Compartmental model with loss of immunity: analysis and parameters estimation for Covid-19
Cristiane M. Batistela, Diego P. F. Correa, Átila M Bueno, José R. C. Piqueira
aa r X i v : . [ q - b i o . P E ] A ug Compartmental model with loss of immunity: analysisand parameters estimation for Covid-19
Cristiane M. Batistela a , Diego P. F. Correa b , Átila M Bueno c , José R. C.Piqueira a, ∗ a Polytechnic School of University of Sao Paulo - EPUSP, São Paulo, SP, Brazil. b Federal University of ABC - UFABC, São Bernardo do Campo, SP, Brazil c São Paulo State University - UNESP, Sorocaba, SP, Brazil
Abstract
The outbreak of Covid-19 led the world to an unprecedent health and economicalcrisis. In an attempt to responde to this emergency researchers worldwide areintensively studying the Covid-19 pandemic dynamics. In this work, a SIRSicompartmental model is proposed, which is a modification of the known classicalSIR model. The proposed SIRSi model considers differences in the immunizationwithin a population, and the possibility of unreported or asymptomatic cases.The model is adjusted to three major cities of São Paulo State, in Brazil, namely,São Paulo, Santos and Campinas, providing estimates on the duration and peaksof the outbreak.
Keywords:
Covid-19, Compartmental models, Equilibrium analysis,Parameter fitting.
1. Introduction
The Wuhan Municipal Health Commission reported a cluster of 27 pneumo-nia cases on 31st December 2019, in the Wuhan city, Hubei Province in China.On 1st January 2020 the World Health Organization (WHO) set up the IncidentManagement Support Team putting the organization to an emergency level for ∗ Corresponding author
Email addresses: [email protected] (Cristiane M. Batistela), [email protected] (Diego P. F. Correa), [email protected] (Átila MBueno), [email protected] (José R. C. Piqueira)
Preprint submitted to Chaos, Solitons and Fractals August 7, 2020 ealing with the outbreak. On 5th January 2020 WHO published the first out-break news on the new virus. On 7th January the causative agent was identifiedand named Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2)(WHO named the disease Covid-19). On 13th January 2020 the first case out-side China was reported in Thailand. On 22nd January 2020 WHO statedthat there was evidence of human-to-human transmission, and approximatelyseven weeks later, on 13th March 2020, WHO characterized the Covid-19 as apandemic [1, 2].In Brazil, the first confirmed Covid-19 case was reported on the 26th Febru-ary 2020, and up to 25th June 2020 there was 1,188,631 confirmed cases with53,830 deaths [3]. Globally, according to [4] there was 9,292,202 confirmed cases,and 479,133 deaths up to 25th June 2020 .Most cases are asymptomatic carriers and are spontaneously resolved, how-ever, some developed various fatal complications, notably for patients with co-morbidities [2], and, allied to the fastly spread of covid-19, the worldwide emer-gency state brings up with important, and yet unanswered, questions related tothe contagion dynamical behavior and its mitigation and control strategies. Asa result, strategies to contain the contagion such as social distancing, quarantineand complete lockdown of areas have been studied [5, 6, 7, 8, 9, 10]In Mexico, on 18th March 2020, the Mexican Health Secretariat reportedthat the pandemic was going to last 90 days, with 250,656 expected cases. Onthe next day, the Health Secretariat informed that approximately 9.8% (24,594)of the cases would need hospitalization, and 4.2% (10,528) of the total caseswould be critical patients, needing intensive care. On the same date the numberof available Health Care units at that time was 4,291 with 2,053 ventilators [11].The situation in Mexico led to the adoption of non-pharmaceutical interventions,such as washing hands, social distancing, cough/sneeze etiquette, and so on.In Rio de Janeiro, Brazil, the social distancing started on 17 March 2020[8], and the government of the state of São Paulo, Brazil, decreed quarantine on Reported on 2:25pm CEST. ≈ ≈
6% of the recovered patients) the NAbs levels wereundetectable, 25 ( ≈ S ick ) population compartment representing nodes of the network that manifestthe symptoms of the disease. The proposed Susceptible - Infected - Removed -Sick (SIRSi) model also considers the birth and death of individuals in the givenpopulation. In addition, a feedback from those recovered who did not gain im-munity or loss their immunity after a period of time and become susceptibleagain is also introduced.The proposed SIRSi model presents both a disease free and an endemicequilibrium, and the influence of the re-susceptibility feedback is investigatedboth analytically and numerically.The parameters of the proposed SIRSi model are numerically fitted to the4pidemic situation for three cities of the São Paulo State, Brazil, namely, SãoPaulo, the capital of the State; Santos, on the coast and approximately 80Km away from São Paulo; and Campinas, in the interior of the state and ap-proximately 90 Km distant from the capital São Paulo, providing estimates fordiagnosis and forecasting of the Covid-19 epidemics spread.The paper is organized as follows, on section 2 the SIRSi compartmentalmathematical model is presented. The equilibrium points existence and stabilityconditions are discussed in section 3, showing the possibility of both endemicand disease-free equilibrium. The parameter fitting of the SIRSi model for thecities of São Paulo, Santos and Campinas is shown in secton 4, and the numericalexperiments results in section 5. The concluding remarks can be seen in section6.
2. A modified SIR model with birth and death cases
The proposed SIRSi model can be seen in Fig. 1. In this model, the sus-ceptible population S is infected at a rate α when contact infected individualsfrom I . The susceptible compartment also receives a population, at a rate /γ ,who didn’t gain complete immunity after recovering or who loss their immunityafter a period of time.The compartment I represents the infectious population in incubation stageprior the onset of symptoms. Infection transmission during this period hasbeen reported in [29, 30, 31, 32]. The infected population can be asymptomaticor symptomatic. The period between infection and onset of symptoms, /β ,ranges from 3 to 38 days with median of 5.2. Once the infected individual istested positive and the case is documented, the case is moved to the S ick com-partment. Those who don’t develop severe symptoms become asymptomatic.In [33] the estimative of the infections originated from undocumented casesis as high as 86%, which include mild, limited and lack of symptoms infectiousindividuals. In other recent studies [34, 35], it was found that, 20% - 40% ofpositive tested patients were asymptomatic.5 I RS ick λ δ α β δγβ σ β Figure 1: Epidemiological SIRSi model.
In this work, and according to [36], the asymptomatic population is con-sidered as those of under-reporting cases. This population could be as 7 timesbigger than the size of the documented cases. This group, under-reporting cases,become recovered with the period /β .Some of the individuals within this population could eventually developsymptoms. In [37], it has been reported that the average time between in-fection and the onset of symptoms can be 4.6 days. Once the case becomesdocumented it should be moved to the S ick compartment. S ick are those in-fected which seek for medical attention with severe symptoms. This populationare those who tested positive for COVID-19. In [38] it was reported that thispopulation could represent up to 19.9% of the total documented cases, of which13.8% are severe cases and 6.1% require intensive care. The sick populationbecome recovered with the period /β or removed at a rate σ (see Fig. 1).In order to consider the social distancing measure effect in the number ofinfected and deaths toll shown in Fig. 1, the parameter θ is introduced onthe mathematical model (1), where θ is subject to the constraint < θ < .Consequently, given these facts, the mathematical SIRSi model is given by (1),6 S = λ − α (1 − θ ) SI − δS + γR, ˙ I = α (1 − θ ) SI − ( β + β ) I, ˙ S ick = β I − ( β + σ ) S ick ˙ R = β I + β S ick − ( γ + δ ) R (1)where λ and δ are the birth and death rates, respectively.It is important to notice that the number of documented cases is a keyinformation that should emerge, somehow, from (1). The reason is the factthat the accumulated number of confirmed cases is publicly available, and willbe used to fine-tuning the model. The number of daily new infections is alsoavailable, but it tends to be noisy and less representative of the dynamics.
3. Disease-free and endemic equilibrium points
Considering (1), such that ˙ x = f ( x ) , where x = ( S, I, S ick , R ) T , x ∈ U ⊂ ( R +0 ) , f : U → U is the right-hand side of (1), and parameters α , β , β , β , σ , γ , θ ∈ R + .To investigate the influence of the introduction of the feedback from thoserecovered with no immunity which become susceptible again, and dividing thepopulation into groups, the equilibrium points related to the model describedby (1) must be determined and their stability discussed.Assuming α = 0 , i.e., susceptible can be converted into infected, and despitebeing an assumption it is realistic for a spreading disease, the equilibrium pointsare such that f ( x ∗ ) = 0 .Using the Hartman-Grobman Theorem [39] the local stability of the equi-librium points can be determined by the eigenvalues of the Jacobian matrixcomputed on each equilibrium point. The Jacobian J = ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) x ∗ of (1) is givenby (2). 7 = − ( δ + I ∗ α (1 − θ )) − S ∗ α (1 − θ ) 0 γI ∗ α (1 − θ ) S ∗ α (1 − θ ) − ( β + β ) 0 00 β − ( β + σ ) 00 β β − ( δ + γ ) (2) In the following the disease-free (section 3.1) and the endemic (section 3.2)equilibrium points are determined.
The disease-free equilibrium is a state corresponding to the absence of in-fected individuals, i.e. , I ∗ = 0 . Applying this condition to the equilibrium of(1), the point can be determined.Assume that there exist a disease free-equilibrium ( I ∗ = 0 ), x ∗ ∈ U , suchthat f ( x ∗ ) = 0 . This equilibrium point x ∗ = ( S ∗ , I ∗ , S ∗ ick , R ∗ ) T with x ∗ in thefirst octant of R is given by: P = ( S ∗ , I ∗ , S ∗ ick , R ∗ ) T = ( λ/δ, , , T . (3)Considering P , the corresponding linear system Jacobian [39] J P = ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) x ∗ is given by (4). J P = − δ − α (1 − θ )( λ/δ ) 0 γ α (1 − θ )( λ/δ ) − ( β + β ) 0 00 β − ( β + σ ) 00 β β − ( δ + γ ) . (4)By the Laplace determinant development [40], the eigenvalues of (4) are theelements in the diagonal, that is, ξ = − δ , ξ = α (1 − θ )( λ/δ ) − ( β + β ) , ξ = − ( β + σ ) and ξ = − ( δ + γ ) .The eigenvalues above with the Hartman-Grobman Theorem indicates that(1) presents one disease-free equilibrium point, and considering the condition8iven by the eigenvalue ξ = α (1 − θ ) λ/δ − ( β + β ), if α (1 − θ ) λ/δ < ( β + β )the eigenvector associated indicates an asymptotically stable direction. Con-sequently, if α (1 − θ ) λ/δ > ( β + β ) the equilibrium point P is unstable,indicating a bifurcation in the parameter space. The endemic equilibrium points are characterized by the existence of infectedpeople in the population, that is, ( I ∗ = 0 ).Therefore, assuming the existence of an endemic equilibrium point, with x ∗ ∈ U , such that f ( x ∗ ) = 0 , the equilibrium point P = x ∗ = ( S ∗ , I ∗ , R ∗ , S ∗ ick ) T in the first octant of R is given by (6), S ∗ = β + β α (1 − θ ) I ∗ = 1 φ ( δ + γ ) ( β + σ ) [ α (1 − θ ) λ − ( β + β ) δ ] S ∗ ick = 1 φ β ( δ + γ ) [ λα (1 − θ ) − ( β + β ) δ ] R ∗ = 1 φ ( β β + β β + β σ ) [ α (1 − θ ) λ − ( β + β ) δ ] , (5)where φ = α (1 − θ ) ( β β δ + β β δ + β δσ + β δσ + β γσ ) .Accordingly, the existence condition of a positive endemic equilibrium P = x ∗ = ( S ∗ , I ∗ , R ∗ , S ∗ ick ) T ∈ R + is given by (6). α (1 − θ ) λδ > β + β . (6)Note that condition (6) reflects the fact that, in order to the endemic equilib-rium exists, the rate α at which, the people flux rate, represented by λ/δ , becomeinfected, has to be greater than the rate at which infected population leave thecompartment I , either overcoming the disease or becoming symptomatic.It is important to highlight that λ/δ could also represent the total peopleflux commuting from a different city in a multi-population model.The linearization A = J P = ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) x ∗ at the endemic equilibrium is:9 = − ( δ + I ∗ α (1 − θ )) − ( β + β ) 0 γI ∗ α (1 − θ ) 0 0 00 β − ( β + σ ) 00 β β − ( δ + γ ) . (7)The characteristic polynomial det ( A − I d ξ ) = 0 is ξ + a ξ + a ξ + a ξ + a = 0 , (8)with a = β + 2 δ + γ + σ + I ∗ α (1 − θ ); a = I ∗ α (1 − θ ) ( β + β + β + δ + γ + σ ) + 2 δ ( β + σ )+ γ ( β + δ + σ ) + δ ; a = I ∗ α (1 − θ )[ β β + β β + ( β + β + β ) δ + ( β + β ) γ +( β + β + δ + γ ) σ ] + β δ + δ σ + β δ γ + δ γ σ ; a = I ∗ α (1 − θ ) ( β β δ + β β δ + β δ σ + β δ σ + β γ σ ) . (9)Any further effort to analytically analyze ξ eigenvalues, becomes quite diffi-cult due to the coefficients complexity. A possible alternative approach is to gofor numerical calculations.However, some insight for the model with feedback γ can be obtained, interms of bifurcations and stability, if we analyze the eigenvalues when γ = 0 and γ = 0 . γ = 0 Note that in this case, the endemic equilibrium is still possible. Computingthe eigenvalues results, 10 = − δ,ξ = − ( β + σ ) ,ξ = 12 ( β + β ) ( − α (1 − θ ) λ + √ ∆) ,ξ = 12 ( β + β ) ( − α (1 − θ ) λ − √ ∆) , (10)such that ∆ = 4 δ ( β + β ) + ( α (1 − θ )) λ − λα (1 − θ )( β + β ) .The eigenvalues ξ and ξ can be either complex conjugate stable, or bothreal. The eigenvalue ξ needs to be further studied due to the possibility ofbifurcation.Analysing the eigenvalue ξ , if α (1 − θ ) λ > √ ∆ , P is stable, and conse-quently (6) holds true.On the other hand, if α (1 − θ ) λ < √ ∆ , the endemic equilibrium pointis unstable, and the existence condition (6) is not satisfied, consequently, theendemic equilibrium point P , if existing, is stable. γ → ∞ Another insight can be obtained by looking with the case γ → ∞ . In thiscase, the endemic equilibrium becomes: S ∗ = β + β α (1 − θ ) ,I ∗ → ( β + σ ) α (1 − θ ) β σ [ α (1 − θ ) λ − ( β + β ) δ ] S ∗ ick → β + β ) α (1 − θ ) ,R ∗ → , (11)which is subject to the same existence condition given in equation (6).Under the assumption γ → ∞ , the characteristic polynomials coefficients inequation (9) can be approximated by (12):11 ≈ γ,a ≈ γ ( I ∗ α (1 − θ ) + β + δ + σ ) = γb ,a ≈ γ ( I ∗ α (1 − θ ) ( β + β + σ ) + δ ( β + σ )) = γb ,a = γI ∗ α (1 − θ ) β σ = γb , (12)the characteristic polynomial (8) can be rewritten as in (13): ξ + γξ + γb ξ + γb ξ + γb = 0 ,ξ + γ ( ξ + b ξ + b ξ + b ) = 0 , (13)assuming that at least one root | ξ | goes to infinity as γ → ∞ , we rewrite thepolynomial as ξ + γξ = 0 ,ξ ( ξ + γ ) = 0 , (14)then, ξ = − γ, and γ → ∞ , so, one eigenvalue seem to be going to −∞ as γ → ∞ . To find an approximationto the other three roots, the characteristic polynomial is rewritten in equation(13). It can be also assumed that the other three roots are finite, ξ + γξ + γb ξ + γb ξ + γb = 0 , γ ξ + ξ + b ξ + b ξ + b = 0 , ≈ ξ + b ξ + b ξ + b = 0 . (15)Looking for insight numerical experiments are performed.
4. Parameters fitting by the least-squares method
In this section the parameters of the proposed SIRSi model (1) (see Fig. 1)are numerically adjusted to fit the curve for confirmed symptomatic infected12ases of three major cities in the state of São Paulo - Brazil, using public avail-able data from the State Data Analysis System - SEADE (
Sistema Estadual deAnálise de Dados ). The total population for the cities was obtained from thesame source and it is shown in table 1. City Total population in 2020
São Paulo 11.869.660Campinas 1.175.501Santos 428.703
Table 1: Total population collected from SEADE.
To calculate birth and death rates, λ and δ respectively, linear interpolationwas necessary, as the data from the public repository was out of date, resultsare shown in Fig. 2. Figure 2: Birth and death rates per 1000 inhabitants for São Paulo, Santos and Campinas.Public data is shown in solid lines and interpolated values are shown in dashed lines.
One of the first actions against the spread of the virus was the impositionof a social distancing measure which was first decreed in São Paulo on May17. This intervention, represented by θ in the model, focuses on reducing the α (1 − θ ) . The time series correspondingto the daily measures of this index along with their mean for each one of thecities considered are shown in figure 3. Although this time variation resemblesa 7-day periodic function, especially on the second half of the register, we useas a first approximation the mean of this measure as a representative value. Figure 3: Percentage of social distancing [41].
By the time we write, there is no much of information about the durationof the antibody response to SARS-CoV-2, however what it is known so far isthat protection after recovery wanes over time. Recent research highlight thatimmune response and antibodies protection after recovery may depend on the14everity of the infection, in some cases this protection can last for as little as 12weeks while in others few cases no antibodies protection is obtained at all [42,43, 44, 45, 46, 47]. To assess the influence of this behavior on the disease spread,we set the feedback parameter γ at constant values γ ∈ { , . , . , . , . } in order to map possible scenarios especially those related to possible secondwaves of infection.The transmission rate of symptomatic individuals prior hospitalization isestimated between 1.12 to 1.19 while for asymptomatic cases this rate rangesbetween 0.1 to 0.6, in model (1) this parameter is represented by the product α (1 − θ ) , thus, we let α ∈ [0 . , . . The mean time between infection and onset-of-symptoms for confirmed cases, /β , is 2 days, thus we let β ∈ [0 . , . . Thetime from onset-of-infection to fully recovery, /β , is considered to be betweena few days up to two weeks (5 to 15 days). The period of time it takes fora symptomatic patient to overcome the disease, /β , and the time betweenhospitalization to death, /σ , are both considered to be between 5 to 20 days[48, 37, 38, 49].The trust-region reflective least-squares algorithm [50, 51, 52] along with a4th-order Runge-Kutta integrator were used to fit parameters in model (1) toactual data collected from public repositories for each one of the three cities intoconsideration. All parameters and initial conditions computed are normalizedwith respect to the total population in each case. For the fitting, we set S ∈ [0 , , I ∈ [0 , with initial guess S i = 99 . and I i = 0 . , the initialcondition for S ick and R was set to zero. Results are shown in tables 2 to 7,and the temporal evolution of the S ick compartment for each city, for both, thefitted model and the public data, is show in figures 6, 8 and 10.In order to assess the influence of the parameter γ in the endemic equilibrium,the eigenvalues were plotted for the set of fitted parameters found, for γ ∈ [0 , .In figures 4 are shown the eigenvalues for the endemic equilibrium for each citycomputed for each one of the fitted sets. At γ = 0 eigenvalues are stable forSão Paulo e Santos, as γ increases, eigenvalues move towards the left-hand sideof the complex plane, whereas for Campinas eigenvalues are unstable for γ = 0 .15 it 1 Fit 2 Fit 3 Fit 4 Fit 5 α β β β σ λ δ θ γ Table 2: Fitted parameter for São Paulo.
Fit 1 Fit 2 Fit 3 Fit 4 Fit 5 S I S ick R Table 3: Fitted initial conditions for São Paulo.
Fit 1 Fit 2 Fit 3 Fit 4 Fit 5 α β β β σ λ δ θ γ Table 4: Fitted parameter for Santos.
In 5 a closer view of the eigenvalues around the origin are shown.16 it 1 Fit 2 Fit 3 Fit 4 Fit 5 S I S ick R Table 5: Fitted initial conditions for Santos.
Fit 1 Fit 2 Fit 3 Fit 4 Fit 5 α β β β σ λ δ θ γ Table 6: Fitted parameter for Campinas.
Fit 1 Fit 2 Fit 3 Fit 4 Fit 5 S I S ick R Table 7: Fitted initial conditions for Campinas.
5. Numerical experiments
In this section the numerical experiments are conducted using the MATLAB-Simulink [53] for two different conditions. Firstly, the SIRSi model is fitted withthe real data for the S icks population, and for different values of the parameter γ . In the sequence, the simulation for the infected population I , that can beinferred from the proposed model, is carried out.17 Real -202 I m ag i na r y -3 São Paulo -2.5 -2 -1.5 -1 -0.5 0
Real -202 I m ag i na r y -3 Santos -2.5 -2 -1.5 -1 -0.5 0
Real -101 I m ag i na r y Campinas
Figure 4: Eigenvalues for γ ∈ { , . } , for the three cities into consideration. The numerical experiments, as in section 4, were conducted for three majorcities in the state of São Paulo, namely, São Paulo, Campinas, and Santos.The initial condition is x = ( S , I , S ick , R ) T , where S and I are thenormalized susceptible and infected populations, which are considered free pa-rameters in the sense that they can be modified by the fitting algorithm. In Fig.6 it can be seen that the SIRSi model is adjusted for the confirmedcases of infected people data.Considering that the acquired immunity is permanent, i.e. , γ = 0 , and thatthe isolation rate is constant, the peak of the infection occurs soon after July2020, and until the end of the the same year, the disease will not persist, sincethe number of confirmed cases will go down to zero.18 Real -4 -202 I m ag i na r y -3 São Paulo -2.5 -2 -1.5 -1 -0.5 0
Real -4 -202 I m ag i na r y -3 Santos -6 -5 -4 -3 -2 -1 0 1 2
Real -3 -0.1-0.0500.05 I m ag i na r y Campinas
Figure 5: Eigenvalues for γ ∈ { , . } , closer view around the origin. On the other hand, assuming that immunity is not permanent and adoptinga reinfection rate γ = 0 . , meaning that every 100 days a recovered personbecomes susceptible again, the model predicts a decrease in the confirmed casesand a new wave of infection in the first half of 2022.Decreasing the time interval to 50 days in which a recovered person becomessusceptible ( γ = 0 . ), the model indicates a second wave of infection in thefirst half of 2021. For the situation in which a recovered system is susceptibleto each 25 days ( γ = 0 . ), the model simulation shows that by the end of thisyear the number of confirmed infected will reduce by almost two thirds and thatthe number of infected will continue decreasing over time, but the number ofconfirmed cases will remain higher than the other simulated curves.In Fig.7, the infected compartment I inferred from the SIRSi model is pre-sented, showing that the peak of infection is close to July 2020.19 an 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Jul 2022 Jan 2023 days C on f i r m ed c a s e s i n S ão P au l o c i t y Real data=0=0.01=0.02=0.03=0.04
Figure 6: Time evolution for confirmed cases ( S ick population) in São Paulo. Increasing γ will reduce the time for a recovered person to become suscep-tible again, causing the peaks in Fig. 7 to increase, when compared to thecurves for lower values of γ . This behavior, however, cannot be observed inFig.6, indicating that the increase the re-susceptibility feedback gain γ possiblycontributes to the increase of asymptomatic or unreported infected cases.In addition, it appears that if those recovered acquire permanent immunity γ = 0 , the number of infected people tends to zero by the end of 2020. For γ = 0 . , it appears that there is a small increase in January 2022. For γ = 0 . a new wave of confirmed cases can be seen in Fig.6, and accordingly, the increasein the infected population is also observed in Fig.7.For São Paulo, the numerical experiments show that considering any reinfec-tion rate, there will be confirmed infected cases and unreported infected cases20 an 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Jul 2022 Jan 2023 days I n f e c t ed popu l a t i on i n S ão P au l o c i t y =0=0.01=0.02=0.03=0.04 Figure 7: Time evolution for infected population in São Paulo. until 2023, indicating the need for a control strategy, being necessary and thestudy of preventive inoculation.
Fig.8 shows the SIRSi model adjusted to the confirmed cases of infectedpeople data for Santos. Assuming that the immunity acquired is permanent, γ = 0 , and that the isolation rate is constant, the peak of the confirmed cases inSantos will occur very close to July 2020. and similarly to São Paulo (see Fig.6), until the end of the same year, the disease will not persist with the numberof confirmed cases going down to zero.Adopting a nonzero reinfection rate, such that one person every 100 daysbecomes susceptible again ( γ = 0 . ), a second wave of infection is seen in thecoastal city around July 2021, one year before the second wave predicted for21 an 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Jul 2022 Jan 2023 days C on f i r m ed c a s e s i n S an t o s c i t y Real data=0=0.01=0.02=0.03=0.04
Figure 8: Time evolution for confirmed cases (sick population) in Santos.
São Paulo with the same value for the re-susceptibility feedback gain γ .Considering γ = 0 . , for which an infected person becomes susceptibleagain in a time interval of 50 days, the second wave of confirmed cases occursat the beginning of the first half of 2021 and the number of confirmed infectedis reduced to one third of the peak value.These situations should be analyzed with caution and it is suggested to studythe influence of the flow of people between these cities, since in the city of Santosthe second waves of infections occur before the city of São Paulo.For γ = 0 . , after the peak of the confirmed cases, a second wave can beobserved in the numerical results before the end of 2020, delaying the reductionof the confirmed cases.For Santos, the numerical experiments show that for γ = 0 and for γ = 0 . ,the numbers of confirmed cases tend to zero in the beginning of 2023.22he infected compartment I inferred from the SIRSi model is presented inFig. 9, showing that the peak of infection is close to July 2020. Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Jul 2022 Jan 2023 days I n f e c t ed popu l a t i on i n S an t o s c i t y =0=0.01=0.02=0.03=0.04 Figure 9: Time evolution for infected population in Santos.
The increase in re-susceptibility feedback gain γ , will reduction the time foran infected person to be susceptible again, causing higher peaks when comparedwith the curves for lower values of γ . This behavior does not occur in Fig.8indicating that the increase in feedback possibly contributes to the increase inasymptomatic or unreported infected cases. This situation is similar to what isobserved for São Paulo.In addition, for γ = 0 , the number of the infected people I tends to zerobefore the end of the 2020 (see the curve for γ = 0 in Fig.9).For γ = 0 . , a new wave of infection in 2021 can be seen, and for γ = 0 . the peak of the second wave of infection is near January 2021 (see Fig.9, γ = 0 . and γ = 0 . ). 23nlike São Paulo, the highest peak of infection among the unreported occurswhen γ = 0 . and this behavior suggests a more detailed study of the dynam-ics, because together with the situation in which the infected person acquirespermanent immunity , these rates suggest that the saving of confirmed cases(see Fig.8 and asymptomatic infected individuals tends to zero more quickly.In the situation in which a recovered person is liable to a new susceptibilityin 25 days, it is observed that the infection persists in the population for alonger time, as shown by the curve with γ = 0 . in Fig.9 and justifies thepolicy strategies public policies, including vaccination. In Fig. 10 the SIRSi model adjusted to the confirmed cases of infected peopledata in Campinas is shown.
Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Jul 2022 Jan 2023 days C on f i r m ed c a s e s i n C a m p i na s c i t y Real data=0=0.01=0.02=0.03=0.04
Figure 10: Time evolution for confirmed cases (sick population) in Campinas.
For permanent acquired immunity, γ = 0 , and constant isolation rate, the24eak of confirmed infection cases occur in the beginning of the second half of2020.Considering the re-susceptibility feedback gain γ > , in Fig. 10, it seemsthat with the increasing of γ the time necessary for the number of confirmedinfected cases go down to zero is slightly bigger, unlike the other two citiesstudied. In addition, Campinas does not present a second wave of infection,even with the variation γ .The general behavior of Campinas, concerning the sensitivity analysis for γ , present results that differ from Santos and São Paulo. It can be noticed inFig. 10 that the observed data are far from the peak of infection predicted bythe SIRSi model. At this point, more data is needed for any further qualitativeanalysis.Observing the eigenvalues for the city of Campinas (See Figs. 4 and 5) it canbe noticed that they are all real, indicating that there is no oscillatory behaviorin the dynamics of the model. Depending on the new data this situation mightchange.The infected compartment of Campinas presents the peak of infection closeto the beginning of the second half of 2020, Fig.11.The increase in the reinfection parameter, causes the peak to increase andthis occurs in the figure Fig.10 indicating that the increase in feedback possiblycontributes to the increase in asymptomatic or unreported infected.
6. Conclusions
The proposed SIRSi model was fitted to publicly available data of the Covid-19 outbreak, providing estimates on the duration and peaks of the outbreak.In addition, the model allows to infer information related to unreported andasymptomatic cases.The proposed model with feedback adjusted to the confirmed infection data,suggests the possibility of the recovered ones having temporary immunity γ > or even permanent γ = 0 . 25 an 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Jul 2022 Jan 2023 days I n f e c t ed popu l a t i on i n C a m p i na s c i t y =0=0.01=0.02=0.03=0.04 Figure 11: Time evolution for infected population in Campinas.
Considering the situation in which immunity is temporary, there is a secondwave of infection which, depending on the time interval for a recovered personto be susceptible again, indicates a second wave with a greater or lesser numberof reinfected persons.If the time interval is shorter (larger γ ), the second wave of infection will havea greater number of infected people when compared to a shorter time intervalof the feedback.The qualitative behavior for São Paulo and Santos are similar in terms thesensitivity analysis of the re-susceptibility feedback gain γ . The bigger the γ theshorter the time for a recovered person to become susceptible again to infection,increasing unreported or asymptomatic cases.For the city of Campinas, it is suggested to collect more data, because asthe data of the confirmed infected presents a certain distance from the peak ofthe infection, the dynamics of the model may undergo some significant change,26iven the sensitivity of the model to disturbances.
7. Availability of data and materials
Data are publicly available with [54, 41].
8. Declaration of competing interest
There is no conflict of interest between the authors.
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