ENHANCEMENT AND NUMERICAL ASSESSMENT OF NOVEL SARS-COV-2 VIRUS TRANSMISSION MODEL Anonymous authors Paper under double-blind review

Abstract

Recent pandemic of the coronavirus started in December 2019, which has affected almost all groups of humankind. In this regard, accurate epidemic models are not only crucial for demonstrating the mitigation of the current pandemic but also helpful for forecasting their future dynamics. In this work, we propose a model for SARS-CoV-2 virus transmission to forecast the temporal dynamics of the novel coronavirus disease by considering the characteristics of the disease and the recent literature. Due to the nondeterministic and stochastic nature of the novel-coronavirus disease, we present the model with the aid of stochastic differential equations by considering two infectious phases: pre-symptomatic and symptomatic, because both are significant in the spread of SARS-CoV-2 virus transmission. We ensure that the model is well-posed and identify the necessary conditions for disease eradication by proving the existence, uniqueness, and extinction analysis. The efficacy of the model and the importance of the current study are demonstrated using the actual data. Finally, the model will be simulated using Euler-Maruyama and Milstein's numerical schemes to support the theoretical findings and show the significance of the results obtained.

1. INTRODUCTION

Diseases are mainly categorized into two groups: infectious and non-infectious diseases. Infectious are those caused by viruses, fungi, parasites, and bacteria, and usually transferred in numerous ways while causing fifty thousand deaths approximately every day all over the world. Some infectious diseases can be directly communicated, while many transfer indirectly. Infectious diseases like hepatitis B, seasonal influenza, HIV (human immunodeficiency virus), Middle East Respiratory Syndrome (MERS), and SARS-CoV-2 are major health issues, affecting millions of populations around the globe (Altamimi et al., 2020; Holmdahl & Buckee, 2020; Mann & Roberts, 2011; Reich et al., 2019; Park et al., 2021) . SARS-CoV is a family of viruses that usually cause illnesses such as MERS and severe respiratory syndrome coronavirus (Syed, 2020) . Coronaviruses are zoonotic diseases, and the novel one is a new strain known as the SARS-CoV-2 virus, which broke out in 2019 and spread throughout the world (Rabaan et al., 2020) . Bats are the most plausible ecological reservoirs for SARS-CoV-2, but it's also possible that the virus infected humans via an intermediate animal host. This intermediate animal host could be an unidentified domesticated food animal, a wild animal, or a domesticated wild animal. Millions of people are infected and face consequences due to the novel disease of coronavirus (Shereen et al., 2020) . Novel coronavirus transmits from one person to another by direct contact with an infected individual and indirect with objects used by the infected person (Chan et al., 2020) . A novel coronavirus disease has multiple phases of infections, pre-symptomatic, asymptomatic, environmental, and symptomatic (He et al., 2020) . Especially, the pre-symptomatic and symptomatic phases are very significant because 47% and 38% of cases are reported respectively, by contact with these individuals (Ferretti et al., 2020) . Generally, preasymptomatic individuals have no symptoms while transmitting the disease to others. Therefore, the immigration of pre-symptomatic and symptomatic patients from one place to another place leads to a major source of novel coronavirus transmission. So, most countries around the globe restricted air traffic and announced a lockdown to use the precautionary measure to minimize human lives as much as possible. Also, every country tried to reduce unnecessary traveling, which helped in the reduction of the newly reported cases. Many countries are badly affected by the pandemic of SARS-CoV-2. Moreover, the economy of different countries has been influenced by the novel deadly virus badly. Many organizations and companies have stopped their production. As a result, the ratio of unemployment as well as poverty has increased in various countries. Besides, the health systems of many developed and powerful countries collapsed due to the consequences of the SARS-CoV-2 virus. Mathematical modeling is one of the best tools to show disease mitigation and design control mechanism. The epidemiology of infectious disease has a rich literature, see for instance 2021)). Very recently, a stochastic epidemiological model has been analyzed for the dynamics of novel disease of corona virus by Khan et al. (2021) . Models with appropriate structure and accurate dynamics are not only important to show disease mitigation but also to forecast the future dynamics of the disease, and thus are helpful for public health planning. Indeed, the above-reported studies yielded some interesting outputs. However, their main objectives are to forecast disease dynamics using deterministic differential equations. While the model related to the stochastic analysis of SARS-CoV-2 virus transmission is reported in Khan et al. ( 2021), many factors are to be improved. For example, they considered the random fluctuation only in the disease transmission rate, but the other parameters such as mobility, vaccination, the occurrence of death, etc. also have stochastic nature. Moreover, the classification of various infection phases of SARS-CoV-2 virus disease has not been considered, such that important roles played by pre-symptomatic and symptomatic individuals were neglected in the pandemic trend of novel corona virus disease. In fact, a small number of pre-symptomatic individuals will lead to a major outbreak, because they have no symptoms while transmitting the disease to others. Further, the pandemic of SARS-CoV-2 rises due to human interaction, but initial sources of transmission were a reservoir that has been ignored. Our goal is to enhance the model reported in Khan et al. (2021) by incorporating the missing parameters and characteristics of SARS-CoV-2 virus disease that can influence the disease transmission. To this end, we use various sources of randomness using different Brownian motions in each population group to include the stochastic effect in every parameter as well as in every group of population. We divide the total infected population into two sub-classes, namely pre-symptomatic and symptomatic, according to the characteristics of the disease. We will also assume that both the pre-symptomatic and symptomatic individuals will contribute to producing the reservoirs. To do this, first, we formulate the model and discuss its biological and mathematical feasibility to show the well-posedness of the problem. For this purpose, we will use the combination of stochastic Lyapunov function theory and the Itô formula. Further, the disease extinction of the model will be discussed to find the conditions for disease elimination. We then develop the algorithms for the proposed epidemic problem to discuss the numerical assessment and verify our analytical findings by using Euler-Maruyama and Milstein's methods. To show the effectiveness and justify the proposed epidemic problem, we fit the model to real data of the SARS-CoV-2 virus reported in the United Arab Emirates in the period of March, 21st 2022 to Jun, 21st 2022. Finally, we compare our results to show the significance of the model solution.

2. PRELIMINARIES

In this section, we introduce some of the fundamental concepts and notations that will be helpful in getting our main results, including a multidimensional Itô formula, the strong law of large numbers, and some other results.



, Mann & Roberts (2011); Hattaf et al. (2012); Shi et al. (2015); Alaa & van der Schaar (2019); Kamarthi et al. (2021); Yin et al. (2021). Various models have been used extensively by researchers to study the temporal dynamics of different infectious diseases(Wang et al., 2014; Khan et al., 2018). Moreover, many mathematicians and biologists also studied the dynamical behaviors of SARS-CoV-2 transmission. More precisely, a model study has been reported to forecast the spread of a novel coronavirus outbreak in Wuhan(Wu et al., 2020).Guo et al. (2020)  studied the prediction of host and infectivity of novel diseases using deep learning algorithm. Selvam et al. (2021) analyzed the spread of corona virus diseases using mathematical modeling and performed stability analysis. Further, a spatial-temporal model has been formulated to study the dynamic risk assessment of SARS-CoV-2 accounting for community virus exposure(Chen et al., 2021). Similarly, many other studies have been reported by various authors to forecast the pandemic trend and control of novel disease (SARS-CoV-2) transmissions (see for more details, Arik et al. (2020); Kucharski et al. (2020); Wang et al. (2020); Flaxman et al. (2020); Chen et al. (2021); Zhang et al. (

