# SIRSi-Vaccine dynamical model for Covid-19 pandemic

Cristiane M. Batistela<EMAIL_ADDRESS>Diego P. F. Correa
<EMAIL_ADDRESS>Átila M Bueno<EMAIL_ADDRESS>José R. C.
Piqueira<EMAIL_ADDRESS>São Paulo University, Polytechnic School -
EPUSP, São Paulo, SP, Brazil.  Federal University of ABC - UFABC, São Bernardo
do Campo, SP, Brazil  São Paulo State University - UNESP, Sorocaba, SP, Brazil

###### Abstract

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 ($\omega\times\theta$) vaccine-effort and social distancing
parameter space. The model is adjusted to the data from São Paulo, Santos and
Campinas, three major cities in the State of São Paulo, Brazil.

###### keywords:

Covid-19 , Compartmental models , Equilibrium analysis , Parameter fitting.

††journal: Chaos, Solitons and Fractals

##  1 Introduction

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 efficacy111The vaccine efficacy is the percentage of reduction of a
disease in the vaccinated group in a clinical trial. 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 short-
term 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.

##  2 SIRSi and SIRSi-Vaccine mathematical models

A constant population model with four compartments is proposed, see Fig. 1.
The compartments are defined as: Susceptible $(S)$, Infected $(I)$, Infected
symptomatic positive tested $(S_{ick})$ and Recovered $(R)$.

In Fig 1, $I$ is the infectious population compartment representing the
population in the incubation stage, i.e., prior to the onset of symptoms.
Transmission during this period has been reported in [31, 32, 33].

$S$$I$$R$$S_{ick}$$\mu$$\mu$$\alpha$$\beta_{1}$$\mu$$\gamma$$\beta_{2}$$\mu$$\mu$$\beta_{3}$
Figure 1: SIRSi model with constant population.

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 a $1/\beta_{1}$ period [30].

The incubation period for those who are sympotmatic is $1/\beta_{2}$. 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.

After a period $1/\beta_{3}$ the $S_{ick}$ population that recovers is moved
to the compartment $R$.

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 $\gamma$.

The model is developed considering the population size constraint of Eq. 1. In
addition, the population growth and death rates $\mu$, 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.

$N=S(t)+I(t)+S_{ick}(t)+R(t)$ (1)

From the foregoing considerations the SIRSi mathematical model is given by Eq.
2.

$\begin{split}\frac{d}{dt}S(t)&=\mu N-\frac{\alpha(1-\theta)}{N}S(t)I(t)-\mu
S(t)+\gamma R(t);\\\
\frac{d}{dt}I(t)&=\frac{\alpha(1-\theta)}{N}S(t)I(t)-(\beta_{1}+\beta_{2})I(t)-\mu
I(t);\\\ \frac{d}{dt}S_{ick}(t)&=\beta_{2}I(t)-\beta_{3}S_{ick}(t)-\mu
S_{ick}(t);\\\ \frac{d}{dt}R(t)&=\beta_{1}I(t)+\beta_{3}S_{ick}(t)-\gamma
R(t)-\mu R(t);\end{split}$ (2)

In Eq. 2, the effect of social distancing measure, which is a form of NPI, is
introduced by the parameter $\theta$, $0<\theta<1$. Where $\theta=0$ means
that no social distancing measure is under consideration, and $\theta=1$ means
a complete lockdown.

A normalization with respect to the population size $N$ is performed in Eq. 1,
resulting in Eq. 3,

$1=s(t)+i(t)+s_{ick}(t)+r(t)$ (3)

where $s(t)=S(t)/N$, $i(t)=I(t)/N$, $s_{ick}(t)=S_{ick}(t)/N$ and
$r(t)=R(t)/N$. Replacing Eq. 3 into the system of Eq. 2, results Eq. 4.

$\begin{split}\frac{d}{dt}s(t)&=\mu-\alpha(1-\theta)si-\mu s+\gamma r;\\\
\frac{d}{dt}i(t)&=\alpha(1-\theta)si-(\beta_{1}+\beta_{2})i-\mu i;\\\
\frac{d}{dt}s_{ick}(t)&=\beta_{2}i-\beta_{3}s_{ick}-\mu s_{ick};\\\
\frac{d}{dt}r(t)&=\beta_{1}i+\beta_{3}s_{ick}-\gamma r-\mu r;\end{split}$ (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,

$r(t)=1-s(t)-i(t)-s_{ick}(t),$ (5)

consequently, the solutions
$\Omega=\\{(s(t),i(t),s_{ick}(t))\in\mathbb{R}^{3}_{+}\,|\,s(t)+i(t)+s_{ick}(t)\leq
1\\}$ of the system of Eq. 4 can be studied in the system of Eq. 6,

$\begin{split}\frac{d}{dt}s(t)&=\mu+\gamma-\alpha(1-\theta)si-(\mu+\gamma)s-\gamma
i-\gamma s_{ick}\\\
\frac{d}{dt}i(t)&=\alpha(1-\theta)si-(\beta_{1}+\beta_{2}+\mu)i\\\
\frac{d}{dt}s_{ick}(t)&=\beta_{2}i-(\beta_{3}+\mu)s_{ick}\end{split}$ (6)

with reproduction basic number,

$R_{0}=\dfrac{\alpha(1-\theta)}{\beta_{1}+\beta_{2}+\mu}.$ (7)

A vaccination intervention is introduced consisting of susceptible individuals
vaccinated at a rate $\omega$. 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.

$\begin{split}\frac{d}{dt}s(t)&=\mu-\alpha(1-\theta)si-\mu s+\gamma r-\omega
s\\\ \frac{d}{dt}i(t)&=\alpha(1-\theta)si-(\beta_{1}+\beta_{2})i-\mu i\\\
\frac{d}{dt}s_{ick}(t)&=\beta_{2}i-\beta_{3}s_{ick}-\mu s_{ick}\\\
\frac{dr}{dt}r(t)&=\beta_{1}i+\beta_{3}s_{ick}-\gamma r+\omega s-\mu
r\end{split}$ (8)

$\begin{split}\frac{d}{dt}s(t)&=\mu+\gamma-\alpha(1-\theta)si-(\mu+\gamma+\omega)s-\gamma
i-\gamma s_{ick}\\\
\frac{d}{dt}i(t)&=\alpha(1-\theta)si-(\beta_{1}+\beta_{2}+\mu)i\\\
\frac{d}{dt}s_{ick}(t)&=\beta_{2}i-(\beta_{3}+\mu)s_{ick}\end{split}$ (9)

$S$$I$$R$$S_{ick}$$\mu$$\mu$$\alpha$$\beta_{1}$$\mu$$\gamma$$\omega$$\beta_{2}$$\mu$$\beta_{3}$$\mu$
Figure 2: SIRSi model with vaccination intervention.

##  3 Disease-free and endemic equilibrium points

In this section the equilibrium points for the mathematical models in Eqs. 6
and 9 are determined and their stabilities are analysed.

###  3.1 SIRSi equilibrium points

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 $P_{df1}$ is given by Eq. 10.

$P_{df1}=\left[\begin{array}[]{c}s^{\ast}\\\ i^{\ast}\\\
s^{\ast}_{ick}\end{array}\right]=\left[\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right]$ (10)

On the other hand, the endemic equilibrium point $P_{e1}$ is given by Eq. 11

$P_{e1}=\left[\begin{array}[]{c}s^{\ast}\\\ i^{\ast}\\\
s^{\ast}_{ick}\end{array}\right]=\left[\begin{array}[]{c}\frac{\beta_{1}+\beta_{2}+\mu}{\alpha\left(1-\theta\right)}\\\
(\beta_{3}+\mu)\varphi\\\ \beta_{2}\varphi\end{array}\right],$ (11)

where

$\varphi=\frac{\left(\gamma+\mu\right)\left(\alpha(1-\theta)-(\beta_{1}+\beta_{2}+\mu)\right)}{\alpha\left(1-\theta\right)\left((\beta_{1}+\beta_{2}+\mu)(\beta_{3}+\mu)+(\beta_{2}+\beta_{3}+\mu)\gamma\right)}.$
(12)

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.

$J=\left[\begin{array}[]{@{}c@{}c@{}c@{}}-\alpha(1-\theta)i^{\ast}-(\mu+\gamma)&-\alpha(1-\theta)s^{\ast}-\gamma&-\gamma\\\
\alpha(1-\theta)i^{\ast}&\alpha(1-\theta)s^{\ast}-(\beta_{1}+\beta_{2}+\mu)&0\\\
0&\beta_{2}&-(\beta_{3}+\mu)\end{array}\right]$ (13)

The eigenvalues associated with $P_{df1}$, the disease free equilibrium point,
are given by Eq. 14.

$\begin{split}\lambda_{1}&=-(\mu+\gamma)\\\
\lambda_{2}&=\alpha(1-\theta)-(\beta_{1}+\beta_{2}+\mu)\\\
\lambda_{3}&=-(\beta_{3}+\mu)\end{split}$ (14)

The eigenvalue $\lambda_{2}$ shows the possibility of a bifurcation since it
can change sign depending on the parameters values. Therefore, the disease
free equilibrium point $P_{df1}$ is stable if

$\alpha(1-\theta)<\beta_{1}+\beta_{2}+\mu.$ (15)

Comparing Eqs. 7 and Eq. 15 it can be noticed that if $R_{0}<1$ then $P_{df1}$
is locally asymptotically stable. On the other hand, if $R_{0}>1$, then
$P_{df1}$ 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.

$\mathcal{P}_{e1}(\lambda)=\lambda^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3},$
(16)

$\begin{split}a_{1}&=\varphi\alpha(1-\theta)\left(\beta_{3}+\mu\right)+\beta_{3}+\gamma+2\mu\\\
a_{2}&=\left(\beta_{3}+\mu\right)\left[\gamma+\mu+\varphi\alpha(1-\theta)\left(\beta_{1}+\beta_{2}+\beta_{3}+\gamma+2\mu\right)\right]\\\
a_{3}&=\varphi\alpha\left(1-\theta\right)\left(\beta_{3}+\mu\right)\left(\begin{array}[]{@{}c@{}}\beta_{1}\beta_{3}+\beta_{1}\mu+\beta_{2}\beta_{3}+\beta_{2}\gamma\\\
+\beta_{2}\mu+\beta_{3}\gamma+\beta_{3}\mu+\gamma\mu+\mu^{2}\end{array}\right)\end{split}$
(17)

The coefficients $a_{1}$, $a_{2}$ and $a_{3}$, in Eq. 17, are positive real
numbers if $\varphi>0$ (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.

$b_{1}=a_{2}-\frac{a_{3}}{a_{1}}>0$ (18)

The proof that expression 18 holds true demands some mathematical reasoning,
which results in Eq. 19.

$\begin{split}b_{1}&=\frac{1}{a_{1}}\left(\beta_{3}+\mu\right)\left[\varphi^{2}c_{1}^{2}c_{2}+\varphi
c_{1}c_{3}+c_{4}\right],\end{split}$ (19)

where

$\begin{split}c_{1}&=\alpha(1-\theta),\\\
c_{2}&=(\beta_{1}+\beta_{2})(\beta_{3}+\mu)+\beta_{3}^{2}+\beta_{3}\gamma+3\beta_{3}\mu+\gamma\mu+2\mu^{2},\\\
c_{3}&=\beta_{1}\gamma+\beta_{1}\mu+\beta_{2}\mu+\beta_{3}^{2}+2\beta_{3}\gamma+4\beta_{3}\mu+\gamma^{2}+4\gamma\mu+4\mu^{2},\\\
c_{4}&=\beta_{3}\gamma+\beta_{3}\mu+\gamma^{2}+3\gamma\mu+2\mu^{2}.\end{split}$

The expression inside brackets in Eq. 19 is positive if $\varphi>0$.
Consequently, for $\varphi>0$ the expression 18 holds true, and $P_{3}$ is
asymptotically stable. In addition, as it can be seen in Eq. 12, $\varphi>0$
implies that $R_{0}>1$.

###  3.2 SIRSi-Vaccine equilibrium points

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
$s^{\ast}>0$.

$P_{df2}=\left[\begin{array}[]{c}s^{\ast}\\\ i^{\ast}\\\
s_{ick}^{\ast}\end{array}\right]=\left[\begin{array}[]{c}\frac{\mu+\gamma}{\mu+\gamma+\omega}\\\
0\\\ 0\end{array}\right]$ (20)

The endemic equilibrium point $P_{e2}$ is given by Eq. 21

$P_{e2}=\left[\begin{array}[]{c}s^{\ast}\\\ i^{\ast}\\\
s_{ick}^{\ast}\end{array}\right]=\left[\begin{array}[]{c}\frac{\beta_{1}+\beta_{2}+\mu}{\alpha(1-\theta)}\\\
(\beta_{3}+\mu)\psi\\\ \beta_{2}\psi\end{array}\right],$ (21)

where

$\psi=\frac{\alpha(1-\theta)(\gamma+\mu)-\left(\beta_{1}+\beta_{2}+\mu\right)\,\left(\gamma+\mu+\omega\right)}{\alpha(1-\theta)((\beta_{1}+\beta_{2}+\gamma+\mu)(\beta_{3}+\mu)+\beta_{2}\,\gamma)}.$
(22)

The local stability of the equilibrium points are determined with the Jacobian
matrix shown in Eq. 23.

$J=\left[\begin{array}[]{@{}c@{}c@{}c@{}}-\alpha(1-\theta)i^{\ast}-(\mu+\gamma+\omega)&-\alpha(1-\theta)s^{\ast}-\gamma&-\gamma\\\
\alpha(1-\theta)i^{\ast}&\alpha(1-\theta)s^{\ast}-(\beta_{1}+\beta_{2}+\mu)&0\\\
0&\beta_{2}&-(\beta_{3}+\mu)\\\ \end{array}\right]$ (23)

The disease free equilibrium point $P_{df2}$ is associated with the
eigenvalues $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ in Eqs. 24, 25 and
26, respectively.

$\displaystyle\lambda_{1}$ $\displaystyle=-(\gamma+\mu+\omega)$ (24)
$\displaystyle\lambda_{2}$
$\displaystyle=-(\beta_{1}+\beta_{2}+\mu)+\alpha(1-\theta)\dfrac{\mu+\gamma}{\mu+\gamma+\omega}$
(25) $\displaystyle\lambda_{3}$ $\displaystyle=-(\beta_{3}+\mu)$ (26)

From Eq. 24 and 26 it can be seen that $\lambda_{1}$ and $\lambda_{3}$ are
negative. However, $\lambda_{3}$ is negative only if the condition in 27 holds
true.

$\omega>\left(\frac{\alpha(1-\theta)}{\beta_{1}+\beta_{2}+\mu}-1\right)(\mu+\gamma)=\left(R_{0}-1\right)(\mu+\gamma)$
(27)

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.

$\omega\leq\left(\dfrac{\alpha(1-\theta)}{\beta_{1}+\beta_{2}+\mu}-1\right)(\mu+\gamma)=\left(R_{0}-1\right)(\mu+\gamma)$
(28)

It is interesting to notice that the existence condition depends on the
vaccination rate $\omega$. Besides, for $R_{0}<1$, $P_{e2}$ exists only for
negative values of $\omega$, 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 $(\theta)$ and the
vaccination rate $(\omega)$ 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 $P_{df2}$ 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.

##  4 SIRSi-Vaccine model parametric fitting

In this section, the parameters of the SIRSi model (Eq. 6) are numerically
determined by parametric fitting using public available data covering a one
year period of the Covid-19 pandemic, specifically, ranging from March, 2020
to March, 2021.

The SIRSi-Vaccine model considers the social distancing influence on the
Covid-19 infectiousness. Recently, a number of works have reported that
Covid-19 epidemiological models are strongly sensitive to social distancing
index [40, 41, 34, 30].

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.

Figure 3: Isolation index of Santos. Figure 4: Isolation index of Campinas.
Figure 5: Isolation index of São Paulo.

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.

The free parameters in the fitting process are
$\left(\gamma,\alpha,\beta_{1},\beta_{2},\beta_{3}\right)$ (see Fig. 2), and
$s(0)$ which is the normalized amount of susceptible individuals prior the
pandemic. The other normalized initial conditions are $s_{ick}(0)=0$ and
$i(0)=1-s(0)$. In addition, for the lower and upper bounds of the parameters,
epidemiological and clinical characteristics of Covid-19 reported in the
literature where considered [45, 46, 47, 48]

The confirmed cases are considered to be the $s_{ick}(t)$ compartment
population in the model of Eq. 9. The results of the least-squares TRR fitting
comparing the confirmed cases and the respective $s_{ick}$ compartment
simulation can be seen in Figs. 6, 7 and 8.

Figure 6: Covid-19 infection confirmed cases in Santos [38] compared to the
respective $s_{ick}$ compartment. Figure 7: Covid-19 infection confirmed cases
in Campinas [38] compared to the respective $s_{ick}$ compartment. Figure 8:
Covid-19 infection confirmed cases in São Paulo [38] compared to the
respective $s_{ick}$ compartment.

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.

Parameter | Value
---|---
$\mu$ | 0.000027
$\gamma$ | 0.100000
$\alpha$ | 0.775985
$\theta$ | 0.415355
$\beta_{1}$ | 0.200000
$\beta_{2}$ | 0.200000
$\beta_{3}$ | 0.047847
$s_{0}$ | 0.999754
$i_{0}$ | 0.000246
Table 1: Fitted parameters for Santos. Parameter | Value
---|---
$\mu$ | 0.000034
$\gamma$ | 0.038255
$\alpha$ | 0.776520
$\theta$ | 0.414454
$\beta_{1}$ | 0.200000
$\beta_{2}$ | 0.200000
$\beta_{3}$ | 0.067000
$s_{0}$ | 0.999883
$i_{0}$ | 0.000117
Table 2: Fitted parameters for Campinas. Parameter | Value
---|---
$\mu$ | 0.000036
$\gamma$ | 0.032774
$\alpha$ | 0.811656
$\theta$ | 0.444603
$\beta_{1}$ | 0.200000
$\beta_{2}$ | 0.200000
$\beta_{3}$ | 0.058792
$s_{0}$ | 0.999800
$i_{0}$ | 0.000200
Table 3: Fitted parameters for São Paulo.

##  5 Simulation results

In this section, the numerical simulations of the SIRSi-Vaccine model (Eq. 9)
are performed. The parameters used for the SIRSi-Vaccine model simulations are
the same parameters determined for the SIRSi model in Section 4, and shown in
Tables 1, 2 and 3.

In the simulations the social distancing index $\theta$ and the vaccination
rate $\omega$ are kept constant over time. However, they are set up with
different values in each performed simulation.

Two different scenarios are considered. Firstly, the vaccination rate is set
to $\omega=0$, and the numerical simulations are performed for different
values of the social distancing index, ranging from $\theta=0.0$, i.e., no
social distancing intervention, to $\theta=0.7$ which means that $70$% of the
population is locked home, keeping social distancing.

In the next scenario, the social distancing index $\theta$ is set up equal to
the average value of the measured social distancing data. The values of
$\theta$ used for each case can be seen in the Tables 1, 2 and 3. In this
scenario, the simulations are performed considering different values of
vaccination rate $\omega$ in order to assess the needed vaccination effort to
stop, or, at least, to mitigate the spread of the Covid-19.

###  5.1 Simulation results for Santos

The simulations in Fig. 9 show that without vaccination the Covid-19 spread do
not decay for a social distancing index less than $\theta=0.5$. On the other
hand, for $\theta=0$ the number of cases nearly triple, which is a potential
threath to health care units. With $\theta=0.7$ the infected population
rapidly decay until the mid July, a behavior that is observed also for
$\theta=0.5$, 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 $\theta$, in fact, presents a moving average value behavior
combined with a 7-days periodic oscillation.

Figure 9: Simulation of infection confirmed cases for Santos with $\omega=0$
and different values of social distancing index.

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 $\omega<0.1$ in Fig. 10, the disease free equilibrium
point $P_{df2}$ 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 $\omega=0.1$ the situation is the opposite. In this case, the disease free
point $P_{df2}$ 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.

Figure 10: Simulation of infection confirmed cases for Santos considering
different values of vaccination rate $\omega$.

In Fig. 11 the number of confirmed cases for Santos as a function of the
social distancing index $\theta$ and of the vaccination rate $\omega$ is
shown. In addition, Fig. 11 is also a bifurcation diagram, showing the stable
region (dark-blue) in the parameter space of the disease free equilibrium
point. The light-blue and green areas indicate the region where the disease
free point is unstable. The same region indicates the existence of the endemic
equilibrium.

Figure 11: Steady state response of the number of confirmed cases for Santos.

In addition, Fig. 11 shows the relation between the social distancing index
$\theta$ and the vaccination rate $\omega$, indicating how to develop a
vaccination strategy, combined with NPI for the city of Santos.

In this case, with an isolation index $\theta>0.5$ and a small vaccination
rate, the number of confirmed cases decay to a disease free equilibrium. On
the other hand, for $\theta<0.3$, the vaccination rate must be much higher,
for instance $\omega>0.4$, otherwise the number of cases do not decay,
remaining constant over time, in a situation compatible with an stable endemic
equilibrium.

###  5.2 Simulation results for Campinas

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.

Figure 12: Simulation of infection confirmed cases for Campinas with
$\omega=0$ and different values of social distancing index. Figure 13:
Simulation of infection confirmed cases for Campinas considering different
values of vaccination rate $\omega$.  Figure 14: Steady state response of the
number of confirmed cases for Campinas.

In Fig. 12 the number of confirmed cases is shown considering $\omega=0$, and
for different values of social distancing index $\theta$.

As a result, it can be seen that for social distancing index higher than
$0.5$, i.e, $\theta>0.5$, the disease free equilibrium is stable - condition
27 holds true - and the number of cases decay to zero over time.

Differently, for $\theta<0.5$, the disease free equilibrium is unstable. The
endemic equilibrium exists - condition 28 holds true - and the number of cases
do not decay over time. It is important to notice that for $\theta<0.3$ the
number of confirmed cases abruptly increases, nearly quadrupling in a very
short period. This indicates the importance of NPI to mitigate the Covid-19
spread and to prevent health care units from collapsing.

The simulations of infected confirmed cases for Campinas, considering a
constant social distancing index $\theta$, see Table 2, and different values
for the vaccination rate $\omega$, are shown in Fig. 13. In this case, a
vaccination rate $\omega=0.1$ 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 generating
stable disease free responses is considerably smaller for Campinas than for
Santos. This behavior is probably due to the re-susceptibility feedback gain
$\gamma$, 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 $\gamma$.

###  5.3 Simulation results for São Paulo

Again, for São Paulo, the qualitative general behavior of the simulations
shown in Figs. 15, 16 and 17, is similar to the behavior of the simulations
presented in the Sections 5.1 and 5.2.

Figure 15: Simulation of infection confirmed cases for São Paulo with
$\omega=0$ and different values of social distancing index. Figure 16:
Simulation of infection confirmed cases for São Paulo considering different
values of vaccination rate $\omega$.  Figure 17: Steady state response of the
number of confirmed cases for São Paulo.

In Fig. 15 the simulations of the infection confirmed cases for $\omega=0$ and
for different values of the social distancing index $\theta$ are shown. It can
be seen that for $\theta>0.5$ the disease free equilibrium point is stable, i.
e., condition 27 holds true, consequently the number of cases decay to zero
over time.

On the other hand, for simulations with $\theta<0.5$ 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.

For $\theta<0.3$ the number of infection confirmed cases increases very fast,
threatening the health care units of collapsing in a short period of time.

The simulations of infection confirmed cases shown in Fig. 16, indicate that
for $\omega\geq 0.1$ the disease free equilibrium is stable. On the contrary,
for the simulated values of $\omega<0.1$, the endemic equilibrium is stable
while the disease free equilibrium is unstable. As a result, in this case, the
number of infection confirmed cases do not decay over time.

Comparing Figs. 17, 14 and 11, it is clear that the simulation results
obtained for São Paulo and Campinas are very similar concerning the behavior
related to the social distancing index $\theta$ and the vaccination rate
$\omega$. In order to stop the Covid-19 spread in Santos, the vaccination rate
need to be higher.

The least-square fitting was performed differently for Campinas and Santos due
to the dispersion of the social distancing index. The influence of this
procedure is still not clear, and further investigation may be needed in order
improve the SIRSi-Vaccine model response.

##  6 Conclusions

A compartmental model of the Covid-19 pandemic is proposed consisting of four
compartments, namely: Susceptible - Infected - Recovered - Sick with
vaccination (SIRSi-Vaccine), considering a vaccination strategy and the
possibility of recovered patients become susceptible again, that is,
individuals loose their acquired immunity either after recovery or
vaccination.

The proposed SIRSi-Vaccine model has both disease free and endemic equilibrium
points, presenting interchangeable stability, that is, the condition that
assures stability for the disease free equilibrium imply the nonexistence of
the endemic equilibrium. On the other hand, the endemic equilibrium existence
condition imply on the instability of the disease free equilibrium. In
addition, the existence and stability conditions of expressions 27 and 28
depend on both the reproduction basic number $R_{0}$ and on the vaccination
rate $\omega$.

The proposed SIRSi-Vaccine model is least-squares fitted to Covid-19 pandemic
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
$\gamma$, 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 $\theta$. Small
variations 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.

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## Availability of data and materials

Data are publicly available with [49, 38].

## Declaration of competing interest

There is no conflict of interest between the authors.

## Acknowledge

JRCP is supported by the Brazilian Research Council (CNPq), grant number:
302883/2018-5.