key: cord-0500739-r369bs8i authors: Batistela, Cristiane M.; Correa, Diego P. F.; Bueno, 'Atila M; Piqueira, Jos'e R. C. title: SIRSi-Vaccine dynamical model for Covid-19 pandemic date: 2021-04-15 journal: nan DOI: nan sha: 1db962868f20ce0ff7838b4bc0a4afca6773f1dc doc_id: 500739 cord_uid: r369bs8i The Severe Acute Respiratory Syndrome Corona Virus 2 (SARS-CoV-2), or Covid-19, burst into a pandemic in the beginning of 2020. An unprecedented worldwide effort involving academic institutions, regulatory agencies and industry is facing the challenges imposed by the rapidly spreading disease. Emergency use authorization for vaccines were granted in the beginning of December 2020 in Europe and nine days later in the United States. The urge for vaccination started a race, forcing governs and health care agencies to take decisions on the fly regarding the vaccination strategy and logistics. So far, the vaccination strategies and non-pharmaceutical interventions, such as social distancing and the use of face masks, are the only efficient actions to stop the pandemic. In this context, it is of fundamental importance to understand the dynamical behavior of the Covid-19 spread along with possible vaccination strategies. In this work a Susceptible - Infected - Removed - Sick with vaccination (SIRSi-Vaccine) model is proposed. In addtion, the SIRSi-Vaccine model also accounts for unreported, or asymptomatic, cases and the possibility of temporary immunity, either after infection or vaccination. Disease free and endemic equilibrium points existence conditions are determined in the (! ? ?) vaccine-effort and social distancing parameter space. The model is adjusted to the data from S~ao Paulo, Santos and Campinas, three major cities in the State of S~ao Paulo, Brazil. In 30 January 2020 the World Health Organization (WHO) declared a Public Health Emergency of International Concern (PHEIC). Four weeks before, the Wuhan Municipal Health Commission had reported a cluster of pneumonia cases, whose causative agent was soon after discovered to be the Severe Acute Respiratory Syndrome Corona Virus 2 (SARS-CoV-2), or Covid-19. Despite the WHO alert, the Covid-19 became pandemic weeks later, in March 2020 [1, 2] . It is of great concern the fact that worldwide governs and Health Care Agencies were unable to stop the Covid-19 pandemic in its beginning. As a result, the rapidly spreading disease is challenging health care angencies, academic institutions and industry to develop and deploy efficient drug treatments and vaccination. A number of drugs have been tried in patients with Covid-19 disease. Unfortunately, studies [3, 4, 5, 6, 7] found that the drugs had little or no effect on the overall mortality, initiation of ventilation, duration of hospital stay or viral clearance even for patients with chronic use of some of the tried drugs. On the other hand, non-pharmaceutical interventions (NPI), such as physical distancing, use of face masks and eye protection, reduce the virus transmission [8, 9] . Worldwide populations have been compliant to NPI, however, in some countries, there is a controversy over its effectiveness [10] . In Brazil, the closure of non-essential activities lasted only for a short time, and the cities lifted NPI in an uncoordinated manner [9] . As a result, in the beginning of November 2020, the downward trend reversed and the number of Covid-19 cases started to raise in a second wave of infection [11] . The development of vaccines usually takes many years, even decades, and is also very expensive. The fast development of Covid-19 vaccines was only possible since researchers had been working for years on vaccines for similar viruses, such as, SARS (Severe Acute Respiratory Syndrome) and MERS (Middle East Respiratory Syndrome). In addition, the experience gained with the Ebola vaccine showed that the development of new vaccines can be accelerated with a worldwide effort, including academic institutions, industry and health care regulatory agencies, without compromising safety [12, 13, 14] . There are currently 78 vaccines in clinical trial on humans, and other 77 are under investigation in animals. Emergency use authorization for vaccines were granted in the beginning of December 2020 by Britain's National Health Service, and later on by the U.S. Food and Drug Administration (FDA). In january 2021 the Brazilian national health surveillance agency (Anvisa) also granted emergency use for vaccines. Up to now, in several countries, 6 vaccines have been granted emergency use, and 7 approved for full use [15, 16] . The Covid-19 pandemic, and the consequent urge for vaccination, started a race, with countries trying to vaccinate populations as fast as possible. In [17] the immunization results of a nationwide mass vaccination in Israel is reported, strengthening the expectation of Covid-19 pandemic effects mitigation. However, at the current pace, and for most countries, a successful vaccination strategy consists of a long journey ahead [18] . Logistics plays an important role in vaccination strategies deployment, since the efficacy 1 of the vaccines currently in development ranges from 50.38% to 95.0% [15] , and that different vaccines need different vaccination strategies, given the number of doses, the number of weeks apart each dose, specific storage facility needs, and so on. Besides, Health Care Agencies must take decisions considering the Covid-19 new variants, vaccination strategy with varying NPI adherence among the population, or consider to delay the second dose in order to speed up the one dose vaccinated population [19, 20] . In addition, important factors are vaccine efficacy and population vaccination cover. Vaccines have to have at least 70% efficacy with population cover of 75% to prevent a pandemic without any other intervention. To stop the ongoing pandemic the needed vaccines efficacy raises to at least 80%, for 75% population cover [21] . Given such complex context it is important to understand the dynamical behavior of the Covid-19 pandemic, considering possible vaccination strategies in order to mitigate of even to extinguish the ongoing pandemic. Many scientific works have been developed since the beginning of the pandemic, aiming to stop, or mitigate, the pandemic effects, consisting of control systems strategies, and deep learning modeling. Other works, consisting of mathematical modeling, analysis, and numerical simulations, in order to predict the Covid-19 contagious behavior, have also been published. In [22] time-series models -Auto-Regressive Integrated Moving Average (ARIMA) and Seasonal Auto-Regressive Integrated Moving Average (SARIMA) are developed for the COVID-19 pandemic, with data for the countries where 70% -80% of global cumulative cases are located. The results show exponential growth of confirmed cases and deaths. In [23] a two parameter model is used to predict the pandemic evolution in France. A Susceptible -Infected -Removed model is used to predict the Covid-19 pandemic in Kuwait [24] . A SIR-type model with non-constant population is developed In [25] . The model is calibrated to the rates of infection and deaths for Italy, Spain and the USA. In [26] the death toll due to the Covid-19 pandemic is modeled via sigmoid curve, predicting mortality, and estimating practical interest values such as the number of new cases per infection, i.e., the basic reproduction number. In [27] a mathematical model of Covid-19 is developed. A control system is proposed aiming to reduce the number of the susceptible individuals in a population applying NPI consisting of social distancing. A number of works consider deep learning to predict the number of Covid-19 cases. In [28] a prediction model of confirmed and death cases of Covid-19 is developed, based on a deep learning algorithm with two long short-term memory layers, considering the available data for Covid-19 in India. In [29] , multilayer deep learning models with perceptron, random forest and Long shortterm memory, are trained and used to forecast the Covid-19 pandemic in Iran. In [30] a Susceptible -Infected -Removed -Sick (SIRSi) model is proposed, modeling unreported or asymptomatic cases, and considering the effects of temporary acquired immunity on the Covid-19 spread. In this work, the model proposed in [30] is modified, and a vaccination strategy is included, consisting of a vaccination rate of susceptible individuals. The proposed SIRSi-Vaccine model is composed of four compartments, namely, Susceptible, Infected, Sick and Recovered. The focus is to assess the effects of vaccination associated with social distancing as a NPI in the Covid-19 spread dynamics. The proposed SIRSi-Vaccine model presents both disease free and endemic equilibrium points. The equilibrium points stabilities are studied both analytically and numerically, and, as a result, the found equilibrium existence condition relates the effects of the social distancing to the vaccination rate. In addition, the SIRSi-Vaccine model is numerically adjusted to the epidemic situation in São Paulo, Santos and Campinas, three major cities in the State of São Paulo connecting the shores with the interior of the São Paulo state. The numerical results show that in order to stop the Covid-19 spread different strategies may be necessary for different cities, concerning social distancing NPI and vaccination rates. The paper is organized as follows; Section 2 presents the mathematical model. In Section 3 the equilibrium points are determined and their stability are investigated. The parametric fitting is presented in Section 4. In Section 5 the simulations results for São Paulo, Santos and Campinas are presented. The final remarks are shown in Section 6. The infected population can be asymptomatic or symptomatic, and according to [34] the incubation period is assumed to be of 5.1 days. Infectiousness occur 12 hours prior to the symptoms onset for the symptomatic. On the other hand, for those who are asymptomatic, the infectiousness is assumed to occur 4.6 days after infection. The average time between infection and infectiousness is 6.5 days. Those who are asymptomatic or do not develop severe symptoms, i.e., neither tested nor documented cases, are moved to the R (Recovered) compartment after The incubation period for those who are sympotmatic is 1/β 2 . Once the infected individual is tested positive and the case is documented, it is moved to the S ick compartment, which consists of those patients with severe symptoms seeking medical attention. partment R. The possibility of temporary immunity for recovered individuals, as addressed in [35, 36, 37, 30] , is also considered. As it can be seen in Fig. 1 , the recovered R population becomes susceptible S again at a rate γ. The model is developed considering the population size constraint of Eq. 1. In addition, the population growth and death rates µ, including deaths due to Covid-19, are similar in the State of São Paulo [38] , therefore, in this work, they are considered to be equal. From the foregoing considerations the SIRSi mathematical model is given In Eq. 2, the effect of social distancing measure, which is a form of NPI, is introduced by the parameter θ, 0 < θ < 1. Where θ = 0 means that no social distancing measure is under consideration, and θ = 1 means a complete lockdown. A normalization with respect to the population size N is performed in Eq. 1, resulting in Eq. 3, where Replacing Eq. 3 into the system of Eq. 2, results Eq. 4. In addition, from Eq. 3, the recovered compartment r(t) can be written as a linear combination of the other compartments (or state variables), as shown in Eq. 5, consequently, the solutions with reproduction basic number, A vaccination intervention is introduced consisting of susceptible individuals vaccinated at a rate ω. The vaccinated individuals are considered to become instantaneously immune after vaccination, being moved to the recovered compartment. The proposed SIRSi-Vaccine model is shown in Fig. 2 . Including the vaccination intervention in Eqs. 4 and 6 results in Eqs. 8 and 9, respectively. In this section the equilibrium points for the mathematical models in Eqs. 6 and 9 are determined and their stabilities are analysed. The system of Eq. 6 has two equilibrium points, one is consistent with a disease-free situation, and the other representing an endemic equilibrium. The disease-free equilibrium point P df 1 is given by Eq. 10. On the other hand, the endemic equilibrium point P e1 is given by Eq. 11 where In order to analyze the local stability of the point, the Jacobian matrix J of the system in Eq. 6 is shown in Eq. 13. The eigenvalues associated with P df 1 , the disease free equilibrium point, are given by Eq. 14. The eigenvalue λ 2 shows the possibility of a bifurcation since it can change sign depending on the parameters values. Therefore, the disease free equilibrium Comparing Eqs. 7 and Eq. 15 it can be noticed that if R 0 < 1 then P df 1 is locally asymptotically stable. On the other hand, if R 0 > 1, then P df 1 is unstable. Considering the endemic equilibrium point P e1 , Eqs. 11 and 12, and the Jacobian matrix, Eq. 13, the characteristic polynomial of P e1 is given by Eq. 16 , where the coefficients are given by Eq. 17. The coefficients a 1 , a 2 and a 3 , in Eq. 17, are positive real numbers if ϕ > 0 (see Eq. 12). In this case, in order to determine whether the characteristic polynomial in Eq. 16 has unstable or stable roots the Routh-Hurwitz stability criterion [39] is used. Consequently, the characteristic polynomial in Eq. 16 is stable if expression 18 holds. The proof that expression 18 holds true demands some mathematical reasoning, which results in Eq. 19. where The expression inside brackets in Eq. 19 is positive if ϕ > 0. Consequently, for ϕ > 0 the expression 18 holds true, and P 3 is asymptotically stable. In addition, as it can be seen in Eq. 12, ϕ > 0 implies that R 0 > 1. Similarly to what was found in Section 3.2, the SIRSi-Vaccine mathematical model of Eq. 9 has two equilibrium points, the first one is consistent with a disease-free situation, and the other with an endemic equilibrium. The disease free equilibrium point of Eq. 9 is shown in Eq. 20, for s * > 0. The endemic equilibrium point P e2 is given by Eq. 21 where The local stability of the equilibrium points are determined with the Jacobian matrix shown in Eq. 3.2. The disease free equilibrium point P df 2 is associated with the eigenvalues λ 1 , λ 2 and λ 3 in Eqs. 24, 25 and 26, respectively. From Eq. 24 and 26 it can be seen that λ 1 and λ 3 are negative. However, λ 3 is negative only if the condition in 27 holds true. Therefore, considering the foregoing relations, and if condition 27 holds, the point P e2 is locally asymptotically stable. Now, regarding the endemic equilibrium point P e2 , it exists only if the condition expressed in 28 holds true. It is interesting to notice that the existence condition depends on the vaccination rate ω. Besides, for R 0 < 1, P e2 exists only for negative values of ω, which is a case of no physical concern. The stability of the endemic equilibrium point P e2 is conditioned to parameters combinations, including the social distancing (θ) and the vaccination rate (ω) parameters. The mathematical reasoning to derive and prove such stability conditions is cumbersome, for that reason, numerical analysis is used to infer the stability of the point. The simulations results presented in Section 5, show that if the disease free equilibrium point P df 2 is unstable, then the trajectories in the vicinity of P e2 are stable. These results can be seen from the numerical analysis represented by the bifurcation graphs in Section 5. In this section, the parameters of the The social distancing index is time-varying, presenting an oscillatory behavior with approximately 7-days period, a feature that impairs the numerical parametric adjustment of the SIRSi-Vaccine model. In order to avoid numerical problems, the approach is to use a polynomial function adjusted to the social distancing data as the social distancing index. The data is obtained using privacy-protected mobile phone device location information available in SEADE [42] for São Paulo, Campinas and Santos. For the least-squares parameters adjustment raw data was used for São Paulo. On the other hand, for Campinas and Santos, instead of the raw data, a 21-day moving average filter was used in order to smooth the data prior to the least-squares fitting. In spite of that, the filtered data preserves both monthly trend and general behavior. The social distancing index polynomial fitting results can be seen in Figs. 3, 4 and 5. For all the other parameters, the Least-squares Trust Region Reflective (TRR) algorithm [43, 44] , which is a robust real-time optimization method, was used to fit the number of Covid-19 confirmed cases. in Figs. 6, 7 and 8. The parameters determined with the least-squares TRR algorithm for Santos, Campinas, and São Paulo, are shown in the Tables 1, 2 and 3, respectively. In this section, the numerical simulations of the SIRSi-Vaccine model (Eq. The simulations in Fig. 9 show that without vaccination the Covid-19 spread do not decay for a social distancing index less than θ = 0.5. On the other hand, for θ = 0 the number of cases nearly triple, which is a potential threath to health care units. With θ = 0.7 the infected population rapidly decay until the mid July, a behavior that is observed also for θ = 0.5, but slower in this last case. It is important to point out that this hypothetical behavior offers an insight to what could be the result in such situation, since the social distancing data shows that θ, in fact, presents a moving average value behavior combined with a 7-days periodic oscillation. The simulations in Fig. 10 are performed adopting constant social distancing index (see Table 1 ). As it can be seen, the simulations results are consistent with the equilibrium points stability and existing conditions in Section 3.2. For the vaccination rate ω < 0.1 in Fig. 10 , the disease free equilibrium point P df 2 is unstable according to condition 27, and the endemic equilibrium point P e2 exists according to condition 28. The simulation also suggests that P e2 is stable due to the steady number of confirmed cases over time. For ω = 0.1 the situation is the opposite. In this case, the disease free point P df 2 is stable, and the endemic equilibrium point P e2 does no exist, according to condition 28, and the number of confirmed cases decay to zero. In Fig. 11 the number of confirmed cases for Santos as a function of the social distancing index θ and of the vaccination rate ω is shown. In addition, In addition, Fig. 11 shows the relation between the social distancing index θ and the vaccination rate ω, indicating how to develop a vaccination strategy, combined with NPI for the city of Santos. In this case, with an isolation index θ > 0.5 and a small vaccination rate, the number of confirmed cases decay to a disease free equilibrium. On the other hand, for θ < 0.3, the vaccination rate must be much higher, for instance ω > 0.4, otherwise the number of cases do not decay, remaining constant over time, in a situation compatible with an stable endemic equilibrium. The qualitative general behavior of the simulations for Campinas, shown in Figs. 12, 13 and 14, is similar to what is presented in the Section 5.1. In Fig. 12 the number of confirmed cases is shown considering ω = 0, and for different values of social distancing index θ. As a result, it can be seen that for social distancing index higher than 0.5, i.e, θ > 0.5, the disease free equilibrium is stable -condition 27 holds true -and the number of cases decay to zero over time. Differently, for θ < 0.5, the disease free equilibrium is unstable. prevent health care units from collapsing. The simulations of infected confirmed cases for Campinas, considering a constant social distancing index θ, see Table 2 , and different values for the vaccination rate ω, are shown in Fig. 13 . In this case, a vaccination rate ω = 0.1 is enough to generate a stable disease free response. For smaller vaccination rates, however, the endemic equilibrium exists and the simulations suggests it is stable. Comparing Figs. 11 and 14 , the vaccination rate that is capable of generat- ing stable disease free responses is considerably smaller for Campinas than for Santos. This behavior is probably due to the re-susceptibility feedback gain γ, since the value found for Santos is nearly 2.6 times bigger than for Campinas, see Tables 1 and 2 . Another relevant information is the slightly bigger social distancing data dispersion for Santos, that influences on the infectiousness of the Covid-19 spread, and probably on the re-susceptibility feedback gain γ. In Fig. 15 On the other hand, for simulations with θ < 0.5 the disease free equilibrium is unstable, and since the condition 28 holds true, the endemic equilibrium exists. Therefore, the number of cases do not decay over time. the SIRSi-Vaccine model response. demic publicly available data for Santos, Campinas and São Paulo. The qualitative general behavior of the simulations is compatible with the analytical results. However, comparing the simulations of Santos with Campinas and São Paulo, a slightly different behavior is found for Santos. The difference is probably due to the re-susceptibility feedback gain γ, determined by the least-squares algorithm, that is 2.6 times higher for Santos, when compared with Campinas and São Paulo. This result indicates that for Santos, a successful vaccination strategy imply a higher vaccination rate. Another result is the influence of the social distancing index θ. Small vari-ations on this index can generate abrupt increase in the number of infected confirmed Covid-19 cases in a short period of time. This behavior is of special concern due to the possible threat to health care units. Why did the world's pandemic warning system fail when COVID hit? World Health Organization. WHO Timeline -COVID-19 Repurposed antiviral drugs for covid-19 -interim WHO solidarity trial results Hydroxychloroquine with or without azithromycin for treatment of early sars-cov-2 infection among high-risk outpatient adults: A randomized clinical trial Effect of hydroxychloroquine and azithromycin on qt interval prolongation and other cardiac arrhythmias in covid-19 confirmed patients Covid-19 pneumonia in patients on chronic hydroxychloroquine therapy: Three cases of covid-19 pneumonia Use of hydroxychloroquine and chloroquine in patients with COVID-19: a meta-analysis of randomized clinical trials, Pathogens and Global Health Physical distancing, face masks, and eye protection to prevent person-to-person transmission of sars-cov-2 and covid-19: a systematic review and meta-analysis Dataset on SARS-CoV-2 non-pharmaceutical interventions in Brazilian municipalities Commentary: Physiological and psychological impact of face mask usage during the covid-19 pandemic Ministério da Saúde do Brasil The lightning-fast quest for covid vaccines -and what it means for other diseases Applying lessons from the ebola vaccine experience for sars-cov-2 and other epidemic pathogens Ebola vaccine success shows how to beat covid-19, Vaccines Today Coronavirus vaccine tracker Por unanimidade, anvisa aprova uso emergencial de vacinas contra covid-19 Bnt162b2 mRNAcovid-19 vaccine in a nationwide mass vaccination setting Coronavirus (covid-19) vaccinations SARS-CoV-2 Vaccines and the Growing Threat of Viral Variants Delayed second dose versus standard regimen for covid-19 vaccination Vaccine efficacy needed for a covid-19 coronavirus vaccine to prevent or stop an epidemic as the sole intervention Forecasting the dynamics of cumulative covid-19 cases (confirmed, recovered and deaths) for top-16 countries using statistical machine learning models: Auto-regressive integrated moving average (arima) and seasonal auto-regressive integrated moving average (sarima) The ockham's razor applied to covid-19 model fitting french data Building a sensible sir estimation model for covid-19 outspread in kuwait Seoane-Sepúlveda, A sirtype model describing the successive waves of covid-19 Effective epidemic model for covid-19 using accumulated deaths An extended state observer based u-model control of the covid-19 Real-time measurement of the uncertain epidemiological appearances of covid-19 infections Covid-19 in iran: Forecasting pandemic using deep learning Sirsi compartmental model for covid-19 pandemic with immunity loss Sars-cov-2 viral load in upper respiratory specimens of infected patients Familial cluster of COVID-19 infection from an asymptomatic Characteristics of COVID-19 infection in Beijing Impact of non-pharmaceutical interventions (npis) to reduce covid-19 mortality and healthcare demand The dynamics of humoral immune responses following sars-cov-2 infection and the potential for reinfection The time course of the immune response to experimental coronavirus infection of man Longitudinal profile of antibodies against sars-coronavirus in sars patients and their clinical significance Automatic Control Systems, Tenth Edition Modelling the covid-19 epidemic and implementation of population-wide interventions in italy The effect of control strategies to reduce social mixing on outcomes of the covid-19 epidemic in wuhan, china: a modelling study SP contra o novo coronavírus adesão ao isolamento social em sp Computing a trust region step Epidemiological and clinical characteristics of the covid-19 epidemic in brazil Effective reproductive number estimation for initial stage of covid-19 pandemic in latin american countries Early transmission dynamics of covid-19 in a southern hemisphere setting Forecasting covid-19 pandemic: A data-driven analysis Portal estatísticas do Estado de São Paulo Availability of data and materials Data are publicly available with Acknowledge JRCP is supported by the Brazilian Research Council (CNPq), grant number: 302883