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      Analysis of COVID-19 using a modified SLIR model with nonlinear incidence

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          Abstract

          Infectious diseases kill millions of people each year, and they are the major public health problem in the world. This paper presents a modified Susceptible-Latent-Infected-Removed (SLIR) compartmental model of disease transmission with nonlinear incidence. We have obtained a threshold value of basic reproduction number ( R 0 ) and shown that only a disease-free equilibrium exists when R 0 < 1 and endemic equilibrium when R 0 > 1 . With the help of the Lyapunov-LaSalle Invariance Principle, we have shown that disease-free equilibrium and endemic equilibrium are both globally asymptotically stable. The study has also provided the model calibration to estimate parameters with month wise coronavirus (COVID-19) data, i.e. reported cases by worldometer from March 2020 to May 2021 and provides prediction until December 2021 in China. The Partial Rank Correlation Coefficient (PRCC) method was used to investigate how the model parameters’ variation impact the model outcomes. We observed that the most important parameter is transmission rate which had the most significant impact on COVID-19 cases. We also discuss the epidemiology of COVID-19 cases and several control policies and make recommendations for controlling this disease in China.

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          WHO Declares COVID-19 a Pandemic

          The World Health Organization (WHO) on March 11, 2020, has declared the novel coronavirus (COVID-19) outbreak a global pandemic (1). At a news briefing, WHO Director-General, Dr. Tedros Adhanom Ghebreyesus, noted that over the past 2 weeks, the number of cases outside China increased 13-fold and the number of countries with cases increased threefold. Further increases are expected. He said that the WHO is “deeply concerned both by the alarming levels of spread and severity and by the alarming levels of inaction,” and he called on countries to take action now to contain the virus. “We should double down,” he said. “We should be more aggressive.” Among the WHO’s current recommendations, people with mild respiratory symptoms should be encouraged to isolate themselves, and social distancing is emphasized and these recommendations apply even to countries with no reported cases (2). Separately, in JAMA, researchers report that SARS-CoV-2, the virus that causes COVID-19, was most often detected in respiratory samples from patients in China. However, live virus was also found in feces. They conclude: “Transmission of the virus by respiratory and extrarespiratory routes may help explain the rapid spread of disease.”(3). COVID-19 is a novel disease with an incompletely described clinical course, especially for children. In a recente report W. Liu et al described that the virus causing Covid-19 was detected early in the epidemic in 6 (1.6%) out of 366 children (≤16 years of age) hospitalized because of respiratory infections at Tongji Hospital, around Wuhan. All these six children had previously been completely healthy and their clinical characteristics at admission included high fever (>39°C) cough and vomiting (only in four). Four of the six patients had pneumonia, and only one required intensive care. All patients were treated with antiviral agents, antibiotic agents, and supportive therapies, and recovered after a median 7.5 days of hospitalization. (4). Risk factors for severe illness remain uncertain (although older age and comorbidity have emerged as likely important factors), the safety of supportive care strategies such as oxygen by high-flow nasal cannula and noninvasive ventilation are unclear, and the risk of mortality, even among critically ill patients, is uncertain. There are no proven effective specific treatment strategies, and the risk-benefit ratio for commonly used treatments such as corticosteroids is unclear (3,5). Septic shock and specific organ dysfunction such as acute kidney injury appear to occur in a significant proportion of patients with COVID-19–related critical illness and are associated with increasing mortality, with management recommendations following available evidence-based guidelines (3). Novel COVID-19 “can often present as a common cold-like illness,” wrote Roman Wöelfel et al. (6). They report data from a study concerning nine young- to middle-aged adults in Germany who developed COVID-19 after close contact with a known case. All had generally mild clinical courses; seven had upper respiratory tract disease, and two had limited involvement of the lower respiratory tract. Pharyngeal virus shedding was high during the first week of symptoms, peaking on day 4. Additionally, sputum viral shedding persisted after symptom resolution. The German researchers say the current case definition for COVID-19, which emphasizes lower respiratory tract disease, may need to be adjusted(6). But they considered only young and “normal” subjecta whereas the story is different in frail comorbid older patients, in whom COVID 19 may precipitate an insterstitial pneumonia, with severe respiratory failure and death (3). High level of attention should be paid to comorbidities in the treatment of COVID-19. In the literature, COVID-19 is characterised by the symptoms of viral pneumonia such as fever, fatigue, dry cough, and lymphopenia. Many of the older patients who become severely ill have evidence of underlying illness such as cardiovascular disease, liver disease, kidney disease, or malignant tumours. These patients often die of their original comorbidities. They die “with COVID”, but were extremely frail and we therefore need to accurately evaluate all original comorbidities. In addition to the risk of group transmission of an infectious disease, we should pay full attention to the treatment of the original comorbidities of the individual while treating pneumonia, especially in older patients with serious comorbid conditions and polipharmacy. Not only capable of causing pneumonia, COVID-19 may also cause damage to other organs such as the heart, the liver, and the kidneys, as well as to organ systems such as the blood and the immune system. Patients die of multiple organ failure, shock, acute respiratory distress syndrome, heart failure, arrhythmias, and renal failure (5,6). What we know about COVID 19? In December 2019, a cluster of severe pneumonia cases of unknown cause was reported in Wuhan, Hubei province, China. The initial cluster was epidemiologically linked to a seafood wholesale market in Wuhan, although many of the initial 41 cases were later reported to have no known exposure to the market (7). A novel strain of coronavirus belonging to the same family of viruses that cause severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS), as well as the 4 human coronaviruses associated with the common cold, was subsequently isolated from lower respiratory tract samples of 4 cases on 7 January 2020. On 30 January 2020, the WHO declared that the SARS-CoV-2 outbreak constituted a Public Health Emergency of International Concern, and more than 80, 000 confirmed cases had been reported worldwide as of 28 February 2020 (8). On 31 January 2020, the U.S. Centers for Disease Control and Prevention announced that all citizens returning from Hubei province, China, would be subject to mandatory quarantine for up to 14 days. But from China COVID 19 arrived to many other countries. Rothe C et al reported a case of a 33-year-old otherwise healthy German businessman :she became ill with a sore throat, chills, and myalgias on January 24, 2020 (9). The following day, a fever of 39.1°C developed, along with a productive cough. By the evening of the next day, he started feeling better and went back to work on January 27. Before the onset of symptoms, he had attended meetings with a Chinese business partner at his company near Munich on January 20 and 21. The business partner, a Shanghai resident, had visited Germany between January 19 and 22. During her stay, she had been well with no signs or symptoms of infection but had become ill on her flight back to China, where she tested positive for 2019-nCoV on January 26. This case of 2019-nCoV infection was diagnosed in Germany and transmitted outside Asia. However, it is notable that the infection appears to have been transmitted during the incubation period of the index patient, in whom the illness was brief and nonspecific. The fact that asymptomatic persons are potential sources of 2019-nCoV infection may warrant a reassessment of transmission dynamics of the current outbreak (9). Our current understanding of the incubation period for COVID-19 is limited. An early analysis based on 88 confirmed cases in Chinese provinces outside Wuhan, using data on known travel to and from Wuhan to estimate the exposure interval, indicated a mean incubation period of 6.4 days (95% CI, 5.6 to 7.7 days), with a range of 2.1 to 11.1 days. Another analysis based on 158 confirmed cases outside Wuhan estimated a median incubation period of 5.0 days (CI, 4.4 to 5.6 days), with a range of 2 to 14 days. These estimates are generally consistent with estimates from 10 confirmed cases in China (mean incubation period, 5.2 days [CI, 4.1 to 7.0 days] and from clinical reports of a familial cluster of COVID-19 in which symptom onset occurred 3 to 6 days after assumed exposure in Wuhan (10-12). The incubation period can inform several important public health activities for infectious diseases, including active monitoring, surveillance, control, and modeling. Active monitoring requires potentially exposed persons to contact local health authorities to report their health status every day. Understanding the length of active monitoring needed to limit the risk for missing infections is necessary for health departments to effectively use resources. A recent paper provides additional evidence for a median incubation period for COVID-19 of approximately 5 days (13). Lauer et al suggest that 101 out of every 10 000 cases will develop symptoms after 14 days of active monitoring or quarantinen (13). Whether this rate is acceptable depends on the expected risk for infection in the population being monitored and considered judgment about the cost of missing cases. Combining these judgments with the estimates presented here can help public health officials to set rational and evidence-based COVID-19 control policies. Note that the proportion of mild cases detected has increased as surveillance and monitoring systems have been strengthened. The incubation period for these severe cases may differ from that of less severe or subclinical infections and is not typically an applicable measure for those with asymptomatic infections In conclusion, in a very short period health care systems and society have been severely challenged by yet another emerging virus. Preventing transmission and slowing the rate of new infections are the primary goals; however, the concern of COVID-19 causing critical illness and death is at the core of public anxiety. The critical care community has enormous experience in treating severe acute respiratory infections every year, often from uncertain causes. The care of severely ill patients, in particular older persons with COVID-19 must be grounded in this evidence base and, in parallel, ensure that learning from each patient could be of great importance to care all population,
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            The construction of next-generation matrices for compartmental epidemic models.

            The basic reproduction number (0) is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of (0) where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterization of (0). We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number (0). Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that (0) defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that (0) > 1 if and only if r > 0, and (0) = 1 if and only if r = 0.
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              First known person-to-person transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in the USA

              Summary Background Coronavirus disease 2019 (COVID-19) is a disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), first detected in China in December, 2019. In January, 2020, state, local, and federal public health agencies investigated the first case of COVID-19 in Illinois, USA. Methods Patients with confirmed COVID-19 were defined as those with a positive SARS-CoV-2 test. Contacts were people with exposure to a patient with COVID-19 on or after the patient's symptom onset date. Contacts underwent active symptom monitoring for 14 days following their last exposure. Contacts who developed fever, cough, or shortness of breath became persons under investigation and were tested for SARS-CoV-2. A convenience sample of 32 asymptomatic health-care personnel contacts were also tested. Findings Patient 1—a woman in her 60s—returned from China in mid-January, 2020. One week later, she was hospitalised with pneumonia and tested positive for SARS-CoV-2. Her husband (Patient 2) did not travel but had frequent close contact with his wife. He was admitted 8 days later and tested positive for SARS-CoV-2. Overall, 372 contacts of both cases were identified; 347 underwent active symptom monitoring, including 152 community contacts and 195 health-care personnel. Of monitored contacts, 43 became persons under investigation, in addition to Patient 2. These 43 persons under investigation and all 32 asymptomatic health-care personnel tested negative for SARS-CoV-2. Interpretation Person-to-person transmission of SARS-CoV-2 occurred between two people with prolonged, unprotected exposure while Patient 1 was symptomatic. Despite active symptom monitoring and testing of symptomatic and some asymptomatic contacts, no further transmission was detected. Funding None.
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                Author and article information

                Journal
                Results Phys
                Results Phys
                Results in Physics
                Published by Elsevier B.V.
                2211-3797
                21 June 2021
                August 2021
                21 June 2021
                : 27
                : 104478
                Affiliations
                [a ]Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh
                [b ]School of Computing and Mathematics, Charles Sturt University, Wagga Wagga, NSW 2678, Australia
                Author notes
                [* ]Corresponding author.
                Article
                S2211-3797(21)00592-1 104478
                10.1016/j.rinp.2021.104478
                8222049
                34183903
                8adb86ca-f80c-4d7d-a1e3-88c9607af972
                © 2021 Published by Elsevier B.V.

                Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre - including this research content - immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.

                History
                : 1 February 2021
                : 17 June 2021
                : 17 June 2021
                Categories
                Article

                epidemic model,nonlinear incidence,stability analysis,covid-19,simulations

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