Forecasting the daily and cumulative number of cases for the COVID-19 pandemic in India
aa r X i v : . [ q - b i o . P E ] J un Forecasting the daily and cumulative number of cases for the COVID-19 pandemic inIndia
Subhas Khajanchi ∗ Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India.
Kankan Sarkar † Department of Mathematics, Malda College, Malda, West Bengal 732101, India (Dated: June 26, 2020)The ongoing novel coronavirus epidemic has been announced a pandemic by the World HealthOrganization on March 11, 2020, and the Govt. of India has declared a nationwide lockdownfrom March 25, 2020, to prevent community transmission of COVID-19. Due to absence of specificantivirals or vaccine, mathematical modeling play an important role to better understand the diseasedynamics and designing strategies to control rapidly spreading infectious diseases. In our study, wedeveloped a new compartmental model that explains the transmission dynamics of COVID-19. Wecalibrated our proposed model with daily COVID-19 data for the four Indian provinces, namelyJharkhand, Gujarat, Andhra Pradesh, and Chandigarh. We study the qualitative properties of themodel including feasible equilibria and their stability with respect to the basic reproduction number R . The disease-free equilibrium becomes stable and the endemic equilibrium becomes unstablewhen the recovery rate of infected individuals increased but if the disease transmission rate remainshigher then the endemic equilibrium always remain stable. For the estimated model parameters, R > In India, 173,763 confirmed cases, 7,964 con-firmed new cases and 4,971 confirmed deaths dueto COVID-19 were reported as of May 30, 2020.As the ongoing COVID-19 outbreak is quicklyspreading throughout India and globe, short-termmodeling predictions give time-critical statisticsfor decisions on containment and mitigation poli-cies. A big problem for short-term predictionis the evaluation of important parameters andhow they alter when the first interventions re-veal an effect. In the absence of any therapeuticsor licensed vaccine, antivirals, isolation of pop-ulations diagnosed with COVID-19 and quaran-tine of populations feared exposed to COVID-19were used to control the rapid spread of infec-tion. During this alarming situation, forecastingis of utmost priority for health care planning andto control the SARS-CoV-2 virus with limitedresources. We proposed a mathematical modelthat monitors the dynamics of six compartments,namely susceptible (S), asymptomatic (A), re-ported symptomatic (I), unreported symptomatic(U), quarantine (Q), and recovered (R) individ-uals, collectively termed SAIUQR, that predictsthe course of the epidemic. Our SAIUQR modeldiscriminates between reported and unreportedinfected individuals, which is important as theformer are typically isolated and hence less likelyto spread the infection. A detailed theoretical ∗ [email protected] † [email protected] analysis has been done for our SAIUQR modelin terms of the basic reproduction number R .All the analytical findings are verified numeri-cally for the estimated model parameters. Wehave calibrated our SAIUQR model with real ob-served data on the COVID-19 outbreak in thefour provinces of India. The basic reproduc-tion number for all the provinces is greater thanunity, which resulted in a substantial outbreakof COVID-19. Based on the simulation, ourSAIUQR model predict that on June 13, 2020,the daily new COVID-19 cases will be around 15,454, 12, 96, and cumulative number of COVID-19 cases will be around 661, 23955, 514, 4487,in Jharkhand, Gujarat, Chandigarh and AndhraPradesh, respectively. I. INTRODUCTION
After a novel strain of COVID-19, was detected inWuhan, the city of Hubei province, China, in Decem-ber 2019 [1], an exponentially increasing number of pa-tients in mainland China were identified with SARS-CoV-2, immediately the Chinese Health authorities toinitiate radical measures to control the epidemic coron-avirus. In spite of these radical measures, a SARS-CoV-2coronavirus pandemic ensued in the subsequent monthsand China became the epicenter. SARS-CoV-2 virusesare enveloped non-segmented positive-sense RNA virusesthat belong to the Coronaviridae family and the orderNidovirales, and are extensively disseminated among hu-mans as well as mammals [2]. The COVID-19 is re-sponsible for a range of symptoms together with fever,dry cough, breathing difficulties, fatigue and lung in-filtration in severe cases, similar to those created bySARS-CoV (severe acute respiratory syndrome coron-avirus) and MERS-CoV (Middle East respiratory syn-drome coronavirus) infections [3]. SARS-CoV-2 has al-ready crossed the earlier history of two coronavirus epi-demics SARS-CoV and MERS-CoV, posing the substan-tial threat to the world people health problem as well aseconomical problem after the second world war [4]. Ac-cording to the World Health Organization report, dated29 May 2020 reported 5,596,550 total cases and 353,373deaths worldwide [5].Till date, there is no licensed vaccine, drugs and effec-tive therapeutics available for SARS-CoV-2 or COVID-19. Due to absence of pharmaceutical interventions,Govt. of various countries are adopting different strate-gies to control the outbreak and the most common oneis the nation-wide lockdown. It was started with thelocal Govt. of Wuhan by temporarily prevention of allpublic traffics within the city on January 23, 2020 andsoon followed by other cities in Hubei province [6]. Inthe absence of drug or specific antivirals for SARS-CoV-2 viruses, maintaining social distancing is the only wayto mitigate the human-to-human transmission for coron-avirus diseases, and thus the other countries also incor-porated the strict lockdown, quarantines and curfews.In India, the first coronavirus case was reported in Ker-ala’s Thrissur district on January 30, 2020 when a stu-dent returned back from Wuhan, the sprawling capitalof China’s Hubei province [7]. The Govt. of India hasimplemented a complete nationwide lockdown through-out the country on and from March 25, 2020 for 21 days,following one day ‘Janata Curfew’ on March 22, 2020 tocontrol the coronavirus or SARS-CoV-2 pandemic in In-dia [8]. Due to massive spread of coronavirus diseases,the Govt. of India has extended the lockdown and it isgoing on Phase 4; from May 18, 2020 to May 31, 2020.Besides the implementation of nationwide lockdown, theMinistry of Health and Family Welfare (MOHFW) of In-dia, recommended different individual hygiene measures,for example, frequent hand washing, social distancing,use of mask, avoid gathering and touching eyes, mouthand nose etc. [9].The Govt. also ceaselessly using different media andsocial network to aware the public regarding coronavirusdiseases and its precautions. Albeit, the factors such asdiverse and huge population, the unavailability of spe-cific therapeutics, drugs or licensed vaccines, inadequateevidences regarding the mechanism of disease transmis-sion make it strenuous to combat against the coron-avirus diseases throughout India. To control the trans-mission of COVID-19, lockdown is a magnificent measurebut testing is also an important factor to identify thesymptomatic and asymptomatic individuals. The symp-tomatic individuals should be reported by the publichealth agencies to separate them from the uninfected orasymptomatic individuals for their ICU (Intensive Care Unit) treatment. Also, from an economic viewpoint, thestrict lockdown may be the cause of a substantial finan-cial crisis in near future. In particular, the lockdown inhigh dense countries can mitigate the disease transmis-sion rate, although entirely control not be obtainable.Thus, to survive the economical status of a country, astrict lockdown for a long period is not advisable at allin any situations. Hence there should be a acceptablebalance between the two different characteristics of gov-ernmental strategies: strict lockdown and healthy eco-nomical situations. Albeit, few questions remains to an-swer whether this cluster containment policy can be ef-fective in mitigating SARS-CoV-2 transmission or not ?If not then what can be the possible solutions to miti-gate the transmission of SARS-CoV-2 viruses ? Thesequestions can only be replied by investigating the dy-namics and forecasting of a mechanistic compartmentalmodel for SARS-CoV-2 transmission and comparing theoutcomes with real scenarios.A plenty of mathematical models has been investi-gated to study the transmission dynamics and forecast-ing of COVID-19 outbreak [10–20]. Kucharski and col-leagues [10] performed a model-based analysis for SARS-CoV-2 viruses and calculate the reproduction number R = 2 .
35, where the authors have taken into account allthe positive cases of Wuhan, China, till March 05, 2020.Wu and colleagues [12] studied a susceptible-exposed-infectious-recovered (SEIR) model to simulate the epi-demic in Wuhan city and compute the basic reproduc-tion number R = 2 .
68 and predict their model based onthe data recorded from December 31, 2019 to January28, 2020. Tang and colleagues [11] developed a com-partmental model to study the transmission dynamics ofCOVID-19 and calculate the basic reproduction number R = 6 .
47, which is very high for the infectious diseases.Recently, Fanelli & Piazza [13] analyzed and predictedthe characteristics of SARS-CoV-2 viruses in the threemostly affected countries till March 2020 with an aid ofthe mathematical modeling. Stochastic based regressionmodel also been studied by Ribeiro and Colleagues [14]to predict the scenarios of the most affected states ofBrazil. Chakraborty & Ghosh [15] investigate a hybridARIMA-WBF model to predict the various SARS-CoV-2 affected countries throughout the world. Khajanchiand Colleagues [18] developed a compartmental modelto forecast and control of the outbreak of COVID-19 inthe four states of India and the overall India. Sarkar& Khajanchi [16] developed a mathematical model tostudy the model dynamics and forecast the SARS-CoV-2viruses in seventeenth provinces of India and the over-all India. A discrete-time SIR model introducing deadcompartment system studied by Anastassopoulou et al.[19] to portray the dynamics of SARS-CoV-2 outbreak.Giordano and colleagues [20] established a new mathe-matical model for COVID-19 pandemic and predict thatrestrictive social distancing can mitigate the widespreadof COVID-19 among the human. A couple of seminalpapers has been investigated to study the transmissiondynamics of COVID-19 or SARS-CoV-2 viruses for differ-ent countries, including Mexico city, Chicago and Wuhan,the sprawling capital of Central China’s Hubei province[21–24]. Short-term prediction is too important as itgives time-critical information for decisions on contain-ment and mitigation strategies [18, 25]. A major problemfor short-term predictions is the evaluation of importantepidemiological parameters and how they alter when firstinterventions reveal an effect.The main objective of this work is to develop a newmathematical model that describes the transmission dy-namics and forecasting of COVID-19 or SARS-CoV-2pandemic in the four different provinces of India, namelyJharkahnd, Andhra Pradesh, Chandigarh, and Gujarat.We estimated the model parameters of four differentstates of India and fitted our compartmental model tothe daily confirmed cases and cumulative confirmed casesreported between March 15, 2020 to May 24, 2020. Wecompute the basic reproduction number R for the fourdifferent states based on the estimated parameter values.We also perform the short-term predictions of the fourdifferent states of India from May 25, 2020 to June 13,2020, and it shows the increasing trends of COVID-19pandemic in four different provinces of India.The remaining part of this manuscript has been orga-nized in the following way. In the Section 2, we describethe formulation of the compartmental model for COVID-19 and its basic assumptions. Section 3 describes thetheoretical analysis of the model, which incorporates thepositivity and boundedness of the system, computationof the basic reproduction number R , existence of thebiologically feasible singular points and their local stabil-ity analysis. In the same section, we perform the globalstability analysis for the infection free equilibrium point E and the existence of transcritical bifurcation at thethreshold R = 1 . In the Section 4, we conduct somemodel simulations to validate our analytical findings byusing the estimated model parameters for Jharkhand, thestate of India. The parameters are estimated for the realworld example on COVID-19 for four different states ofIndia and perform a short-term prediction based on theestimated parameter values. A discussion in the Section5 concludes the manuscript.
II. MATHEMATICAL MODEL
A compartmental mathematical model has been devel-oped to study the transmission dynamics of COVID-19outbreak in India and throughout the world. We adopta variant that focuses some important epidemiologicalproperties of COVID-19 or SARS-CoV-2 coronavirus dis-eases. Based on the health status, we stratify the to-tal human population into six compartments, namelysusceptible or uninfected ( S ), asymptomatic or pauci-symptomatic infected ( A ), symptomatic reported in-fected ( I ), unreported infected ( U ), quarantine ( Q ), andrecovered ( R ) individuals, collectively termed SAIUQR. At any instant of time, the total population is denotedby N = S + A + I + U + Q + R. Depending on the six statevariables, we aim to develop an autonomous system usingfirst order nonlinear ordinary differential equations.In the model formulation, quarantine refers to the sep-aration of coronavirus infected population from the gen-eral population when the individuals are infected butclinical symptoms has not yet developed, whereas isola-tion refers to the separation of coronavirus infected popu-lation when the population already identified the clinicalsymptoms. Our mathematical model introduces somedemographic effects by assuming a proportional naturalmortality rate δ > s per unit time. This parameter represents newbirth, immigration and emigration. The parameter β s represents the probability of disease transmission rate.However the disease transmission from vulnerable to in-fected individuals (for our model, the class is A) dependon various factors, namely safeguard precautions (use ofmask, social distancing, etc.) and hygienic safeguard (useof hand sanitizer) taken by the susceptible individuals aswell as infected population. In our model formulation,we incorporate the asymptomatic or pauci-symptomaticinfected (undetected) individuals, which is important tobetter understand the transmission dynamics of COVID-19, which also studied by Giordano et al. [20] and Xiao-Lin et al. [21].In our model formulation, we assumed that theCOVID-19 virus is spreading when a vulnerable per-son come into contact with an asymptomatic infectedindividuals. The uninfected individuals decreases af-ter infection, obtained through interplays among a sus-ceptible population and an infected individuals whomay be asymptomatic, reported symptomatic and un-reported symptomatic. For these three compartmentsof infected population, the transmission coefficients are β s α a , β s α i , and β s α u respectively. We consider β s asthe disease transmission rate along with the adjustmentfactors for asymptomatic α a , reported symptomatic α i and unreported symptomatic α u individuals. The in-terplays among infected populations (asymptomatic, re-ported symptomatic, and unreported symptomatic) andsusceptible individuals can be modeled in the form of to-tal individuals using standard mixing incidence [26–29].The quarantined population can either move to thesusceptible or infected compartment (reported and unre-ported), depending on whether they are infected or not[30], with a portion ρ s . Here, γ q is the rate at whichthe quarantined uninfected contacts were released intothe wider community. Asymptomatic individuals wereexposed to the virus but clinical symptoms of SARS-CoV-2 viruses has not yet developed. The asymptomaticindividuals decreases due to contact with reported andunreported symptomatic individuals at the rate γ a witha portion θ ∈ (0 , ξ a . Also, the asymptomatic individuals become recov-ered at the rate η a and has a natural mortality rate δ .A fraction of quarantine individuals become reported in-fected individuals at the rate γ q with a portion ρ s (where ρ s ∈ (0 , θ ) and unreported symptomatic individuals(1 − θ ). The reported symptomatic individuals separatedfrom the general populations and move to the isolatedclass or hospitalized class for clinical treatment.Also, it can be noticed that once an individual re-covered from the SARS-CoV-2 diseases has a very littlechance to become infected again for the same disease [31].Therefore, we assume that none of the recovered individ- uals move to the susceptible or uninfected class again. Inour mathematical model formulation, we assume that thereported infected individuals ( I ) are unable to spread ortransmit the viruses as they are kept completely isolatedfrom the susceptible or uninfected individuals. As thereported infected individuals are moved to the hospitalor Intensive Care Unit (ICU) [32]. For our modeling per-spective, we are mainly interested in predictions over arelatively short time window within which the temporaryimmunity is likely still to be in placed, and the possibilityof reinfection would negligibly affect the total number ofuninfected populations and so there would be no consid-erable difference in the evolution of the epidemic curveswe consider. Social mixing patterns are introduced intoour contagion parameters in an average fashion over theentire individuals, irrespective of age. Based on thesebiological assumptions, we develop the following mathe-matical model using a system of nonlinear ordinary dif-ferential equations to study the outbreak of COVID-19or SARS-CoV-2 coronavirus diseases: S ′ ( t ) = Λ s − β s S (cid:18) α a AN + α i IN + α u UN (cid:19) + ρ s γ q Q − δS,A ′ ( t ) = β s S (cid:18) α a AN + α i IN + α u UN (cid:19) − ( ξ a + γ a ) A − η a A − δA,I ′ ( t ) = θγ a A + (1 − ρ s ) γ q Q − η i I − δI,U ′ ( t ) = (1 − θ ) γ a A − η u U − δU,Q ′ ( t ) = ξ a A − γ q Q − δQ,R ′ ( t ) = η u U + η i I + η a A − δR, (1)the model is supplemented by the following initial values: S ( t ) = S ≥ , A ( t ) = A ≥ , Q ( t ) = Q ≥ ,I ( t ) = I ≥ , U ( t ) = U ≥ , R ( t ) = R ≥ . (2)In our model t ≥ t is the time in days, t representsthe starting date of the outbreak for our system (1). Thetransmission dynamics of the COVID-19 is illustrated inthe Figure 1. The description of the model parametersare presented in the Table I. III. SAIUQR MODEL ANALYSIS
In this section, we provide the basic properties of theSAIUQR model (1), including positivity and bounded-ness of the solutions, basic reproduction number andthe biologically feasible singular points and their sta-bility analysis, subject to the non-negative initial values( S , A , Q , I , U , Q , R ) ∈ R . Theorem III.1
The solutions of the SAIUQR system(1) with the initial values (2) are defined on R remainpositive for all t > . Proof III.1
The proof of this theorem is given in theAppendix A.
Theorem III.2
The solutions of the SAIUQR system(1) with the initial conditions (2) are uniformly boundedin the region Ω . Proof III.2
The proof of this theorem is given in theAppendix B.
A. Basic reproduction number
In any infectious disease modeling, the basic reproduc-tion number is the key epidemiological parameter for de-scribing the characteristics of the diseases. The basicreproduction number symbolized by R and is definedas “the number of secondary infected individuals causedby a single infected individuals in the entire susceptibleindividuals” [33]. The dimensionless basic reproductionnumber R quantifies the expectation of the disease dieout or the spreading of the diseases. For, R < R > (1- (cid:1) s ) (cid:2) q (cid:0) s (cid:3) q (cid:4) (cid:5) a (cid:6)(cid:7) a (cid:8) (cid:9) i (cid:10) s (cid:11) s (S/N)( (cid:12) a A+ (cid:13) i I+ (cid:14) u U) (cid:15) a (cid:16) (cid:17) (1- (cid:18) ) (cid:19) a (cid:20) u (cid:21)(cid:22) FIG. 1. A schematic representation of the mechanisticSAIUQR model for the transmission dynamics of COVID-19or SARS-CoV-2. The interaction among different stages of in-dividuals is shown in the graphical scheme: S, susceptible oruninfected population; A, asymptomatic infected population;I, COVID-19 reported symptomatic infected individuals; U,COVID-19 unreported symptomatic infected individuals; Q,quarantine individuals; and R, COVID-19 recovered individ-uals. Biological interpretations of the model parameters aregiven in the Table I. describes each infected individuals spread on an averagemore than 1 new infection, and the disease can spreadthroughout the population. Various techniques can beused to compute the basic reproduction number R foran epidemic outbreak. In our present study, we use thenext generation matrix to evaluate R [33]. In our com-partmental model, the following classes are explicitly re-lated to the outbreak of the novel coronavirus diseases: A , I , U , and Q . Thus, from the SAIUQR model system(1), we get the matrices F for the new infection and V for the transition terms are given by, respectively F = β s S (cid:18) α a AN + α i IN + α u UN (cid:19) , V = ( ξ a + γ a + η a + δ ) A − θγ a A − (1 − ρ s ) γ q Q + ( η i + δ ) I − (1 − θ ) γ a A + ( η u + δ ) U − ξ a A + ( γ q + δ ) Q . The variational matrix for the SAIUQR system (1),can be evaluated at an infection-free singular point E ( S , A , Q , I , U , R ) = ( Λ s δ , , , , , F = β s α a SN β s α i SN β s α u SN
00 0 0 00 0 0 00 0 0 0 ,V = ξ a + γ a + η a + δ − θγ a ( η i + δ ) 0 − (1 − ρ s ) γ q − (1 − θ ) γ a η u + δ ) 0 − ξ a γ q + δ . The basic reproduction number R = ρ ( F V − ), where ρ ( F V − ) represents the spectral radius for a next gener- ation matrix F V − . Therefore, from the SAIUQR model(1), we get the basic reproduction number R as ρ ( F V − ) = R = β s α a ξ a + γ a + η a + δ + (1 − θ ) β s α u γ a ( η u + δ )( ξ a + γ a + η u + δ )+ β s α i ( θγ a ( γ q + δ ) + (1 − ρ s ) ξ a γ q )( ξ a + γ a + η a + δ )( γ q + δ )( η i + δ ) . B. Equilibria
The SAIUQR model (1) has two biologically feasibleequilibrium points, namely(i) infection free steady state E ( S , A , Q , I , U , R ) = ( Λ s δ , , , , , E ∗ ( S ∗ , A ∗ , Q ∗ , I ∗ , U ∗ , R ∗ ), where S ∗ = Λ s δ − ˆ SA ∗ , I ∗ = ˆ IA ∗ , U ∗ = ˆ U A ∗ , Q ∗ = ˆ QA ∗ and R ∗ = ˆ RA ∗ . The expression of A ∗ isgiven by A ∗ = Λ s ( R − δ { ˆ S ( R − I + ˆ U + ˆ Q + ˆ R } , whereˆ S = δ h ξ a + γ a + η a + δ − ξ a ρ s γ q γ q + δ i ,ˆ I = θγ a ( γ q + δ )+(1 − ρ s ) ξ a γ q ( η i + δ )( γ q + δ ) , ˆ U = (1 − θ ) γ a η u + δ , ˆ Q = ξ a γ q + δ andˆ R = δ (cid:20) (1 − θ ) γ a η u η u + δ + η i { θγ q ( γ q + δ )+(1 − ρ s ) γ q ξ a } ( γ q + δ )( η i + δ ) + η a (cid:21) .It can be observed the the infection free singular point E ( S , A , Q , I , U , R ) is always feasible and the en-demic equilibrium point E ∗ ( S ∗ , A ∗ , Q ∗ , I ∗ , U ∗ , R ∗ ) is fea-sible if the following condition holds:( i ) R > , ( ii ) 1 ξ a + γ a + η a + δ (cid:20) Λ s A ∗ + ξ a ρ s γ q γ q + δ (cid:21) > . C. Stability analysis
In the present subsection, we investigate the linear sta-bility analysis for the SAIUQR model (1) for the two fea-sible steady states. By using the techniques of lineariza-tion, we investigate the local dynamics of the complicatedsystem of the coronavirus compartmental model. Gen-erally, we linearize the SAIUQR model around each ofthe feasible steady states and perturb the compartmen-tal model by a very small amount, and observe whetherthe compartmental model returns to that steady statesor converges to any other steady states or attractor.The local stability analysis aids in understanding thequalitative behavior of the complex nonlinear dynami-cal system. By using the following theorem, we provethe local stability of the infection free singular point E ( S , A , Q , I , U , R ): Theorem III.3
The infection free steady state E is lo-cally asymptotic stable if R < and unstable if R > . Proof III.3
The proof of this theorem is given in Ap-pendix C.
Theorem III.4
The infection free steady state E isglobally asymptotic stable for R < in the bounded re-gion Ω . Proof III.4
The proof of this theorem is given in Ap-pendix D.
Theorem III.5
The SAIUQR model system (1) admitsa locally asymptotic stable around the endemic equilib-rium point E ∗ for R > . Also, the system (1) experi-ences forward bifurcation at R = 1 . Proof III.5
The proof of this theorem is given in Ap-pendix E.
IV. NUMERICAL SIMULATION
In this section, we conduct some numerical illustrationsto validate our analytical findings. Analytically, we per-form the local stability analysis for infection free steadystate E and a unique endemic equilibrium point E ∗ . Wealso perform the transcritical bifurcation at the threshold R = 1 and the global stability analysis for disease-freesteady state E . In order to validate the analytical cal-culations, we used the estimated parameter values forJharkhand, the state of India and the techniques for pa-rameter estimation are described in the subsection IV A. A. Model calibration β s , α a , α i , α u , γ a and γ q , out of fourteen system parame-ters for the system (1) by using least square method [35].The values of these parameters and the initial populationsize plays an important role in the model simulation. Theparameters are estimated by assuming the initial popu-lation size. The initial population are presented in theTable III. Three days moving average filter has been ap-plied to the daily COVID-19 cases to smooth the data.The daily reported confirmed COVID-19 cases are fit-ted with the model simulation by using the least squaremethod. The estimated parameter values are listed inthe Table II. Different set of parameter values locallyminimizes the Root Mean Square Error (RMSE) and wehave considered the set of parameter values, which gives the realistic value of the basic reproduction number R .RMSE is the measure of the accuracy of the fitting dataand the RMSE is defined as follows: RM SE = r Σ ni =1 ( O ( i ) − M ( i )) n , where n represents the size of the observed data, O ( i )is the reported daily confirmed COVID-19 cases and M ( i ) represents the model simulation. The Figure 2shows the daily confirmed COVID-19 cases (first col-umn), cumulative confirmed COVID-19 cases (secondcolumn) and model simulations has been shown in theblue curve for all the four provinces of India. The val-ues of RMSE and basic reproduction number R for allthe four provinces are presented in the inset of the fig-ure. The SAIUQR model performs well for the threeprovinces, namely Jharkhand, Chandigarh, and AndhraPradesh. The RMSE for Gujarat is higher than the otherprovinces as the number of daily confirmed COVID-19cases are higher than the other provinces. The valueof the basic reproduction number R for Jharkhand,Gujarat, Chandigarh, and Andhra Pradesh are 1.6877,1.8803, 1.4775, and 1.2435, respectively and the trend ofdaily confirmed COVID-19 cases is increasing. This in-creasing trend of the daily new COVID-19 cases for allthe four provinces of India are captured by our modelsimulation. In all the four provinces R >
1, so thedisease free equilibrium point E is unstable. The ba-sic reproduction number for the four states are greaterthan unity, which indicates the substantial outbreak ofthe COVID-19 in the states. B. Validation of analytical findings
In this section, we have validated our analytically find-ings by using numerical simulations for the parametervalues in the Table I, and the estimated parameter valuesin the Table II for our SAIUQR model for the coron-avirus diseases. The parameter values are estimatedfor the observed COVID-19 data for the three states ofIndia, namely Jharkhand, Gujarat and Andhra Pradeshand for the city Chandigarh. Our analytical findingsstated in the Theorem III.3, shows that the disease freeequilibrium point E is locally asymptotically stablewith R <
1, and the Theorem III.5 stated that a uniqueendemic equilibrium point E ∗ is locally asymptoticallystable for R >
1. The numerical simulations of theSAIUQR model system (1) have been presented inthe Figure 3 for all the six individuals and for thedifferent values of the disease transmission rate β s .The value of the parameters considered for numericalsimulations are α a = 0 . , α i = 0 . , α u = 0 . ,γ a = 0 . , γ q = 0 . , δ = 0 . , Λ s = 1200 and theother model parameter values are listed in the TableI. Six initial population sizes are considered for themodel simulation, namely (39402 , , , , , TABLE I. Table of the biologically relevant parameter values and their description for the SAIUQR model system (1).Symbol Biological interpretations Values & SourceΛ s birth rate of the susceptible individuals Table III β s probability of the disease transmission coefficient Estimated α a modification factor for asymptomatic infected individuals Estimated α i modification factor for symptomatic infected individuals Estimated α u modification factor for unreported infected individuals Estimated ρ s fraction of quarantine individuals that become susceptible individuals 0.5 (0, 1) Fixed γ q rate at which the quarantined individuals becomes susceptibleindividuals Estimated δ natural death rate of entire individuals 0.1945 × − [18] ξ a rate at which the asymptomatic individuals become quarantined 0.07151 [16] γ a rate of transition from the asymptomatic individuals to infectedindividuals Estimated η a average recovery rate of asymptomatic individuals . [16] θ fraction of asymptomatic individuals that become reported infectedindividuals 0.8 (0, 1) Fixed η i average recovery rate of reported symptomatic infected individuals [32] η u average recovery rate of unreported symptomatic infected individuals [32] (31402 , , , , , , , , , , , , , , , , , , , ,
0) and(15000 , , , , , β s = 1 .
10 (red curves in the Figure 3)and β s = 0 .
55 (blue curves in the Figure 3). The valuesof R are 1 . . β s = 1 .
10 and β s = 0 . E (40000 , , , , , R = 0 . <
1. Our SAIUQRmodel system (1) converges to the endemic equilibriumpoint E ∗ (31035 . , . , . , . , . , .
4) for β s = 1 .
10 and R = 1 . > R = 1. We have plotted the COVID-19 re-ported symptomatic individuals ( I ) in the ( R , I ) planeby gradually increasing the disease transmission rate β s (see the Figure 4). The model parameter values are α a = 0 . , α i = 0 . , α u = 0 . , γ a = 0 . ,γ q = 0 . , δ = 0 . , Λ s = 1200 and the other pa-rameter values listed in the Table I. We vary the diseasetransmission rate β s from 0.67 to 1.10 and computed thebasic reproduction number R and the COVID-19 re-ported symptomatic individuals ( I ). The numericallycomputed values are presented in the Figure 4, whichclearly shows that the system (1) experiences transcriti-cal bifurcation at the threshold R = 1. The blue curverepresents the stable endemic equilibrium point E ∗ , black line represents the stable disease free equilibrium point E and the red line represents the unstable branch of thedisease free equilibrium point E . Hence, the Figure 4shows that disease free equilibrium point E is stable forthe reproduction number R < E ∗ is stable for the reproduction number R >
1. From the biological point of view, it can bedescribed that the model system (1) will be free fromCOVID-19 for the reproduction number R < R > . Figure 5(a) represents the reproduction number R decreases as the recovery rate η i of reported infectedindividuals increases and the reproduction number R becomes less than one for β s = 0 .
85 and β s = 0 . E switches the stability of the model system (1) as η i changes. But the reproduction number R remainsgrater than one for the disease transmission rate β s =1 .
10 and β s = 0 .
95, that is, for large β s unique endemicequilibrium point remains locally asymptotically stableeven if η i changes. In terms of COVID-19 diseases thisinterprets that if the rate of recovery for infected individ-uals ( η i ) be increased, which can be done by vaccineesor specific therapeutics, the model system (1) changesits stability to disease free equilibrium E from endemicequilibrium E ∗ but if the transmission of the disease ( β s )is high enough then by vaccines or specific therapeutics,the system (1) can not change its stability from endemicequilibrium to disease free equilibrium.Figure 5(b) represents the reproduction number R increases as γ a (transition rate from asymptomatic in- / /
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FIG. 2. Model estimation based on the observed data. Model simulations fitted with the daily new cases and the cumulativeconfirmed cases of COVID-19 for four provinces of India, namely Jharkhand, Gujarat, Chandigarh, and Andhra Pradesh.Observed data points are shown in the red circle while the blue curve represents the best fitting curve for the SAIUQR model.The first column represents the daily new cases and the second column represents the cumulative confirmed cases of COVID-19.The estimated parameter values are listed in the Table II and other parameter values are listed in the Table I. The initial valuesused for this parameter values are presented in the Table III. The RMSE and the value of R for each provinces are mentionedin the inset. dividuals to symptomatic individuals) increases but thereproduction number R remains less than one for thedisease transmission rate β s = 0 .
55 and β s = 0 .
45. For β s = 0 .
66 and β s = 0 .
76, the basic reproduction num-ber R becomes grater than one and the SAIUQR modelsystem (1) loses the stability of disease free equilibriumpoint E . Thus, to flatten the COVID-19 curve in anyof the four provinces of India, reduction of the transmis-sion of the COVID-19 disease is utmost priority even ifthe recovery rate increased by medication. Biologically it means that to mitigate the COVID-19 diseases, the peo-ple must have to maintain the social distancing, contacttracing by avoiding the mass gathering.The predictive competency for the SAIUQR model sys-tem (1) requires valid estimation of the system parame-ters γ a (rate of transition from asymptomatic to symp-tomatic infected individuals), γ q (the rate that the quar-antine become susceptible), θ (fraction of asymptomaticinfectious that become reported symptomatic infectious), ξ a (rate at which asymptomatic individuals become quar-antined). In the Figure 6(a), we plot the reproductionnumber R as a function of ξ a and γ q for the parametervalues in the Table I and estimated parameters for thestate Jharkhand, to encapsulate the significance of thesevalues in the evolution of COVID-19 outbreak. Fromthe Figure 6(a), it can be observed that the parametershave a little influence on the outbreak of the coronavirusdiseases as the parameters ξ a and γ q has a little contri-bution for the reproduction number R . In the Figure6(b), we plot the reproduction number R as a functionof θ and γ a for the parameter values in the Table I andestimated parameters for the state Jharkhand, to encap-sulate the significance of these values in the evolution ofCOVID-19 outbreak. The Figure 6(b) shows that the pa-rameters θ and γ a are more influential in increasing thereproduction number R . Thus, to control the outbreakof COVID-19, we must control the parameters θ and γ a .The correctness of these values rely on the input of medi-cal and biological epidemiologists. Thus, the fraction θ ofreported symptomatic infected individuals may be sub-stantially increased by public health reporting measures,with greater efforts to recognize all the present cases. Ourmodel simulation reveals the effect of an increase in thisfraction θ in the value of the reproduction number R , asevident in the Figure 6(b), for the COVID-19 epidemicin the four sates of India. C. Short-term prediction
Mathematical modeling of infectious diseases can pro-vide short-term and long-term prediction of the pandemic[16, 18, 20, 32]. Due to absence of any licensed vaccinesor specific therapeutics, forecasting is of utmost impor-tance for strategies to control and prevention of the dis-eases with limited resources. It should be noted that herewe can predict the epidemiological traits of SARS-CoV-2or COVID-19 for short-term only as the Governmentalstrategies can be altered time to time resulting in thecorresponding changes in the associated parameters ofthe proposed SAIUQR model. Also, it is true that thescientists are working for drugs and/or effective vaccinesagainst COVID-19 and the presence of such pharmaceu-tical interventions will substantially change the outcomes[36]. Thus, in this study we performed a short-term pre-diction for our SAIUQR model system (1) using the pa-rameter values in the Table I and the estimated parame-ter values in the Table II. Using the observed data up toMay 24, 2020, a short-term prediction (for 20 days) hasbeen done for daily new COVID-19 cases (first column)and cumulative confirmed cases (second column) are pre-sented in the Figure 7. The black dot-dashed curve rep-resents the short-term prediction of our SAIUQR modelfrom May 25, 2020 to June 13, 2020. Red shaded regionis the standard deviation band of our SAIUQR modelsimulated curve. The standard deviations are computedfrom the model simulation based on the estimated data.In each of four states, we plot the standard deviation bands at a standard deviation level above and below themodel simulation for different days. Standard deviationband gives an estimation of the deviation of the actualmodel data. The trend of the predicted daily COVID-19 cases is increasing for all the four provinces of India.Prediction of the SAIUQR model should be regarded asan estimation of the daily infected population and cumu-lative confirmed cases of the four states of India. Fromthe SAIUQR model simulation, we can predict that theestimated daily new reported COVID-19 cases on June13, 2020 will be approximately 15, 454, 12, and 96 inJharkhand, Gujarat, Chandigarh, and Andhra Pradesh,respectively (see the left column of the Figure 7). OurSAIUQR model simulation predict that the confirmedcumulative number of cases on June 13, 2020 will beapproximately 661, 23955, 514, and 4487 in Jharkhand,Gujarat, Chandigarh, and Andhra Pradesh, respectively(see the right column of the Figure 7).
TABLE II. The SAIUQR model parameter values esti-mated from the observed daily new COVID-19 cases for fourprovinces of India, namely Jharkhand, Gujarat, Chandigarhand Andhra Pradesh. Six important parameters β s , α a , α i , α u , γ a and γ q are estimated among fourteen system parame-ters.Provinces β s α a α i α u γ a γ q Jharkhand 0.760 0.264 0.760 0.9600 0.0012 0.0015Gujarat 1.006 0.342 0.168 0.1308 0.0004 0.0046Chandigarh 0.750 0.294 0.444 0.4600 0.0010 0.0011Andhra Pradesh 0.431 0.419 0.688 0.7100 0.0006 0.0280TABLE III.
Initial population size and the values of Λ s usedin numerical simulations for four different provinces of In-dia, namely Jharkhand, Gujarat, Chandigarh, and AndhraPradesh. Provinces S (0) A (0) Q (0) I (0) U (0) R (0) Λ s Jharkhand 39402 575 19 1 0 0 1200Gujarat 85402 1525 27 1 0 0 1300Chandigarh 20402 275 10 1 0 0 1200Andhra Pradesh 75401 355 12 1 0 0 970
V. DISCUSSION AND CONCLUSION
The SARS-CoV-2 pandemic in India is a potentialmenace throughout the country due to its exponentialgrowth. Everyday the new cases are reported around 5-6thousands and more than that from different states andterritories of India, which is an alarming situation as withthe second most populated country worldwide [34]. Dueto absence of any licensed vaccine, therapeutics or treat-ment and with a peculiar epidemiological traits of SARS-CoV-2, one would depend on the qualitative control of0
Time (Days) S Time (Days) A Time (Days) Q Time (Days) I Time (Days) U Time (Days) R s =1.10, R =1.2889 s =0.55, R =0.7030 FIG. 3. Stability of the SAIUQR model system (1) around the disease free equilibrium point E and an unique endemicequilibrium point E ∗ . The values of the estimated parameters are α a = 0 . , α i = 0 . , α u = 0 . , γ a = 0 . , γ q = 0 . ,δ = 0 . , Λ s = 1200 and other parameter values are listed in the Table I. Initial population sizes are (39402 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β s = 1 .
10 (red curves) and β s = 0 .
55 (blue curves). The values of R are 1 . . β s = 1 .
10 and β s = 0 .
55, respectively. Disease free equilibrium point E is locally asymptotically stable when R < E ∗ is locally asymptotically stable when R > the diseases rather than complete elimination. Duringthis period of an epidemic when person-to-person trans-mission is confirmed and the reported cases of SARS-CoV-2 viruses are rising throughout the globe, predic-tion is of utmost priority for health care planning andto manage the virus with limited resource. Furthermore,mathematical modeling can be a powerful tool in design-ing strategies to manage exponentially spreading coron-avirus diseases in absence of any antivirals or diagnostictest.In this study, we proposed and analyzed a new com-partmental mathematical model for SARS-CoV-2 virusesto forecast and control the outbreak. In the model for-mulation, we incorporate the transmission variability ofasymptomatic and unreported symptomatic individuals. We also incorporate the symptomatic infected popula-tions who are reported by the public health services.We assume that reported infected individuals will no-longer associated into the infection as they are isolatedand move to the hospitals or Intensive Care Unit (ICU).In our model, we incorporate the constant transmissionrate in the early exponential growth phase of the SARS-CoV-2 diseases as identified in [18, 29]. We model therole of the Govt. imposed restrictions for the public inIndia, beginning on March 25, 2020, as a time-dependentdecaying transmission rate after March 25, 2020. But,due to less stringent lockdown exponentially increasingthe disease transmission rate, we are able to fit with in-creasing accuracy, our model simulations to the Indianreported cases data up to May 24, 2020.1 R I Stable EndemicEquilibrium Stable Disease FreeSteady State Unstable Disease FreeSteady State
FIG. 4. The figure represents the transcritical bifurcationdiagram of the SAIUQR model system (1) with respect tothe basic reproduction number R . The parameter values are α a = 0 . , α i = 0 . , α u = 0 . , γ a = 0 . , γ q = 0 . ,δ = 0 . , Λ s = 1200 and other parameters as listed in theTable I. Stability of the SAIUQR system (1) exchange at thethreshold R = 1. We fit our SAIUQR model for the daily confirmedcases and cumulative confirmed cases of the four differentprovinces of India, namely Jharkhand, Andhra Pradesh,Chandigarh, and Gujarat with data up to May 24, 2020.The estimated model parameters for different states ofIndia are given in the Table II and the correspondinginitial population size are listed in the Table III. It canbe observed that the basic reproduction number for fourdifferent provinces of India, namely Jharkhand, AndhraPradesh, Chandigarh, and Gujarat are 1.6877, 1.2435,1.4775, and 1.8830, respectively, which demonstrates thedisease transmission rate is quite high that basically indi-cates the substantial outbreak of the COVID-19 diseases.This higher value of reproduction number R capturesthe outbreak of COVID-19 phenomena in India. Basedon the estimated parameter values our model simulationsuggest that the rate of disease transmission need to becontrolled, otherwise India will enter in stage-3 of SARS-CoV-2 disease transmission within in a short period oftime.Based on the estimated model parameters we havevalidated our detailed analytical findings. Our pro-posed SAIUQR model has two biologically feasible sin-gular points, namely infection free steady state E and aunique endemic steady state E ∗ and they become locallyasymptotically stable for R < R >
1, respec-tively. Analytically, we have shown that the infectionfree steady state E of the SAIUQR model (1) is glob-ally asymptotically stable for R < . We also showedthat the SAIUQR model (1) experiences transcritical bi-furcation at the threshold parameter R = 1, which hasbeen shown in the Figure 4. The Figure 3 (blue curves)and the Figure 3 (red curves) represents the local asymp- totic stability as well as global asymptotic stability of theinfection free steady state E for R < E ∗ for R >
1, respectively.The calibrated model then utilized for short-termpredictions in the four different states of India. OurSAIUQR model performs well in case of all the fourprovinces of India, namely Jharkahnd, Chandigarh, Gu-jarat, and Andhra Pradesh for daily confirmed cases andcumulative confirmed cases. Albeit, the increasing (orexponential) pattern of daily new cases and cumulativeconfirmed cases of SARS-CoV-2 is well captured by ourproposed model for all the four states of India, whichhas been shown in the Figure 2. Our model simulationshowed a short-term prediction for 20 days (from May25, 2020 to June 13, 2020) for daily confirmed cases andcumulative confirmed cases of the four provinces of In-dia. The short-term prediction for the four provincesof India will demonstrates the increasing pattern of thedaily and cumulative cases in the near future (see theFigure 7). From the simulation, our model predict thaton June 13, 2020 the daily confirmed cases of COVID-19of the four provinces of India, namely Jharkhand, Gu-jarat, Chandigarh, and Andhra Pradesh will be 15, 454,12, and 96, respectively (see the left column of the Fig-ure 7). Similarly, from the simulation, our model predictthat on June 13, 2020 the cumulative confirmed cases ofCOVID-19 of the four provinces of India, namely Jhark-hand, Gujarat, Chandigarh, and Andhra Pradesh will be661, 23955, 514, and 4487, respectively (see the right col-umn of the Figure 7).It is worthy to mention that the scientists or clini-cians are working for effective vaccine or therapeutics toeradicate and/or control the outbreak of SARS-CoV-2diseases and the existence of such pharmaceutical inter-ventions will substantially change the outcomes [36, 37].Thus, in this study, we are mainly focusing on short-term predictions for the COVID-19 pandemic and sub-sequently there would be a very little chance to alter incorresponding parametric space. But the framework ofour present compartmental model provides some signif-icant insights into the dynamics and forecasting of thespread and control of COVID-19. Moreover, our modelsimulation suggest that the quarantine, reported and un-reported symptomatic individuals as well as governmentintervention polices like media effect, lockdown and socialdistancing can play a key role in mitigating the transmis-sion of COVID-19.
ACKNOWLEDGMENTS
This study is supported by Science and EngineeringResearch Board (SERB) (File No. ECR/ 2017/000234),Department of Science & Technology, Government of In-dia. The authors are thankful to the anonymous re-viewers for their careful reading and constructive sug-gestions/comments which helped in better exposition ofthe manuscript.2 i R (a) s =1.10 s =0.95 s =0.85 s =0.76 a R (b) s =0.76 s =0.66 s =0.55 s =0.45 FIG. 5. The figures represents the basic reproduction number R in terms of (a) η i (rate of recovery for infected individuals)and (b) γ a (rate at which asymptomatic individuals develops detected symptomatic infected individuals). Green shaded regionindicates R < R >
1. The parameter values are α a = 0 . , α i = 0 . , α u = 0 . ,γ a = 0 . , γ q = 0 . , δ = 0 . , Λ s = 1200 and the other parameter values are listed in the Table I.FIG. 6. The figures represents the surface plot of the basic reproduction number R in (a) ( γ q , ξ a )-plane and (b) ( γ a , θ )-plane.Red shading plane indicates R = 1. The parameter values for the sub-figure (a): β s = 0 . α a = 0 . , α i = 0 . , α u = 0 . ,γ a = 0 . , δ = 0 . , Λ s = 1200 and for the sub-figure (b): β s = 0 . α a = 0 . , α i = 0 . , α u = 0 . , γ q = 0 . , δ = 0 . , Λ s = 1200 and the other parameter values are listed in the Table I. DATA AVAILABILITY
All the data used in this work has been obtained fromofficial sources [34]. All data supporting the findings ofthis study are in the paper and available from the corre-sponding author on request.
AUTHOR CONTRIBUTIONS STATEMENT
Subhas Khajanchi and Kankan Sarkar designed andperformed the research as well as wrote the paper.3 / /
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Observed Model Simulation Model Prediction
FIG. 7. The figure represents the short-term prediction of the daily new COVID-19 cases (first column) and the cumulativeconfirmed cases (second column) for the three states of India namely Jharkhand, Gujarat, Andhra Pradesh and one city ofIndia namely Chandigarh. The black dot-dashed curve represents the prediction from May 25, 2020 to June 13, 2020 (20 days).The red shaded region is the standard deviation band of the SAIUQR model simulated curve.
COMPETING INTERESTS
The authors declare that they have no conflict of in-terest.
Appendix A: Proof of Theorem III.1
Proof.
To prove the positivity of the system (1), weshow that any solution initiating from the non-negativeoctant R remains positive for all t >
0. In order to dothis, we have to prove that on each hyperplane boundingthe non-negative octant the vector field points into R . For our system (1), we observe that dSdt (cid:12)(cid:12)(cid:12)(cid:12) S =0 = Λ s + ρ s γ q Q ≥ ,dAdt (cid:12)(cid:12)(cid:12)(cid:12) A =0 = β s S (cid:18) α i IN + α u UN (cid:19) ≥ ,dIdt (cid:12)(cid:12)(cid:12)(cid:12) I =0 = θγ a A + (1 − ρ s ) γ q Q ≥ ,dUdt (cid:12)(cid:12)(cid:12)(cid:12) U =0 = (1 − θ ) γ a A ≥ ,dQdt (cid:12)(cid:12)(cid:12)(cid:12) Q =0 = ξ a A ≥ ,dRdt (cid:12)(cid:12)(cid:12)(cid:12) R =0 = η u U + η i I + η a A ≥ . Therefore, the positivity of the solutions starting in the4interior of R is assured. R is positively invariant setof the SAIUQR model system (1). Appendix B: Proof of Theorem III.2
Proof.
To prove the boundedness of the SAIUQRsystem (1), we add all the model equations, which gives N = S + A + I + U + Q + R . Taking the differentiationgives dNdt = Λ s − δN, which gives lim sup t →∞ N ( t ) ≤ Λ s δ . Without any loss of generality, we can assumethat lim sup t →∞ S ( t ) ≤ Λ s δ , lim sup t →∞ A ( t ) ≤ Λ s δ , lim sup t →∞ I ( t ) ≤ Λ s δ , lim sup t →∞ U ( t ) ≤ Λ s δ , lim sup t →∞ Q ( t ) ≤ Λ s δ , and lim sup t →∞ R ( t ) ≤ Λ s δ . Thus,we have a bounded setΩ = (cid:26) ( S, A, I, U, Q, R ) ∈ R : 0 ≤ S, A, I, U, Q, R ≤ Λ s δ (cid:27) , which is also a positively invariant set with respect tothe SAIUQR model (1). Thus, any solution trajectorystarting from an interior point of R ultimately enterinto the region Ω and remains there for all finite time.This results indicates that none of the individuals growunboundedly or exponentially for a finite time window. Appendix C: Proof of Theorem III.3
Proof.
The variational matrix around the infectionfree steady state E for the SAIUQR model system (1)is given by J E = − δ − β s α a − β s α i − β s α u ρ s γ q β s α a − ( ξ a + γ a + η a + δ ) β s α i β s α u θγ a − ( η i + δ ) 0 (1 − ρ s ) γ q
00 (1 − θ ) γ a − ( η u + δ ) 0 00 ξ a − ( γ q + δ ) 00 η a η i η u − δ . The above Jacobian matrix J E has two repeated eigen-values which are - δ , while the other four eigenvaluesare the roots of the following characteristics equation | J E − λI | = 0,( ξ a + γ a + η a + δ + λ )( η i + δ + λ )( η u + δ + λ )( γ q + δ + λ ) − β s α a ( η i + δ + λ )( η u + δ + λ )( γ q + δ + λ ) − θγ a β s α i ( η u + δ + λ )( γ q + δ + λ ) − (1 − θ ) γ a β s α u ( η i + δ + λ )( γ q + δ + λ ) − (1 − ρ s ) γ q ξ a β s α i ( η u + δ + λ ) = 0 , which can be rewritten in the following form: β s α a ξ a + γ a + η a + δ + λ + θγ a β s α i ( ξ a + γ a + η a + δ + λ )( η i + δ + λ )+ (1 − θ ) γ a β s α u ( ξ a + γ a + η a + δ + λ )( η u + δ + λ )+ (1 − ρ s ) γ q ξ a β s α i ( ξ a + γ a + η a + δ + λ )( η i + δ + λ )( γ q + δ + λ ) = 1 . Denote the above expression as the following: m ( λ ) = β s α a ξ a + γ a + η a + δ + λ + θγ a β s α i ( ξ a + γ a + η a + δ + λ )( η i + δ + λ )+ (1 − θ ) γ a β s α u ( ξ a + γ a + η a + δ + λ )( η u + δ + λ )+ (1 − ρ s ) γ q ξ a β s α i ( ξ a + γ a + η a + δ + λ )( η i + δ + λ )( γ q + δ + λ ) , = m ( λ ) + m ( λ ) + m ( λ ) + m ( λ ) (say) . λ = x + iy , and we know that Re ( λ ) ≥
0, thenthe above expression leads to | m ( λ ) | ≤ β s α a | ξ a + γ a + η a + δ + λ | ≤ m ( x ) ≤ m (0) , | m ( λ ) | ≤ θγ a β s α i | ξ a + γ a + η a + δ + λ || η i + δ + λ | ≤ m ( x ) ≤ m (0) , | m ( λ ) | ≤ (1 − θ ) γ a β s α u | ξ a + γ a + η a + δ + λ || η u + δ + λ |≤ m ( x ) ≤ m (0) , | m ( λ ) | ≤ (1 − ρ s ) γ q ξ a β s α i | ξ a + γ a + η a + δ + λ || η i + δ + λ || γ q + δ + λ |≤ m ( x ) ≤ m (0) . Thus, m (0) + m (0) + m (0) + m (0) = m (0) = R < , which gives that m ( λ ) ≤ . Therefore for R <
1, all the eigenvalues ofthe characteristics equation m ( λ ) = 1 are real or havenegative real parts. Thus for R <
1, all the eigenvaluesare negative and hence the infection free steady state E is locally asymptotically stable.Now, if we consider R >
1, that is, m (0) >
1, thenlim λ →∞ m ( λ ) = 0 , then there exists λ ∗ > m ( λ ∗ ) = 1 . This indicates that there exists non-negative eigenvalue λ ∗ > J E . Hence, the infec-tion free steady state E is unstable for R > Appendix D: Proof of Theorem III.4
Proof.
To prove the global asymptotic stability of theinfection free steady state E ( S , A , Q , I , U , R ), we can rewrite the SAIUQR model (1) in the following form: dXdt = F ( X, V ) ,dVdt = G ( X, V ) , G ( X,
0) = 0 , where X = ( S, R ) ∈ R represents (its components) thenumber of susceptible or uninfected individuals and V =( A, I, U, Q ) ∈ R represents (its components) the num-ber of infected individuals incorporating asymptomatic,quarantine, and infectious etc. E = ( X ∗ ,
0) designatesthe infection free steady state for the SAIUQR modelsystem (1). For the compartmental model (1), F ( X, V )and G ( X, V ) are defined as follows: F ( X, V ) = (cid:18) Λ s − β s SN ( α a A + α i I + α u U ) + ρ s γ q Q − δSη u U + η i I + η a A − δR (cid:19) , and G ( X, V ) = β s SN ( α a A + α i I + α u U ) − ( ξ a + γ a ) A − η a A − δAθγ a A + (1 − ρ s ) γ q Q − η i I − δI (1 − θ ) γ a A − η u U − δUξ a A − γ q Q − δQ . From the above expression of G ( X, V ), it is clear that G ( X,
0) = 0 . The following two conditions ( C
1) and ( C
2) must bemet to assure the global asymptotic stability:(C1) For dXdt = F ( X, , X ∗ is globally asymptoticallystable,(C2) G ( X, V ) = BV − b G ( X, V ) , b G ( X, V ) ≥ X, V ) ∈ Ω,where B = D I G ( X ∗ ,
0) is an M-matrix (the non-diagonalcomponents are non-negative) and in the region Ω, theSAIUQR model system (1) is biologically feasible. Thecompartmental model (1) stated in the condition ( C ddt (cid:18) SR (cid:19) = (cid:18) Λ s − δS − δR (cid:19) . Analytically solve the above system of equations, weobtain that S ( t ) = Λ s δ + exp( − δt )( S (0) − Λ s δ ) , and R ( t ) = exp( − δt ) R (0) . Considering t → ∞ , S ( t ) = Λ s δ and R ( t ) → . Thus, X ∗ is globally asymptoticallystable for dXdt = F ( X, . Thus, the first condition ( C B and b G ( X, V ) for the SAIUQRmodel system (1) can be expressed as6 B = − ( ξ a + γ a + η a + δ ) + α a β s α i β s α u β s θγ a − ( η i + δ ) 0 (1 − ρ s ) γ q (1 − θ ) γ a − ( η u + δ ) 0 ξ a − ( γ q + δ ) , b G ( X, V ) = α a β s A (cid:18) − SN (cid:19) + α i β s I (cid:18) − SN (cid:19) + α u β s U (cid:18) − SN (cid:19) . It is clear that B is a M-matrix as all its non-diagonalcomponents are non-negative. Also, b G ( X, V ) ≥ S ( t ) ≤ N ( t ) . Also, we showed that X ∗ is aglobally asymptotically stable steady state of the system dXdt = F ( X, E of the SAIURQ model (1) is globally asymptoticallystable in the region Ω for R < . Appendix E: Proof of Theorem III.5
Proof.
Now, we use the theory of center manifold toinvestigate the local asymptotic stability of the interiorequilibrium point E ∗ ( S ∗ , A ∗ , Q ∗ , I ∗ , U ∗ , R ∗ ) by consid-ering the disease transmission rate β s as a bifurcationparameter, β s = β cs corresponding to R = 1 , is β cs = ( η u + δ )( γ q + δ )( η i + δ )( ξ a + γ a + η a + δ ) α a ( η u + δ )( γ q + δ )( η i + δ ) + (1 − θ ) α u γ a ( γ q + δ )( η i + δ ) + α i { θγ a ( γ q + δ ) + (1 − ρ s ) ξ a γ a } ( η u + δ ) . The variational matrix of the SAIUQR model (1)at β s = β cs , denoted by J E has the right eigen-vector associated to zero is eigenvalue given by ω =[ ω , ω , ω , ω , ω , ω ] T , where ω = ω δ (cid:20) − (1 − θ ) γ a β s α u η u + δ + ρ s γ q ξ a η q + δ − ( ξ a + γ a + η a + δ )+ β s α u (1 − θ ) γ a η u + δ (cid:21) ,ω = ω > , ω = (1 − θ ) γ a ω η u + δ , ω = ξ a ω η q + δ ,ω = ω β s α i (cid:20) ( ξ a + γ a + η a + δ ) − β s α a − β s α u (1 − θ ) γ a η u + δ (cid:21) ,ω = ω δ (cid:20) η a + η u (1 − θ ) γ a η u + δ + η i (( ξ a + γ a + η a + δ ) − β s α a ) β s α i − η i β s α u (1 − θ ) γ a β s α i ( η u + δ ) (cid:21) . Similarly, at the threshold β s = β cs , the variationalmatrix J E has the left eigenvector associated to zeroeigenvalue is given by υ = [ υ , υ , υ , υ , υ , υ ], where υ = 0 , υ = 0 , υ = υ > ,υ = β s α i υ η i + δ , υ = β s α u υ η u + δ ,υ = υ ξ a (cid:20) ( ξ a + γ a + η a + δ ) − β s α a − θγ a β s α i η i + δ − β s α u (1 − θ ) γ a η u + δ (cid:21) . Let us introduce the notations for the SAIUQR modelsystem (1): S = x , A = x , I = x , U = x , Q = x , R = x , and dx i dt = f i , where i = 1 , , ..., . Now,we compute the following nonzero second order partialderivatives of f i at the infection free steady state E andobtain ∂ f ∂x ∂x = − β s ( α a + α i ) Λ s δ , ∂ f ∂x ∂x = − β s ( α a + α u ) Λ s δ ,∂ f ∂x ∂x = − β s α a Λ s δ , ∂ f ∂x ∂x = − β s α a Λ s δ ,∂ f ∂x ∂x = − β s ( α i + α u ) Λ s δ , ∂ f ∂x ∂x = − β s α i Λ s δ ,∂ f ∂x ∂x = − β s α i Λ s δ , ∂ f ∂x ∂x = − β s α u Λ s δ ,∂ f ∂x ∂x = − β s α u Λ s δ , ∂ f ∂x ∂x = − β s α a Λ s δ ,∂ f ∂x ∂x = − β s α i Λ s δ , ∂ f ∂x ∂x = − β s α u Λ s δ . The rest of the partial derivatives at the infection freesteady state E remains zero. Now, we compute the co-efficients a and b due to the well-known Theorem 4.1 byCastillo-Chavez & Song [38] as follows: a = X i,j,k =1 υ k ω i ω j ∂ f k (0 , ∂x i ∂x j , b = X i,k =1 υ k ω i ∂ f k (0 , ∂x i ∂β s . By substituting the values of all the nonzero second-orderpartial derivatives and the left and right eigenvectorsfrom above analysis at the threshold β s = β cs , we have a = − β s υ Λ s δ ( ω ω ( α a + α i ) + ω ω ( α a + α u )+ ω ω α a + ω ω α a + ω ω ( α i + α u ) + ω ω α i + ω ω α i + ω ω α u + ω ω α u + 2 ω α a + 2 ω α i + 2 ω α u ) , and b = υ Λ s δ ( ω α a + ω α i + ω α u ) . From the above expressions, it can be observed that a < b >
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