A proposed fractional dynamic system and Monte Carlo-based back analysis for simulating the spreading profile of COVID-19

Eur. Phys. J. Spec. Top. https://doi.org/10.1140/epjs/s11734-022-00538-1 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article A proposed fractional dynamic system and Monte Carlo-based back analysis for simulating the spreading profile of COVID-19 Arash Sioofy Khoojine1,a , Mojtaba Mahsuli2,b , Mahdi Shadabfar2,c , Vahid Reza Hosseini3,d , and Hadi Kordestani4,e 1 Faculty of Economics and Business Administration, Yibin University, Yibin 644000, China Department of Civil Engineering, Center for Infrastructure Sustainability and Resilience Research, Sharif University of Technology, Tehran 145888-9694, Iran 3 Institute for Advanced Study, Nanchang University, Nanchang 330031, China 4 School of Civil Engineering, Shandong Jianzhu University, Jinan 250101, China 2 Received 29 October 2021 / Accepted 5 March 2022 © The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract This paper presents a dynamic system for estimating the spreading profile of COVID-19 in Thailand, taking into account the effects of vaccination and social distancing. For this purpose, a compartmental network is built in which the population is divided into nine mutually exclusive nodes, including susceptible, insusceptible, exposed, infected, vaccinated, recovered, quarantined, hospitalized, and dead. The weight of edges denotes the interaction between the nodes, modeled by a series of conversion rates. Next, the compartmental network and corresponding rates are incorporated into a system of fractional partial differential equations to define the model governing the problem concerned. The fractional degree corresponding to each compartment is considered the node weight in the proposed network. Next, a Monte Carlo-based optimization method is proposed to fit the fractional compartmental network to the actual COVID-19 data of Thailand collected from the World Health Organization. Further, a sensitivity analysis is conducted on the node weights, i.e., fractional orders, to reveal their effect on the accuracy of the fit and model predictions. The results show that the flexibility of the model to adapt to the observed data is markedly improved by lowering the order of the differential equations from unity to a fractional order. The final results show that, assuming the current pandemic situation, the number of infected, recovered, and dead cases in Thailand will, respectively, reach 4300, 4.5 × 106 , and 36,000 by the end of 2021. 1 Introduction The mathematical modeling of infectious diseases has been the foundation of infectious disease epidemiology for more than a century [1]. In recent years, detailed scrutiny of such diseases has become extensive because of improvements in data analysis, computing methods, diagnostic tests, and genome sequencing. Advances in such methods enable researchers to deploy mathematical models to design practical schemes for disease control, and also develop and test mathematical and statistical methods [2–6]. Mathematical models that describe the spread of diseases are continually being developed and are playing a fundamental role in promoting public health strategies in many countries [7–10]. a e-mail: e-mail: (corresponding author) c e-mail: (corresponding author) d e-mail: e e-mail: b 0123456789().: V,-vol In December 2019, a novel coronavirus, officially called coronavirus disease 2019 (COVID-19) by the World Health Organization (WHO), led to an outbreak of atypical pneumonia, first in Wuhan, the capital city of Hubei Province in China, and then expeditiously flared out in the entire world [11]. As of October 01, 2021, there have been more than 230 million confirmed cases, together with nearly 4.8 million deaths reports in the world [12]. Throughout anti-epidemic struggles, apart from medical, biological, and scientific research, theoretical studies based on mathematical and statistical modeling may thus significantly contribute to the conception of outbreak attributes [13–15]. Forecasting the trajectory of the spread accordingly helps countries make informed decisions on the required actions to mitigate the spread of the disease. Different models have thus far been proposed to analyze COVID-19, but compartmental methods that divide the population into compartments have dominated epidemiological modeling area [16–19]. These models assist in interpreting the way COVID-19 spreads 123 Eur. Phys. J. Spec. Top. and in forecasting regional pinnacles of the pandemic. Compartmental models are broadly based on studying the systems of differential equations. In this respect, differential equations concentrate on the rate of changes in a variable or a group of variables as time proceeds. The most common model is the SIR, which divides a population into susceptible (S), infected (I), and recovered (R). The popularity of this model stems from its simplicity in predicting a small number of parameters. There are a number of studies that predict the spread of COVID-19 by this method, e.g., see [20,21], and [22]. Various models have been similarly extracted from the basic SIR model, which have thus far supplemented additional compartments to the SIR model [23]. One of the valuable models is the SEIR, namely, the susceptible-exposed-infected-recovered model, which has added the exposed compartment to the SIR structure [24]. The exposed group refers to the phase between the susceptible and the infected individuals. It comprises the population exposed to an infection but not yet infected. The SEIR model is thus capable of analyzing the spread of diseases more accurately than the basic SIR model [25]. Of note, a host of studies have analyzed the COVID-19 pandemic through modified versions of the SEIR model, e.g., see [26,27] and [28]. Some studies have proposed the fractional SIR model, where one or, feasibly, both sub-compartments are raised by exponents that are usually less than unity [29]. The reason is that the infection transmits outwards from an infected population to the whole population through the early stages of an epidemic. Within this framework, where susceptible is much greater than infected it is better to scale them as a fractional power. Given the decreasing number of infected people over the general population in real world, the exponent is anticipated to be larger than 1/2. Furthermore, the power of the susceptible population, which is notably higher than that of the infected population, may be negligible at least in the initial stages of an epidemic. Deploying fractional exponents stemmed from a growth model, called the Norton–Simon–Massagué (NSM) model [30]. This model was created to explain the growth of biological organisms through applying determined energy principles. The governing differential equation reads dG(t) = p1 G(t)α(t) − p1 G(t), dt (1) where p1 and p2 measure anabolism growth and defuse, respectively. Equation 1 may be simplified as declaring the net growth rate of a biological organism (...truncated)