 2024 Published 18 July 2024 This method offers explicit estimates of parameters using second-order numerical derivatives to construct empirical parameters. In addition, the method constructs confidence intervals, providing a robust assessment of parameter uncertainty. To validate the accuracy of our method, we applied it to simulated data, in order to demonstrate its effectiveness in accurately estimating the true model parameters. Furthermore, we applied this method to actual COVID-19 case data from the USA, Indonesia, and the Philippines. This application enables the estimation of parameters and reproductive numbers, along with their confidence intervals, thus underscoring the efficacy of our technique. Notably, the parameter estimates obtained through our approach successfully predicted the case numbers in all three countries, confirming its predictive reliability. Our method offers significant advantages in terms of simplicity and accuracy, making it an invaluable tool for epidemiological modeling and public health planning. KEYWORDS SIR model Parameter estimation Empirical parameters Confidence intervals This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonComercial 4.0 International License. Editorial of JJBM: Department of Mathematics, Universitas Negeri Gorontalo, Jln. Prof. Dr. Ing. B. J. Habibie, Bone Bolango 96554, Indonesia. 1. Introduction The Susceptible-Infectious-Recovered (SIR) model, a cornerstone in epidemiology, has played a pivotal role in understanding the dynamics of infectious diseases for almost a century. Initially introduced by Kermack and McKendrick in 1927, this model categorizes a population into susceptible (S), infectious (I), and recovered (R) compartments [1]. A visual representation of how individuals transition between these compartments is provided in Figure 1. In this paper, we consider the proportion of individuals in each class, meaning we are working with the population proportions. Moreover, we consider a simple shortterm model, i.e., no population turnover, or age structure, inhomogeneities and group behavior. This simplify the model to a system of ordinary differential equations: dS(t) = −βS(t)I(t), dt dI(t) = −βS(t)I(t) − γI(t), dt dR(t) = γI(t), dt (1) with nonnegative initial conditions S(0) = S0 , I(0) = I0 , R(0) = R0 , and domain Ω = {(x, y, z) ∈ R3 : x, y, z ≥ 0 and x + y + z = 1} that is positively invariant under System (1). Here, β and γ represent the infection and removal rates, respectively. It is worth noting that in reality, especially in diseases with ∗ Corresponding Author. Email : (N. Susyanto) Homepage : http://ejurnal.ung.ac.id/index.php/JJBM/index / E-ISSN : 2723-0317 © 2024 by the Author(s). a significant mortality rate, individuals in the R compartment can include both those who have recovered and those who have succumbed to the disease. Although the SIR model may seem deceptively simple, it has been proven to be an immensely powerful tool for analyzing various spreading phenomena. Its versatility extends beyond infectious diseases to encompass a range of phenomena, including the spread of diseases such as herpes [2], influenza [3], and the recent COVID-19 pandemic [4–6]. Furthermore, in non-disease contexts, the SIR model finds application in investigating the spread of behaviors like smoking [7] and computer viruses [8]. Extensions of the SIR model, such as SIRD, SIRS, SEIR, SEIRD, SEIRS, and SIRDS, have further broadened its applicability. These extensions are employed in analyzing the spread of diseases like dengue, incorporating vector compartments [9], capturing phenomena involving reinfection [10], exploring the relationship between different diseases [11, 12], assessing the impact of policies on disease control [13], and evaluating the effectiveness of control measures or treatments in epidemic models [14–16]. Research on the analysis of the SIR model from a mathematical perspective abounds in the literature. The investigation of equilibrium points and their stability in the SIR model [17] provides insights into the interaction of different species in biological communities. From a global stability standpoint, [18] offers a comprehensive analysis. The stability of models with delays is explored in [19], while [20] delves into the stability of the SIR model considering vaccination and treatment. In the wake of the COVID-19 pandemic, the SIR model and N. Susyanto and J. P. Arcede – Unveiling SIR Model Parameters: Empirical Parameter Approach for Explicit… S βSI I γI 55 R Figure 1. SIR Diagram Transfer its extensions, coupled with various mathematical tools, have become indispensable for analyzing the spread of the virus. During the early stages of the pandemic, the SIR model was employed to understand the spread of the virus in Indonesia [4, 21], while in France, case analyses w (...truncated)