key: cord-0729064-v84czpww authors: Oluyori, D. A.; Adebayo, H. O. title: Global Analysis of an SEIRS Model for COVID-19 Capturing Saturated Incidence with Treatment Response date: 2020-05-20 journal: nan DOI: 10.1101/2020.05.15.20103630 sha: 340a166b0ad68b9fdfdeef9307ad2c85f40cb479 doc_id: 729064 cord_uid: v84czpww Sequel to [10], who studied the dynamics of COVID-19 using an SEIRUS model. We consider an SEIRS model capturing saturated incidence with treatment response. In this theoretical model, we assumed that the treatment response is proportional to the number of infected as long as the incidence cases are within the capacity of the healthcare system, after which the value becomes constant, when the number of confirmed cases exceed the carrying capacity of the available medical facilities. Thus, we obtain the reproduction number stating that when , the disease free equilibrium is globally asymptotically stable. Also, we studied the existence of the local and global stability of the disease free and endemic equilibria and found that the kind of treatment response and inhibitory measures deployed in tackling the COVID-19 pandemic determines whether the disease will die out or become endemic. The novel coronavirus disease (COVID- 19) was first confirmed in the Chinese city of Wuhan, late December, 2019. The rapidity of its spread in many countries around the globe made the WHO declare it as a global pandemic and public health emergency, raising concerns that if countries with robust healthcare systems to detect and control disease outbreak are having challenges managing the disease, countries with weak healthcare system need to put adequate measures in place to contain the spread [1] . The coronavirus disease caused by the Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) presents clinical features which are similar to the diseases caused by other coronaviruses, Severe Acute Respiratory Syndrome (SARS) and Middle East Respiratory Syndrome such as lower respiratory illness with fever, dry cough, myalgia, shortness of breath etc. The Coronavirus disease is "novel" in the sense that, it is a new strain of zoonotic origin which has not been previously discovered to affect humans. Historically, the COVID-19 pandemic is a major human coronavirus epidemic in the last two decades aside SARS [2] and MERS [3, 4] respectively. The incubation period of COVID-19 is between 2 -14 days with symptoms averagely between 5-7 days. Its basic reproduction number is averaged 2.2 [5] and even more ranging from 1.4 -6.5 in [6] . Globally as at May According to literatures the bilinear and standard incidences has been extensively studied by various authors in [33 -36] and others. Another kind of incidence that is of interest to our work is the Saturation incidence. In 1973, sequel to the study of cholera epidemic which occurred in Bari, Capasso and Serio [22] introduced the saturated incidence denoted as ∞ Matias [20] studied a model for disease transmitted by vector with saturating incidence such that the model assumes a saturating effect in the incidence due to the response of the vector to change in the susceptible and infected densities. The Saturation incidence seems more realistic than the bilinear incidence due to the inclusion of behavioural change and crowding effect of the infective. In the face of the current realities from COVID-19, it is evident that we have high saturation incidence in which useful strategies need to be deployed to contain the spread through various interventions such as good hygiene, physical/social distancing, partial/total lockdown, travel/public gathering ban, good treatments, contact tracing, pool testing etc can help to reduce the high rate of secondary infections as stipulated by the WHO guidelines. It is a general assumption in classical epidemic models that treatment rate of infection is assumed to be proportional to the number of the infective individuals and the recovery rate depends on the medical resources available such as test kits, drugs, isolation centres, ventilators, availability of trained medical personnel, efficiency of treatment. WHO situation reports from many nations have shown how stretched the healthcare systems of countries have been with its attendant high morbidity. Therefore, it is important for countries with increased cases to adopt suitable treatment function. Wang and Ruan [23] introduced a constant treatment in SIR models as follows: is the positive constant and ‫ܫ‬ is the number of infected individuals. Recently, Wang [24] considered a piecewise linear treatment function defined as: The first conditions in (3) explains the proportionality of the treatment response to the number of the infectious people when the number of infectious is less than or equal to a fixed value ‫ܫ‬ , the second typifies an endemic situation, where the number of the infectious has increased to a saturation point where the available medical facilities are stretched beyond capacity and death toll rises in an unprecedented manner. Therefore in many disease outbreak there are different kinds of delays when they spread, such as latent period delay before symptoms surfaces and immunity period delay after recovery. Zhou [25] [26] considering an SEIRS epidemic model with saturated incidence and treatment rate. Badole et'al in [28] taking some cue from [27] studied the global dynamics of an SEIR model with saturated incidence under treatment. Various authors have considered saturated incidence and treatment to study the stability and bifurcation of different dynamic systems in [24, 34, 37 -39 ]. Since the outbreak, many mathematical have appeared in an attempt to assess the dynamics of the COVID-19 epidemic. The first models were dynamic mechanistic models aimed at estimating of the basic reproduction number ܴ [12, 13, 14, 15] , also simple exponential growth models [16, 17] . Other compartmental epidemiological models such as SIRD, SEIR, SEIRD and SEIRUS [21, 19, 18, 10,] has been proposed to estimate other epidemiological parameters such as the transmission rate, local and global stability of the disease-free and endemic equilibria to provide insights for forecasting purposes. Recently, [10] considered an SEIRUS (Susceptible-Exposed-Infected-Recovered-Undetectable-Susceptible) model for COVID-19, where it was predicted that with strict adherence to the guidelines of the WHO on observatory and treatment procedures, the pandemic will soon die out. Based on the motivation from [10, 24, 27, 28 and 29] , we present an SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) model with saturated incidence and treatment functions which prescribes inbihitory measures such as personal hygiene, wearing of face mask, travel/public gathering ban, partial or total lockdown etc and rapid responses such as public enlightenment, pool testing, increased medical facilities and trained medical personnel etc., as . potent means of slowing down the spread of COVID-19. The model can be described as follows: is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103630 doi: medRxiv preprint inhibitory effect, ߚ is the rate of transmission ߝ is the proportion of the removed population that is been observed and will subsequently moved to the susceptible, ߛ is the rate of developing infection/incidence rate. ߮ is the disease induced death rate of the infected population not quarantined ߸ is the fraction of the removed population under observation (the undetected) before moving to the susceptible class, is the saturation incidence parameter, is the inhibitory parameter and ܶ ሺ ‫ܫ‬ ሻ is the treatment response as defined in (3). It follows from (4) that The feasible region for system (4) is Thus is naturally follows that the region Ω is positively invariant with respect to system (4). Hence the system is mathematically and epidemiological well posed in Ω . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. By simple calculation from the system (5) we obtain the equilibrium state where the LHS vanish). Thus the steady state of system (5) satisfy the following algebraic system of equation: At the disease free equilibrium, when no disease outbreak occurs, no one is in the exposed or infected class and as such no one is in the recovered class. Therefore, . On substitution the algebraic in (6) The reproduction number , the disease will be endemic in the population. Next we find the reproduction number, ܴ of the system (5) by obtaining the Jacobian of the system and using the Next Generation Matrix due to Driessche and Watmough [32] . Applying the disease free state condition in the Jacobian matrix, we have . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. (8) Using the next generation matrices it is clear that the reproduction number ܴ is the spectral radius of the next generation matrix derived from the exposed and infected class i.e. , is the next generation matrix. ‫ܨ‬ is derived from the exposed and infected class and ܸ are the remaining terms after ‫ܨ‬ is taken. is the next generation matrix of the system (5). The spectral radius is Hence the reproduction number, . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. We examined the local stability of the equilibrium by the analysis of the eigenvalues of the Jacobian matrices of (5) at the equilibrium using the Routh Hurwitz Criterion. By (11) it is clear that are the two roots of (11) . The other roots of (11) are determined by the equation. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. ‫ז‬ . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103630 doi: medRxiv preprint In this section we analyze the global stability of the disease-free equilibrium and to do this we reduce the system of equation in (5) We consider the geometric approach due to Li and Muldowney [31] , to obtain the global stability of the endemic equilibrium and find that the sufficient conditions for which the endemic equilibrium is globally asymptotically stable. We describe the geometric approach method as follows. We consider Where the matrix (10) …….. (18) . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. It is shown in [31] that, if ‫ܦ‬ is simply connected, the condition ‫ݍ‬ 0 rules out the presence of any orbit that gives rise to a simple closed rectifiable curve that is invariant for (16) , such as periodic orbits, homoclinic orbits and heteroclinic cycles. Moreover, it is robust under ‫ܥ‬ ଵ local perturbations of ݂ near any non equilibrium point that is non-wandering. In particular, the following global stability result is proved in [31] . And its second additive matrix is . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 20, 2020. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 20, 2020. In this section we analyze the global stability of the endemic steady states. After reducing the system of equation in (34) And its second additive matrix is . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 20, 2020. . World Health Organization Coronavirus disease (COVID-2019) situation reports 2020 Severe Acute Respiratory Syndrome Discovery of a novel coronavirus associated with recent pneumonia outbreak in human and its potential bat origin Isolation of a novel coronavirus from a man with pneumonia in Saudi Arabia Pattern of Early Human-to-Human Transmission of Wuhan 2019 Novel Coronavirus The Reproduction Number of COVID-19 is Higher Compared to SARS Coronavirus A Coronavirus Associated with Human Respiratory Disease in China Structure Analysis of the Receptor Binding of 2019 n-CoV. Biochemical and Biophysical Research Communications Identification of Novel Compounds Against Three Targets of SARS-CoV-2 Coronavirus by Combined Virtual Screening and Supervised Machine Learning Mathematical Predictions for COVID-19 as a Global Pandemic Causes and Clinical Manifestation and Diagnosis, Prevention and Control of Coronavirus Disease (COVID-19) During the Early Outbreak Period: a Scoping Review Nowcasting and Forecasting the Potential Domestic and International Spread of the 2019-nCoV Outbreaks Originating in Wuhan, China: A Modeling Study, The Lancet Transmissibility of 2019-nCoV Estimating the Scale of COVID-19 epidemic in the United States: Simlations Based on Air Traffic Directly From Wuhan Data-based Analysis, Modeling and Forecasting of the COVID-19 Outbreak Preliminary Estimation of the Basic Reproduction Number of Novel Coronavirus (2019-nCoV) in China from 2019 to 2020: A Data -driven Analysis in the early phase of the outbreak COVID-19 and Italy: What Next Breaking down of the healthcare system: Mathematical Modeling for Controlling the novel coronavirus (2019-nCoV) Outbreak in Wuhan China Tracing DAY-ZERO and Forecasting the Fade out of the COVID-19 Outbreak in Lombardy, Italy: A Compartmental Modelling and Numerical Optimization Approach Contributions to the Mathematical Theory of Epidemics A Generalization of the Kermack-McKendrick Deterministic Epidemic Model Bifurcations in an epidemic model with constant removal rate of incentives Backward Bifurcation of an Epidemic Model with Treatment Dynamics of an SIR Model with Limited Medical Resources revisited Analysis of an SEIR Epidemic Model with Saturated Incidence and Saturated Treatment Function Global Analysis of an SEIRS Epidemic Model With Saturated Incidence and Saturated Treatment Global Dynamics of an SEIR Epidemic Model with Saturated Incidence Under Treatment Modeling and analysis of an SEIR Epidemic Model with a Limited Resource for Treatment Population Biology of Infectious I A geometric Approach to Global Stability Problems Reproduction Numbers and Subthreshold Endemic Equilibria for Compartmental Models of Disease Transmission An Exact Global Solution for the Classical Epidemic Model Backward Bifurcation of an Epidemic Model with Standard Incidence Rate and Treatment Rate Global Dynamics of a Class of SEIRS Epidemic Model with Time Delay Global Dynamics of a Class of SEIRS Epidemic Models in a Periodic Environment Backward bifurcation of an epidemic modelwith saturated treatment function Bifurcation analysis for a delayed SEIR epidemic model with saturated incidence and saturated treatment function Bifurcation Analysis of an SIR Model with Logistic Growth,Nonlinear Incidence, and Saturated Treatment Integrating both sides simultaneously, we haveThus, by (17) It follows from the system of equation in (34)The feasible region for system (4) isTherefore the system of equation (34) is epidemiologically well-posed. We consider the local stability of the endemic equilibrium point Proof: The Jacobian matrix of the system (34) atFrom which we obtain the characteristic equation pandemic determines whether the disease will die out or become endemic. So how long COVID-19 pandemic stays with us depends on how much we are willing to take responsibility as individuals and government.