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      Qualitative Analysis of a Mathematical Model in the Time of COVID-19

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          Abstract

          In this article, a qualitative analysis of the mathematical model of novel corona virus named COVID-19 under nonsingular derivative of fractional order is considered. The concerned model is composed of two compartments, namely, healthy and infected. Under the new nonsingular derivative, we, first of all, establish some sufficient conditions for existence and uniqueness of solution to the model under consideration. Because of the dynamics of the phenomenon when described by a mathematical model, its existence must be guaranteed. Therefore, via using the classical fixed point theory, we establish the required results. Also, we present the results of stability of Ulam's type by using the tools of nonlinear analysis. For the semianalytical results, we extend the usual Laplace transform coupled with Adomian decomposition method to obtain the approximate solutions for the corresponding compartments of the considered model. Finally, in order to support our study, graphical interpretations are provided to illustrate the results by using some numerical values for the corresponding parameters of the model.

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          Most cited references24

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          A pneumonia outbreak associated with a new coronavirus of probable bat origin

          Since the outbreak of severe acute respiratory syndrome (SARS) 18 years ago, a large number of SARS-related coronaviruses (SARSr-CoVs) have been discovered in their natural reservoir host, bats 1–4 . Previous studies have shown that some bat SARSr-CoVs have the potential to infect humans 5–7 . Here we report the identification and characterization of a new coronavirus (2019-nCoV), which caused an epidemic of acute respiratory syndrome in humans in Wuhan, China. The epidemic, which started on 12 December 2019, had caused 2,794 laboratory-confirmed infections including 80 deaths by 26 January 2020. Full-length genome sequences were obtained from five patients at an early stage of the outbreak. The sequences are almost identical and share 79.6% sequence identity to SARS-CoV. Furthermore, we show that 2019-nCoV is 96% identical at the whole-genome level to a bat coronavirus. Pairwise protein sequence analysis of seven conserved non-structural proteins domains show that this virus belongs to the species of SARSr-CoV. In addition, 2019-nCoV virus isolated from the bronchoalveolar lavage fluid of a critically ill patient could be neutralized by sera from several patients. Notably, we confirmed that 2019-nCoV uses the same cell entry receptor—angiotensin converting enzyme II (ACE2)—as SARS-CoV.
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            Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia

            Abstract Background The initial cases of novel coronavirus (2019-nCoV)–infected pneumonia (NCIP) occurred in Wuhan, Hubei Province, China, in December 2019 and January 2020. We analyzed data on the first 425 confirmed cases in Wuhan to determine the epidemiologic characteristics of NCIP. Methods We collected information on demographic characteristics, exposure history, and illness timelines of laboratory-confirmed cases of NCIP that had been reported by January 22, 2020. We described characteristics of the cases and estimated the key epidemiologic time-delay distributions. In the early period of exponential growth, we estimated the epidemic doubling time and the basic reproductive number. Results Among the first 425 patients with confirmed NCIP, the median age was 59 years and 56% were male. The majority of cases (55%) with onset before January 1, 2020, were linked to the Huanan Seafood Wholesale Market, as compared with 8.6% of the subsequent cases. The mean incubation period was 5.2 days (95% confidence interval [CI], 4.1 to 7.0), with the 95th percentile of the distribution at 12.5 days. In its early stages, the epidemic doubled in size every 7.4 days. With a mean serial interval of 7.5 days (95% CI, 5.3 to 19), the basic reproductive number was estimated to be 2.2 (95% CI, 1.4 to 3.9). Conclusions On the basis of this information, there is evidence that human-to-human transmission has occurred among close contacts since the middle of December 2019. Considerable efforts to reduce transmission will be required to control outbreaks if similar dynamics apply elsewhere. Measures to prevent or reduce transmission should be implemented in populations at risk. (Funded by the Ministry of Science and Technology of China and others.)
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              Outbreak of pneumonia of unknown etiology in Wuhan, China: The mystery and the miracle

              Since December 2019, a total of 41 cases of pneumonia of unknown etiology have been confirmed in Wuhan city, Hubei Province, China. Wuhan city is a major transportation hub with a population of more than 11 million people. Most of the patients visited a local fish and wild animal market last month. At a national press conference held today, Dr Jianguo Xu, an academician of the Chinese Academy of Engineering, who led a scientific team announced that a new‐type coronavirus, tentatively named by World Health Organization as the 2019‐new coronavirus (2019‐nCoV), had caused this outbreak. 1 The 2019‐nCoV has a different coronavirus‐specific nucleic acid sequence from known human coronavirus species, which are similar to some of the beta coronaviruses identified in bats. 2 , 3 The virus‐specific nucleic acid sequences were detected in lung fluid, blood and throat swab samples in 15 patients and the virus that was isolated showed a typical coronavirus appearance under electron microscopy. Further research will be conducted to better understand the new coronavirus to develop antiviral agents and vaccines. 4 We applauded the excellent job that has been done so far. The infection was first described in December. Within 9 days, a special team consisted of physicians, scientists and epidemiologists who ruled out several extremely contagious pathogens including SARS, which killed hundreds of people more than a decade ago, and MERS. This has surely alleviated environmental concerns as Hong Kong authorities had quickly stepped up the disinfection of trains and airplanes and checks of passengers due to this outbreak. Most of the patients visited the fish and wild animal market last month in Wuhan. This fish and wild animal market also sold live animals such as poultry, bats, marmots, and snakes. All patients received prompt supportive treatment in quarantine. Among them, seven patients were in serious condition and one patient died. All of the 42 patients so far confirmed were from China except one Thailand patient who was a traveler from Wuhan. Eight patients have been cured of the disease and were discharged from the hospital last week. The 2019‐nCoV now have been isolated from multiple patients and appears to be the culprit. But the mystery has not been completely solved yet. Until there is a formal published scientific manuscript, the facts can be argued, particularly regarding causality despite these facts having been officially announced. The data collected so far is not enough to confirm the causal relationship between the new‐type coronavirus and the respiratory disease based on classical Koch's postulates or modified ones as suggested by Fredricks and Relman. 5 The viral‐specific nucleic acids were only discovered in 15 patients, and successful virus culture was extremely limited to only a few patients. There remains considerable work to be done to differentiate between colonization, shedding, and infection. Additional strains of the 2019‐nCoV need to be isolated to study their homologies. It is expected that antigens and monoclonal antibodies will be developed so serology can be used to confirm previous and acute infection status. The episode demonstrates further the need for rapid and accurate detection and identification methods that can be used in the local hospitals and clinics bearing the burden of identifying and treating patients. Recently, the Clinical Laboratory Improvement Amendments (CLIA) of 1988 has waived highly sensitive and specific molecular devices known as CLIA‐waived devices so that these devices are gradually becoming available for point of care testing. Finally, the epidemiological similarity between this outbreak and that of SARS in 2002‐2003 6 is striking. SARS was then traced to animal markets 7 and eventually to palm civets. 8 Later bats were identified as animal reservoirs. 9 Could this novel coronavirus be originated from wild animals? The family Coronaviridae includes two subfamilies. 10 One, the subfamily Coronavirinae, contains a substantial number of pathogens of mammals that individually cause a remarkable variety of diseases, including pneumonia. In humans, coronaviruses are among the spectrum of viruses that cause the common cold as well as more severe respiratory disease—specifically SARS and MERS, which are both zoonoses. The second subfamily, Torovirinae, contains pathogens of both terrestrial and aquatic animals. The genus Torovirus includes the type species, equine torovirus (Berne virus), which was first isolated from a horse with diarrhea, and the Breda virus, which was first isolated from neonatal calves with diarrhea. White bream virus from fish is the type species of the genus Bafinivirus. However, there is no evidence so far that the seafood from the fish and animal market caused 2019‐nCoV‐associated pneumonia. This epidemiologic similarity clearly provides a starting point for the further investigation of this outbreak. In the meantime, this fish and animal market has been closed until the epidemiological work determines the animal host of this novel coronavirus. Only then will the miracle be complete.
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                Author and article information

                Contributors
                Journal
                Biomed Res Int
                Biomed Res Int
                BMRI
                BioMed Research International
                Hindawi
                2314-6133
                2314-6141
                2020
                25 May 2020
                : 2020
                : 5098598
                Affiliations
                1Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan
                2Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
                3Department of Medical Research, China Medical University, Taichung 40402, Taiwan
                4Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan
                5College of Engineering and Technology, American University of the Middle East, Kuwait
                6Department of Mathematics, Çankaya University, Ankara 06790, Turkey
                Author notes

                Guest Editor: Sinan Deniz

                Author information
                https://orcid.org/0000-0002-8851-4844
                https://orcid.org/0000-0002-8889-3768
                https://orcid.org/0000-0002-7222-3014
                Article
                10.1155/2020/5098598
                7273369
                33204691
                0e005b49-96cb-44fa-b311-90ee190132a5
                Copyright © 2020 Kamal Shah et al.

                This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

                History
                : 2 April 2020
                : 30 April 2020
                Categories
                Research Article

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