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      A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control

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          Abstract

          Since the first case of 2019 novel coronavirus disease (COVID-19) detected on 30 January, 2020, in India, the number of cases rapidly increased to 3819 cases including 106 deaths as of 5 April, 2020. Taking this into account, in the present work, we have analysed a Bats–Hosts–Reservoir–People transmission fractional-order COVID-19 model for simulating the potential transmission with the thought of individual response and control measures by the government. The real data available about number of infected cases from 14 March, 2000 to 26 March, 2020 is analysed and, accordingly, various parameters of the model are estimated or fitted. The Picard successive approximation technique and Banach’s fixed point theory have been used for verification of the existence and stability criteria of the model. Further, we conduct stability analysis for both disease-free and endemic equilibrium states. On the basis of sensitivity analysis and dynamics of the threshold parameter, we estimate the effectiveness of preventive measures, predicting future outbreaks and potential control strategies of the disease using the proposed model. Numerical computations are carried out utilising the iterative Laplace transform method and comparative study of different fractional differential operators is done. The impacts of various biological parameters on transmission dynamics of COVID-19 is investigated. Finally, we illustrate the obtained results graphically.

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          The continuing 2019-nCoV epidemic threat of novel coronaviruses to global health — The latest 2019 novel coronavirus outbreak in Wuhan, China

          The city of Wuhan in China is the focus of global attention due to an outbreak of a febrile respiratory illness due to a coronavirus 2019-nCoV. In December 2019, there was an outbreak of pneumonia of unknown cause in Wuhan, Hubei province in China, with an epidemiological link to the Huanan Seafood Wholesale Market where there was also sale of live animals. Notification of the WHO on 31 Dec 2019 by the Chinese Health Authorities has prompted health authorities in Hong Kong, Macau, and Taiwan to step up border surveillance, and generated concern and fears that it could mark the emergence of a novel and serious threat to public health (WHO, 2020a, Parr, 2020). The Chinese health authorities have taken prompt public health measures including intensive surveillance, epidemiological investigations, and closure of the market on 1 Jan 2020. SARS-CoV, MERS-CoV, avian influenza, influenza and other common respiratory viruses were ruled out. The Chinese scientists were able to isolate a 2019-nCoV from a patient within a short time on 7 Jan 2020 and perform genome sequencing of the 2019-nCoV. The genetic sequence of the 2019-nCoV has become available to the WHO on 12 Jan 2020 and this has facilitated the laboratories in different countries to produce specific diagnostic PCR tests for detecting the novel infection (WHO, 2020b). The 2019-nCoV is a β CoV of group 2B with at least 70% similarity in genetic sequence to SARS-CoV and has been named 2019-nCoV by the WHO. SARS is a zoonosis caused by SARS-CoV, which first emerged in China in 2002 before spreading to 29 countries/regions in 2003 through a travel-related global outbreak with 8,098 cases with a case fatality rate of 9.6%. Nosocomial transmission of SARS-CoV was common while the primary reservoir was putatively bats, although unproven as the actual source and the intermediary source was civet cats in the wet markets in Guangdong (Hui and Zumla, 2019). MERS is a novel lethal zoonotic disease of humans endemic to the Middle East, caused by MERS-CoV. Humans are thought to acquire MERS-CoV infection though contact with camels or camel products with a case fatality rate close to 35% while nosocomial transmission is also a hallmark (Azhar et al., 2019). The recent outbreak of clusters of viral pneumonia due to a 2019-nCoV in the Wuhan market poses significant threats to international health and may be related to sale of bush meat derived from wild or captive sources at the seafood market. As of 10 Jan 2020, 41 patients have been diagnosed to have infection by the 2019-nCoV animals. The onset of illness of the 41 cases ranges from 8 December 2019 to 2 January 2020. Symptoms include fever (>90% cases), malaise, dry cough (80%), shortness of breath (20%) and respiratory distress (15%). The vital signs were stable in most of the cases while leucopenia and lymphopenia were common. Among the 41 cases, six patients have been discharged, seven patients are in critical care and one died, while the remaining patients are in stable condition. The fatal case involved a 61 year-old man with an abdominal tumour and cirrhosis who was admitted to a hospital due to respiratory failure and severe pneumonia. The diagnoses included severe pneumonia, acute respiratory distress syndrome, septic shock and multi-organ failure. The 2019-nCoV infection in Wuhan appears clinically milder than SARS or MERS overall in terms of severity, case fatality rate and transmissibility, which increases the risk of cases remaining undetected. There is currently no clear evidence of human to human transmission. At present, 739 close contacts including 419 healthcare workers are being quarantined and monitored for any development of symptoms (WHO, 2020b, Center for Health Protection and HKSAR, 2020). No new cases have been detected in Wuhan since 3 January 2020. However the first case outside China was reported on 13th January 2020 in a Chinese tourist in Thailand with no epidemiological linkage to the Huanan Seafood Wholesale Market. The Chinese Health Authorities have carried out very appropriate and prompt response measures including active case finding, and retrospective investigations of the current cluster of patients which have been completed; The Huanan Seafood Wholesale Market has been temporarily closed to carry out investigation, environmental sanitation and disinfection; Public risk communication activities have been carried out to improve public awareness and adoption of self-protection measures. Technical guidance on novel coronavirus has been developed and will continue to be updated as additional information becomes available. However, many questions about the new coronavirus remain. While it appears to be transmitted to humans via animals, the specific animals and other reservoirs need to be identified, the transmission route, the incubation period and characteristics of the susceptible population and survival rates. At present, there is however very limited clinical information of the 2019-nCoV infection and data are missing in regard to the age range, animal source of the virus, incubation period, epidemic curve, viral kinetics, transmission route, pathogenesis, autopsy findings and any treatment response to antivirals among the severe cases. Once there is any clue to the source of animals being responsible for this outbreak, global public health authorities should examine the trading route and source of movement of animals or products taken from the wild or captive conditions from other parts to Wuhan and consider appropriate trading restrictions or other control measures to limit. The rapid identification and containment of a novel coronavirus virus in a short period of time is a re-assuring and a commendable achievement by China’s public health authorities and reflects the increasing global capacity to detect, identify, define and contain new outbreaks. The latest analysis show that the Wuhan CoV cluster with the SARS CoV.10 (Novel coronavirus - China (01): (HU) WHO, phylogenetic tree Archive Number: 20200112.6885385). This outbreak brings back memories of the novel coronavirus outbreak in China, the severe acute respiratory syndrome (SARS) in China in 2003, caused by a novel SARS-CoV-coronavirus (World Health Organization, 2019a). SARS-CoV rapidly spread from southern China in 2003 and infected more than 3000 people, killing 774 by 2004, and then disappeared – never to be seen again. However, The Middle East Respiratory Syndrome (MERS) Coronavirus (MERS-CoV) (World Health Organization, 2019b), a lethal zoonotic pathogen that was first identified in humans in the Kingdom of Saudi Arabia (KSA) in 2012 continues to emerge and re-emerge through intermittent sporadic cases, community clusters and nosocomial outbreaks. Between 2012 and December 2019, a total of 2465 laboratory-confirmed cases of MERS-CoV infection, including 850 deaths (34.4% mortality) were reported from 27 countries to WHO, the majority of which were reported by KSA (2073 cases, 772 deaths. Whilst several important aspects of MERS-CoV epidemiology, virology, mode of transmission, pathogenesis, diagnosis, clinical features, have been defined, there remain many unanswered questions, including source, transmission and epidemic potential. The Wuhan outbreak is a stark reminder of the continuing threat of zoonotic diseases to global health security. More significant and better targeted investments are required for a more concerted and collaborative global effort, learning from experiences from all geographical regions, through a ‘ONE-HUMAN-ENIVRONMENTAL-ANIMAL-HEALTH’ global consortium to reduce the global threat of zoonotic diseases (Zumla et al., 2016). Sharing experience and learning from all geographical regions and across disciplines will be key to sustaining and further developing the progress being made. Author declarations All authors have a specialist interest in emerging and re-emerging pathogens. FN, RK, OD, GI, TDMc, CD and AZ are members of the Pan-African Network on Emerging and Re-emerging Infections (PANDORA-ID-NET) funded by the European and Developing Countries Clinical Trials Partnership the EU Horizon 2020 Framework Programme for Research and Innovation. AZ is a National Institutes of Health Research senior investigator. All authors declare no conflicts of interest.
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            Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts

            Summary Background Isolation of cases and contact tracing is used to control outbreaks of infectious diseases, and has been used for coronavirus disease 2019 (COVID-19). Whether this strategy will achieve control depends on characteristics of both the pathogen and the response. Here we use a mathematical model to assess if isolation and contact tracing are able to control onwards transmission from imported cases of COVID-19. Methods We developed a stochastic transmission model, parameterised to the COVID-19 outbreak. We used the model to quantify the potential effectiveness of contact tracing and isolation of cases at controlling a severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)-like pathogen. We considered scenarios that varied in the number of initial cases, the basic reproduction number (R 0), the delay from symptom onset to isolation, the probability that contacts were traced, the proportion of transmission that occurred before symptom onset, and the proportion of subclinical infections. We assumed isolation prevented all further transmission in the model. Outbreaks were deemed controlled if transmission ended within 12 weeks or before 5000 cases in total. We measured the success of controlling outbreaks using isolation and contact tracing, and quantified the weekly maximum number of cases traced to measure feasibility of public health effort. Findings Simulated outbreaks starting with five initial cases, an R 0 of 1·5, and 0% transmission before symptom onset could be controlled even with low contact tracing probability; however, the probability of controlling an outbreak decreased with the number of initial cases, when R 0 was 2·5 or 3·5 and with more transmission before symptom onset. Across different initial numbers of cases, the majority of scenarios with an R 0 of 1·5 were controllable with less than 50% of contacts successfully traced. To control the majority of outbreaks, for R 0 of 2·5 more than 70% of contacts had to be traced, and for an R 0 of 3·5 more than 90% of contacts had to be traced. The delay between symptom onset and isolation had the largest role in determining whether an outbreak was controllable when R 0 was 1·5. For R 0 values of 2·5 or 3·5, if there were 40 initial cases, contact tracing and isolation were only potentially feasible when less than 1% of transmission occurred before symptom onset. Interpretation In most scenarios, highly effective contact tracing and case isolation is enough to control a new outbreak of COVID-19 within 3 months. The probability of control decreases with long delays from symptom onset to isolation, fewer cases ascertained by contact tracing, and increasing transmission before symptoms. This model can be modified to reflect updated transmission characteristics and more specific definitions of outbreak control to assess the potential success of local response efforts. Funding Wellcome Trust, Global Challenges Research Fund, and Health Data Research UK.
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              A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action

              Highlights • For the ongoing novel coronavirus disease (CODID-19) outbreak in Wuhan, China, the Chinese government has implemented control measures such as city lockdown to mitigate the impact of the epidemic. • We model the outbreak in Wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalisation and quarantine) based on some parameters of the 1918 influenza pandemic in London, United Kingdom. • We show the different effects of individual reaction and governmental action and preliminarily estimate the magnitude of these effects. • We also preliminarily estimate the time-varying reporting ratio.
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                Author and article information

                Contributors
                amjatshaikh@gmail.com
                shaikhiqbaln@gmail.com
                n.sooppy@psau.edu.sa , ksnisar1@gmail.com
                Journal
                Adv Differ Equ
                Adv Differ Equ
                Advances in Difference Equations
                Springer International Publishing (Cham )
                1687-1839
                1687-1847
                22 July 2020
                22 July 2020
                2020
                : 2020
                : 1
                : 373
                Affiliations
                [1 ]Department of Mathematics, AKI’s Poona College of Arts, Science and Commerce, Camp, Pune, India
                [2 ]Department of Chemistry, AKI’s Poona College of Arts, Science and Commerce, Camp, Pune, India
                [3 ]GRID grid.449553.a, Department of Mathematics, College of Arts and Sciences, , Prince Sattam bin Abdulaziz University, ; Wadi Aldawaser, 11991 Saudi Arabia
                Author information
                http://orcid.org/0000-0003-4829-1698
                http://orcid.org/0000-0001-5769-4320
                Article
                2834
                10.1186/s13662-020-02834-3
                7374076
                32226454
                61710b7b-dedb-48e2-b6bc-1188708c3911
                © The Author(s) 2020

                Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

                History
                : 16 April 2020
                : 13 July 2020
                Categories
                Research
                Custom metadata
                © The Author(s) 2020

                35a20,35a22,34a08,35r11,coronavirus,caputo–fabrizio derivative,basic reproduction number,existence and stability,numerical simulations

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