# Simulation of deterministic compartmental models for infectious diseases
dynamics

Antonio M. Batista1,2,3, Silvio L. T. de Souza4, Kelly C. Iarosz3,5,6,∗,
Alexandre C. L. Almeida7, José D. Szezech Jr1,2, Enrique C. Gabrick1, Michele
Mugnaine8, Gefferson L. dos Santos1,5, Iberê L. Caldas3 1Postgraduate Program
in Sciences, State University of Ponta Grossa, 84030-900, Ponta Grossa, PR,
Brazil  2Department of Mathematics and Statistics, State University of Ponta
Grossa, 84030-900, Ponta Grossa, PR, Brazil  3Physics Institute, University of
São Paulo, 05508-090, São Paulo, SP, Brazil  4Federal University of São João
del-Rei, Campus Centro-Oeste, Divinópolis, MG, Brazil  5Faculdade de Telêmaco
Borba, FATEB, Telêmaco Borba, PR, Brazil  6Graduate Program in Chemical
Engineering Federal Technological University of Paraná, Ponta Grossa, PR,
Brazil  7Statistics, Physics and Mathematics Department, Federal University of
São João del-Rei, Ouro Branco, MG, Brazil  8Department of Physics, Federal
University of Paraná, Curitiba, PR, Brazil

###### Abstract

Infectious diseases are caused by pathogenic microorganisms and can spread
through different ways. Mathematical models and computational simulation have
been used extensively to investigate the transmission and spread of infectious
diseases. In other words, mathematical model simulation can be used to analyse
the dynamics of infectious diseases, aiming to understand the effects and how
to control the spread. In general, these models are based on compartments,
where each compartment contains individuals with the same characteristics,
such as susceptible, exposed, infected, and recovered. In this paper, we cast
further light on some classical epidemic models, reporting possible outcomes
from numerical simulation. Furthermore, we provide routines in a repository
for simulations.

###### keywords:

compartmental model , computational simulation , infectious diseases ,
COVID-19

††journal: Rev. Bras. Ensino Fis.

##  1 Introduction

Infectious diseases have caused epidemics with devastating effects, for
instance influenza A virus sub-type H1N1 [1] and smallpox [2]. Infectious
disease with pandemic potential is one of the greatest challenges of the
health system. Recently, the World Health Organization declared COVID-19 [3,
4] as a pandemic. The mortality depends on many factors such as number of
infected people, virulence, and prevention efforts [5]. Various infectious
diseases can come in waves, e.g., the first three waves of avian influenza A
(H7N9) virus circulation [6]. Kissler et al. [7] projected that recurrent
outbreaks of COVID-19 will probably happen.

Epidemiological models [8] have been proposed to analyse the spread of
infectious diseases in host populations [9]. In 1760, Bernoulli [10] proposed
a model to describe the impact of variolation. In 1906, a mathematical model
to explain the epidemic of measles was introduced by Hamer [11].

The SI model [12] describes the evolution of susceptible S and infected I
individuals, respectively. A model was proposed by Ross [13] in 1916 for
malaria. In the Ross model, known as SIS model [14], susceptible become
infected and infected recover without immunity. Kermack and McKendrick [15]
introduced a model, known as SIR model, in which the removed can be recovered,
immune, or dead. The SIRS model [16] was obtained when a waning immunity was
incorporated. In the SEIR model [17] there are four states, where E
corresponds to exposed, namely a latency period is considered. It has been
applied to measles [18] and rubella [19]. The SEIRS model [20] considers
individuals that are transferred from the recovered to the susceptible
compartments.

One of the different types of controls of epidemic is the vaccination. The
vaccination controls have as main objective to remove by immunity the
population from the susceptible state [21]. Gao et al. [22] demonstrated that
pulse vaccination can be an effective strategy for the elimination of
infectious diseases. Investigating the transmission of tuberculosis by means
of epidemiological models, Liu et al. [23] reported that mixed vaccination
gives a rapid control.

When there is no vaccine for the infectious disease, the control strategies
are based on quarantine and isolation, for instance the COVID-19 epidemics
[24]. Law et al. [25] studied a time-varying SIR model for the transmission
dynamics of COVID-19. Prem et al. [26] reported that the magnitude of the
epidemic peak can be reduced through sustained physical distancing. Feng [27]
analysed the final and peak epidemic sizes considering quarantine and
isolation in SEIR models. Recently, Boldog et al. [28] analysed the risk
assessment of novel coronavirus outbreaks outside China by means of an
extension of a standard SEIR model.

Stochastic mathematical models have been considered in prediction of the
spread of several infectious diseases [29]. The law of mass-action has been
used for almost a century in epidemiology to describe the contact rate of
individuals. This law originated in practice and theory of chemical reaction
kinetics [30]. Tomé and Oliveira [14] presented models for epidemic spreading
and showed the analogy between the spreading of a disease with a critical
phase transition. They also analysed the epidemic curve, which is a graphical
representation of the number of identified cases over a period of time.

In this work, we focus on the simulation of different deterministic models
that have been used to describe infectious disease dynamics. We provide
routines in a repository [31] to assist students with their first steps about
computer simulations of mathematical models in epidemiology. The reader can
find mathematical details in Ref. [14]. We show the dynamical behaviours
predict by some mathematical models, such as SI, SIS, SIR, SIRS, SEIR, and
SEIRS. We present an application of the SIR model in COVID-19 for the
parameters obtained from China according to Ref. [32]. Considering the SEIR
model, we show the impact of easing restriction on the infection rate that was
reported in Ref. [33].

##  2 The SI model

The SI model describes the dynamical behaviour of transmitted diseases [34]
with interactions among infected and susceptible people. In this model, vital
processes are not considered, such as rates of birth and mortality. Figure 1
displays the diagram of the SI model, where the population is divided into
susceptible and infected individuals. After infected, the individual does not
return to the susceptible class, for instance herpes that is spread from
person to person by the virus Herpesviridae.

Figure 1: Compartment diagram of the SI model, where S and I are the numbers
of susceptible and infected individuals, respectively, and $\beta$ (in units
of 1/day) is the effective contact rate of the disease.

The SI model is given by

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\beta
SI}{N},$ (1) $\displaystyle\frac{dI}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N},$ (2)

where $N=S+I$ is the total population and $\beta$ is the effective contact
rate of the disease. Figure 2 exhibits the time evolution (in days) of $S$
(blue line) and $I$ (red line) for $\beta=0.1$. With time, every susceptible
individual becomes infected.

Figure 2: Time evolution of $S$ (blue line) and $I$ (red line) for $N=1000$,
$S(0)=995$, $I(0)=5$, and $\beta=0.1$.

##  3 The SIS model

In the SIS model [35, 36], which was studied recently [14, 29], the infected
individuals can become susceptible, however, there is no long-lasting
immunity. Due to this fact, an individual can have recurrent infections, such
as common cold (rhinoviruses) and influenza, as well as sexually transmitted
diseases, for instance chlamydia and gonorrhoea. In Fig. 3, we see a schematic
representation of the compartment diagram related to the SIS model.

Figure 3: Compartment diagram of the SIS model, where S and I are the numbers
of susceptible and infected individuals, respectively, $\beta$ (in units of
1/day) is the infectious rate, and $\delta$ (in units of 1/day) is the rate in
which the infected individuals recover to the susceptible state.

The SIS model is written as

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\beta
SI}{N}+\delta I,$ (3) $\displaystyle\frac{dI}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N}-\delta I,$ (4)

where the infected individual goes to the susceptible state with a rate
$\delta$. In Fig. 4, we consider $\beta=0.1$ and $\delta=0.01$ for $N=1000$.
The SIS model has two stable equilibria, one for $I=0$ and another for
$I=N(1-\delta/\beta)$.

Figure 4: Time evolution of $S$ (blue line) and $I$ (red line) for $N=1000$,
$S(0)=995$, $I(0)=5$, $\beta=0.1$, and $\delta=0.01$.

##  4 The SIR model

In 1927, Kermack and McKendrick [15] proposed the SIR model, a mathematical
model of spread of an infectious disease within a population, studied recently
[14, 29]. They considered three compartments, in which the susceptible
individuals go to the infectious compartment according to an infectious rate
and, depending on the recover rate, the infected individual recover and
develop immunity, as shown in Fig. 5.

Figure 5: Compartment diagram of the SIR model, where S, I, and R are the
numbers of susceptible, infected, and recovered individuals, respectively,
$\beta$ (in units of 1/day) is the infectious rate, and $\gamma$ (in units of
1/day) is the rate in which the infected individuals recover.

The SIR model was introduced to explain the rapid increase of infected
individuals, as verified in epidemics such as the cholera and the plague. The
mathematical model is given by

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\beta
SI}{N},$ (5) $\displaystyle\frac{dI}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N}-\gamma I,$ (6) $\displaystyle\frac{dR}{dt}$
$\displaystyle=$ $\displaystyle\gamma I,$ (7)

where $\gamma$ is the recovery rate and $N=S+I+R$. The behaviour of the
infectious class has a dependence on the parameter $R_{0}=\beta/\gamma$, known
as reproduction ratio. $R_{0}$ is an important threshold quantity that
describes the transmissibility or contagiousness of pathogenic microorganisms
[37]. In a susceptible group of individuals, it gives the expected number of
secondary cases of infections due to an infected individual. The pathogenic
microorganism is able to invade the population of susceptible individuals if
$R_{0}>1$. The births and deaths are not considered due to the fact that the
infection and recovery rates are fast.

Figure 6: Time evolution of $S$ (blue line), $I$ (red line), and $R$ (green
line) for $N=1000$, $S(0)=995$, $I(0)=5$, $\beta=0.2$, and $\gamma=0.05$.

Figure 6 shows the time evolution (in days) of $S$ (blue line), $I$ (red
line), and $R$ (green line) for $N=1000$, $\beta=0.2$, and $\gamma=0.05$. As
initial conditions, we consider $S(0)=995$ and $I(0)=5$. We see that the
infected population reaches a peak while the susceptible individuals decrease
and recovered ones increase. After the peak, the number of infected
individuals decreases.

In 2020, the SIR model was utilised by Cooper et al. [32] to study the spread
of COVID-19 in different communities. They analysed data recorded between
January and June 2020 from China, South Korea, India, Australia, Italy, and
the state of Texas in the USA. It was demonstrated that the SIR model can
provide a theoretical framework to study how the COVID-19 virus spreads within
communities. Figure 7 displays the time evolution of $S$ (blue line), $I$ (red
line), and $R$ (green line) for $N=83132$, $S(0)=83127$, $I(0)=5$,
$\beta=0.35$, and $\gamma=0.035$, where the parameters are selected according
to Ref. [32] for China.

Figure 7: Time evolution of $S$ (blue line), $I$ (red line), and $R$ (green
line) for $N=83132$, $S(0)=83127$, $I(0)=5$, $\beta=0.35$, and $\gamma=0.035$.

##  5 The SIRS model

The SIRS model has been studied by various authors [38, 39] and studied
recently [14, 16, 29]. Figure 8 exhibits the process diagram for the SIRS
model. In this model, the recovery can generate temporary immunity, and as a
consequence the recovered individuals return to the susceptible class after
some time. The loss of immunity is observed in smallpox, tetanus, and
influenza.

Figure 8: Compartment diagram of the SIRS model, where S, I, and R are the
numbers of susceptible, infected, and recovered individuals, respectively,
$\beta$ (in units of 1/day) is the infectious rate, $\gamma$ (in units of
1/day) is the rate in which the infected individuals recover, $\delta$ (in
units of 1/day) is the rate in which the recovered individuals return to the
susceptible class.

The SIRS model is given by

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\beta
SI}{N}+\delta R,$ (8) $\displaystyle\frac{dI}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N}-\gamma I,$ (9) $\displaystyle\frac{dR}{dt}$
$\displaystyle=$ $\displaystyle\gamma I-\delta R,$ (10)

where $\delta$ is the rate in which the recovered individuals return to the
susceptible class after losing the immunity and $N=S+I+R$.

In Fig. 9, the blue, red, and green lines correspond to the time evolution (in
days) of $S$, $I$, and $R$, respectively. We see the persistence of the
infected population due to the transfer of individuals from the recovery class
to the susceptible one.

Figure 9: Time evolution of $S$ (blue line), $I$ (red line), and $R$ (green
line) for $N=1000$, $S(0)=995$, $I(0)=5$, $\beta=0.4$, $\gamma=0.2$, and
$\delta=0.005$. The inset figure corresponds to the magnification of the
infected individuals.

##  6 The SEIR model

One of the compartmental models of infectious diseases is the SEIR model,
studied recently [14, 29]. In this model, the individuals are separated into
four compartments. The SEIR model assumes people in the susceptible (S),
exposed (E), infected (I), and recovered (R) states, as shown in Fig. 10.

Figure 10: Compartment diagram of the SEIR model, where S, E, I, and R are the
numbers of susceptible, exposed, infected, and removed individuals,
respectively, $\beta$ (in units of 1/day) is the infectious rate, $\omega$ (in
units of 1/day) is the coefficient of migration rate, and $\gamma$ (in units
of 1/day) is the rate in which the infected individuals recover.

The people move from S to E due to direct or indirect contact. In the E stage,
the people are infected but are not infectious, namely an latent period. The
infected individuals are recovered. The SEIR model was used by Carcione et al.
[40] to simulate the COVID-19 epidemic.

The SEIR model is written as

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\beta
SI}{N},$ (11) $\displaystyle\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N}-\omega E,$ (12) $\displaystyle\frac{dI}{dt}$
$\displaystyle=$ $\displaystyle\omega E-\gamma I,$ (13)
$\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle\gamma I,$ (14)

where $\beta$ is the coefficient of infection rate, $\omega$ is the
coefficient of migration rate of latency, $\gamma$ is the coefficient of
migration rate, and $N=S+E+I+R$ is the total population.

Figure 11 displays the temporal evolution (in days) of $S$ (blue line), $E$
(black line), $I$ (red line), and $R$ (green line) for $\beta=0.5$,
$\gamma=0.01$, $\omega=0.1$. Initially, $S$ decreases and $E$, $I$, and $R$
increase. $S$ is depleted by the epidemic and goes to zero. Both $E$ and $I$
exhibit peaks and $R$ saturates when they go to zero.

Figure 11: Time evolution of $S$ (blue line), $E$ (black line), $I$ (red
line), and $R$ (green line) for $N=1000$, $S(0)=995$, $I(0)=5$, $\beta=0.5$,
$\omega=0.1$, and $\gamma=0.1$.

Recently, Souza et al. [33] considered the SEIR model to analyse the impact of
easing restriction on the infection rate during COVID-19 pandemic. They
included a parameter related to the restriction $r$ in the SEIR model,
$\beta\rightarrow\beta(1-r)$. By increasing $r$, the peak of infectious is
delayed and the curve peak is flattened, as shown in Fig. 12(a). Figure 12(b)
shows that changes in the value of $r$ can generate a second wave, namely the
increase of the number of infected individuals after few cases.

Figure 12: (a) Time evolution of $I$ for $\omega=0.2$, $\gamma=0.25$, $r=0$
(red line), $r=0.2$ (red dashed line), and $r=0.4$ (red dotted line). (b) Time
evolution of $I$ for $r=0$ ($t<14$), $r=0.6$ ($14\leq t<365$), and $r=0.2$
($t\geq 365$).

##  7 The SEIRS model

In the SEIRS model, the susceptible individuals first go through the exposed
class before infected one. After the infectious, they are transferred to the
recovered compartment and become susceptible again, as shown in Fig. 13.
Denphedtnong et al. [41] used the SEIRS epidemic model to describe the spread
of diseases by transports. They investigated the data of SARS (severe acute
respiratory syndrome) outbreak in 2003.

Figure 13: Compartment diagram of the SEIRS model, where S, E, I, and R are
the numbers of susceptible, exposed, infected, and removed individuals,
respectively, $\beta$ (in units of 1/day) is the infectious rate, $\omega$ (in
units of 1/day) is the coefficient of migration rate, $\gamma$ (in units of
1/day) is the rate in which the infected individuals recover, and $\delta$ (in
units of 1/day) is the rate in which the recovered individuals return to the
susceptible class.

The SEIRS model is given by

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\beta
SI}{N}+\delta R,$ (15) $\displaystyle\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta SI}{N}-\omega E,$ (16) $\displaystyle\frac{dI}{dt}$
$\displaystyle=$ $\displaystyle\omega E-\gamma I,$ (17)
$\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle\gamma I-\delta
R,$ (18)

where $\delta$ is the rate in which the recovered individuals return to the
susceptible class after losing the immunity.

Figure 14: Time evolution of $S$ (blue line), $E$ (black line), $I$ (red
line), and $R$ (green line) for $N=1000$, $S(0)=995$, $I(0)=5$, $\beta=0.5$,
$\omega=0.1$, $\gamma=0.1$, and $\delta=0.03$.

In Fig. 14, we calculate $S$ (blue line), $E$ (black line), $I$ (red line),
and $R$ (green line) for $N=1000$, $S(0)=995$, $I(0)=5$, $\beta=0.5$,
$\omega=0.1$, $\gamma=0.1$, and $\delta=0.03$. We see that the susceptible,
exposed, and infected individuals do not go to the value equal to zero over
time. This occurs due to the fact that recovered individuals become
susceptible again.

##  8 Conclusions

Investigations about infectious diseases play a crucial role in reducing
negative consequences and improving the recovery of individuals. Scientists
have been carried out research in epidemiology to understand how the people
are affected by transmissible diseases over time. Many different mathematical
models were proposed to find epidemiological parameters and manners to prevent
illness.

In this work, we present some mathematical models that have been used to
describe the dynamical behaviour of infectious diseases. The models are based
on individuals that are separated into compartments, such as susceptible (S),
exposed (E), infected (I), and recovered (R). They focus on the prediction of
the population growth or reduction in each compartment. The models depend on
the parameters related to the rate in which individuals are transferred
between the classes.

The SI model considers that the individuals go from the susceptible to
infected states, while the SIS model assumes that there is no immunity and the
infected individuals return to the susceptible compartment. In the SIR and
SIRS model, it is included the class of the recovered individuals. The SEIR
and SEIRS models have not only the susceptible, infected, and recovered
individuals, but also the exposed ones. Recently, the SIR and SEIR models have
been used in studies about the COVID-19 epidemic. The SIR, SIRS, SEIR, and
SEIRS models exhibit a peak of infected people that was observed in different
infection spread.

It has been demonstrated that through the intensity of restrictions associated
with the control policies to reduce the infection spread, it is possible to
flat the curve associated with the temporal evolution of the infected people
[33]. However, the results depend on the duration and specific time in which
the restriction is applied. The peak of infections can be reduced by means of
restrictions or another peak can appear after suspending the restrictions.

We provide routines in a repository for the SI, SIS, SIR, SIRS, SEIR, and
SEIRS models [31].

## Acknowledgements

This work was possible by partial financial support from the following
Brazilian government agencies: CAPES, CNPq ($407543/2018-0$, $302903/2018-6$,
$420699/2018-0$, $407299/2018-1$, $428388/2018-3$, $311168/2020-5$), Fundação
Araucária, and São Paulo Research Foundation (FAPESP $2018/03211-6$). We would
like to thank www. 105groupscience.com.

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