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Bulletin of Biomathematics, 2024, 2(2), 182–197
https://www.bulletinbiomath.org
ISSN Online: 2980-1869 / Open Access
https://doi.org/10.59292/bulletinbiomath.2024008
RESEARCH PAPER
Bifurcations on a discrete–time SIS–epidemic model with
saturated infection rate
Hasan S. Panigoro ID 1,*,‡,Emli Rahmi ID 1,‡,Salmun K. Nasib ID 1,‡,Nur’ain
Putri H. Gawa ID 1,‡ and Olumuyiwa James Peter ID 2,‡
1Biomathematics Research Group, Department of Mathematics, Universitas Negeri Gorontalo, Bone
Bolango 96554, Indonesia, 2Department of Mathematical and Computer Sciences, University of
Medical Sciences, Ondo City, Ondo State, Nigeria
*Corresponding Author
‡hspanigoro@ung.ac.id (Hasan S. Panigoro); emlirahmi@ung.ac.id (Emli Rahmi); salmun@ung.ac.id (Salmun K.
Nasib); aingawa0802@gmail.com (Nur’ain Putri H. Gawa); peterjames4real@gmail.com (Olumuyiwa James Peter)
Abstract
In this paper, we explore the complex dynamics of a discrete-time SIS (Susceptible-Infected-Susceptible)-
epidemic model. The population is assumed to be divided into two compartments: susceptible and
infected populations where the birth rate is constant, the infection rate is saturated, and each recovered
population has a chance to become infected again. Two types of mathematical results are provided
namely the analytical results which consist of the existence of fixed points and their dynamical behav-
iors, and the numerical results, which consist of the global sensitivity analysis, bifurcation diagrams,
and the phase portraits. Two fixed points are obtained namely the disease-free and the endemic
fixed points and their stability properties. Some numerical simulations are provided to present the
global sensitivity analysis and the existence of some bifurcations. The occurrence of forward and
period-doubling bifurcations has confirmed the complexity of the solutions.
Keywords: SIS-epidemic model; saturated infection rate; bifurcation
AMS 2020 Classification: 37N25; 92B05; 39A60; 92D25
1 Introduction
The mathematical modeling using a deterministic approach is a powerful tool to reduce the
impact of the infectious disease [
1
–
3
]. In recent decades, two popular ways are used for the
deterministic approaches namely the continuous-time and the discrete-time models. There are
many epidemiological studies employing differential equations for the operator of the continuous
time model. See [
4
–
9
] and references therein. For the discrete-time model, the difference equation
is used and becomes popular due to the complexity of the dynamical behaviors in epidemiological
➤Received: 13.05.2024 ➤Revised: 15.10.2024 ➤Accepted: 23.10.2024 ➤Published: 31.10.2024
182
Panigoro et al. |183
cases. Most of them propose the complexity of dynamical behaviors such as period-doubling and
Neimark-Sacker bifurcations as well as the existence of chaotic solutions. See [
10
–
17
] and cited
articles therein.
The classical epidemic model is given by Kermack and MacKendrick [18,19] defined by
dS
dt =−βSI,dI
dt =βSI −ρI,dR
dt =ρI, (1)
where
S
is the susceptible compartment,
I
is the infected compartment,
R
is the recovered compart-
ment,
β
is the infection rate, and
ρ
is the recovery rate. Some modifications are applied to include
the real phenomena in nature. For example, Federico et al. [
20
] include the optimal vaccination
and the recovery rate to the susceptible compartment to model
(1)
. In another way, Zhang and
Qiao [
21
] focus on studying the bifurcation analysis of model
(5)
by assuming the infection rate is
saturated and the population has a strong Allee effect. Interesting works were also given by Li
and Eskandari [
1
] which focus on the analytical and numerical results of a discrete-time seasonally
forced SIR epidemic model. On the other hand, Omame et al. [
22
] and Atede et al. [
23
] have
focused on investigating the application of model
(1)
on COVID-19 transmission by involving the
vaccination and the memory effect. Some of them have integrated deterministic and stochastic
approaches to describe the dynamical behaviors in modelling [
24
–
30
]. Following those articles,
in this work, we also focus on the mathematical results of a modified SIR model and do not
specifically discuss an epidemiological case. The model is modified based on some assumptions
as follows:
(i) The constant birth rate is denoted by Λ.
(ii)
The population has a natural death rate denoted by
δ1
,
δ2
, and
δ3
which respectively define
the natural death rate of susceptible, infected, and recovered compartments.
(iii)
The recovered individuals can be infected by disease again with the transfer rate to suscepti-
ble prey denoted by ω.
(iv)
The infection rate term
βSI
is replaced by the saturated infection rate term denoted by
βSI
η+I
.
This infection rate term naturally occurs in the epidemic model since each population can
protect itself from infection so that although the infection population increases, the infection
rate will have a threshold [31–33].
Thus, model (1) becomes
dS
dt =Λ−
βSI
η+I
−δ1S+ωR,dI
dt =βSI
η+I
−ρI−δ2I,dR
dt =ρI−δ3R−ωR. (2)
By assuming all recovered compartments can be infected again, we drop the recovered compart-
ment Rand hence the model (2) is simplified into
dS
dt =Λ−
βSI
η+I+ωI−δ1S,
dI
dt =βSI
η+I
−(ω+δ2)I.
(3)
Now, we adopt similar ways as in [
34
–
36
] to construct the discrete-time model using the forward
184 |Bulletin of Biomathematics, 2024, Vol. 2, No. 2, 182–197
Euler scheme. We get
Sn+1−Sn
h=Λ−
βSnIn
η+In
+ωIn−δ1Sn,
In+1−In
h=βSnIn
η+In
−(ω+δ2)In.
(4)
From model (4), the simplification yields
Sn+1=Sn+hΛ−
βSnIn
η+In
+ωIn−δ1Sn,
In+1=In+hβSnIn
η+In
−(ω+δ2)In.
(5)
Based on the above description, we get the key contributions and the novelty of this research are
given as follows:
(i)
The model is constructed using a saturated infection rate and all recovered individuals
can be infected again. We also use the difference equation for the operator rather than the
differential equation. According to our literature review, although the model is simple, we
cannot find similar works as given by model (5).
(ii)
All possible dynamical behaviors of fixed point are analyzed namely sink, source, saddle,
and non-hyperbolic.
(iii)
The most influential parameter concerning the basic reproduction number and the popula-
tion density for each compartment is identified using the Partial Rank Correlation Coefficient
(PRCC) along with Saltelli sampling to generate the data.
(iv)
More complex dynamics are provided numerically namely the forward and period-doubling
bifurcations.
We organize this article as follows: In Section 1, we give the introduction and model formulation.
In Section 2, we explore the dynamical behaviors of the model by identifying the feasible fixed
points, the basic reproduction number, and their stability properties. In Section 3, some numerical
simulations are provided such as the global sensitivity analysis, forward, and period-doubling
bifurcations by giving the PRCC bar chart, contour plots, PRCC time-series, bifurcation diagrams,
and phase portraits around fixed points. We end this article by presenting a conclusion in Section 4.
2 Analytical results and findings
We start investigating the feasible fixed point of model (5) by solving the following equation
S=S+hΛ−
βSI
η+I+ωI−δ1S,
I=I+hβSI
η+I
−(ω+δ2)I.
(6)
We find two fixed points on the axial and the interior of the model (5) which are discussed in the
next subsections.
Panigoro et al. |185
The disease-free fixed point
The first fixed point is given by the disease-free fixed point (DFF) denoted by
E0=Λ
δ1, 0,
which describes the condition when the disease disappears from the population. By following
[
37
–
41
], we apply the next generation matrix to obtain the basic reproduction number
(R0)
which states the number of secondary infections caused by one primary infection in an entirely
susceptible population. We get
R0=βΛ
(ω+δ2)δ1η. (7)
Now, we give the following theorem to present the dynamical behaviors of DFF.
Theorem 1 Let ha=2
δ1and hb=2ω
(1−R0)(ω+δ2). The DFF E0=Λ
δ1, 0is
(i) a sink (locally asymptotically stable) if R0<1and h <min {ha,hb}; or
(ii) a source if R0>1and h >ha; or if R0<1and h >min {ha,hb}; or
(iii)
a saddle if
h<hb
and
R0>1
; or if
h<ha
and
R0<1
and
h>hb
; or if
h>ha
and
R0<1
and
h<hb; or
(iv) a non-hyperbolic if h =ha; or R0=1; or R0<1and h =hb.
Proof For DFF, we have the following Jacobian matrix:
J(S,I)|E0="1−2h
ha
(δ1ωη−βΛ)h
δ1ωη
0 1 −2h
hb#.
Therefore, we obtain a pair of eigenvalues
λ1=1−2h
ha
and
λ2=1−2h
hb
. By observing
λ1
, we have
the following condition
•|λ1|<1 when h<ha; and
•|λ1|=1 when h=ha; and
•|λ1|>1 when h>ha.
We also have the sign of λ2as follows.
•|λ2|<1 when R0<1 and h<hb; and
•|λ2|=1 when R0=1 or;
when R0<1 and h=hb; and
•|λ2|>1 when R0>1; or
when R0<1 and h>hb.
Following Lemma 1 in [42], all statements given by Theorem 1 are proven.
The endemic fixed point
The next fixed point is given by the endemic fixed point (EFP) defined by
ˆ
E= ( ˆ
S
,
ˆ
I)
where
ˆ
S=(ω+δ2)(η+ˆ
I)
β
and
ˆ
I=(R0−1)(ω+δ2)δ1η
βδ2+(ω+δ2)δ1
. The EFP describes the condition when the disease exists
in the population where the existence condition is given by R0>1. To investigate the dynamics
186 |Bulletin of Biomathematics, 2024, Vol. 2, No. 2, 182–197
−0.8−0.6−0.4−0.2 0.0 0.2 0.4 0.6 0.8
PRCC
β
ω
η
Parameters
positive relationship with R0
negative relationship with R0
0.583
-0.426
-0.583
Figure 1. PRCC results with respect to the value of the basic reproduction number
(R0)
. The infection rate
(β)
becomes the most influential parameter to the value of R0
0.0 0.2 0.4 0.6 0.8 1.0
β×10−1
4
5
6
7
8
9
η
×10−1
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
R0
0.0 0.2 0.4 0.6 0.8 1.0
β×10−1
4.0
4.5
5.0
5.5
6.0
6.5
7.0
ω
×10−1
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
R0
(a) Contour plot on (β,η)−plane (b) Contour plot on (β,ω)−plane
Figure 2. The contour plots of (a)
(β
,
η)
, and (b)
(β
,
ω)
respect to the values of
R0
. The parameter
β
is directly
proportional while ηand ωis inversely proportional to R0
around EFP, we do linearization around EFP. The Jacobian matrix at EFP is given by
J(S,I)|ˆ
E=
1−h(β+δ1)ˆ
I+δ1η
η+ˆ
I−hδ2η−ωˆ
I
η+ˆ
I
βˆ
Ih
η+ˆ
I1−
(ω+δ2)ˆ
Ih
η+ˆ
I
,
and hence, we have eigenvalues
λ1,2 =1
2ξ±qξ2−4ζ, (8)
Panigoro et al. |187
0 10 20 30 40 50
n
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
PRCC respect to S
β=−0.754 ω= 0.581 η= 0.083
(a) PRCC respect to S
0 10 20 30 40 50
n
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
PRCC respect to I
β= 0.755 ω=−0.581 η=−0.083
(b) PRCC respect to I
Figure 3. PRCC results respect to the density of susceptible individuals
(S)
and infected individuals
(I)
. The
parameter βis directly proportional to Iand inversely proportional to Swhile eta and ωare opposite to it
where
ξ=2−
h
η+ˆ
I((β+δ1)−(ω+δ2))ˆ
I+δ1η,
ζ=1−
h
η+ˆ
I((ω+δ2) + (β+δ1))ˆ
I+δ1η
+h2ˆ
I
η+ˆ
I[(ω+δ2)δ1+βδ2].
Let Φ(θ) = θ2−θξ +ζ. We have
Φ(1) = 1−ξ+ζ
=[h((ω+δ2)δ1+βδ2)−2(ω+δ2)]hˆ
I
η+ˆ
I,
188 |Bulletin of Biomathematics, 2024, Vol. 2, No. 2, 182–197
0.00 0.02 0.04 0.06 0.08 0.10 0.12
β
7.4
7.6
7.8
8.0
S
E0−stable
E0−unstable
ˆ
E−stable
β≈0.0525 (R0= 1)
0.00 0.02 0.04 0.06 0.08 0.10 0.12
β
0.0
0.2
0.4
0.6
I
E0−stable
E0−unstable
ˆ
E−stable
β≈0.0525 (R0= 1)
(a) Bifurcation diagram driven by β
(b) Phase portrait for different values of β
Figure 4. Bifurcation diagrams and phase portraits of model
(5)
driven by
β
using parameter values:
Λ=0
.
8
,
η=0.6, ω=0.6, δ1=0.1, δ2=0.1, and h=0.1
which satisfies Φ(1)>0 when h>2(ω+δ2)
(ω+δ2)δ1+βδ2. We also achieve
Φ(−1) = 1+ξ+ζ
=4−
2h
η+ˆ
I(β+δ1)ˆ
I+δ1η+h2ˆ
I
η+ˆ
I[(ω+δ2)δ1+βδ2].
Following Lemmas 1 and 2 in [42], we have the following theorem as the results.
Theorem 2 Let R0>1and h >2(ω+δ2)
(ω+δ2)δ1+βδ2. The EFP is
(i) a sink if Φ(−1)>0and ζ<1; or
(ii) a source if Φ(−1)>0and ζ>1; or
(iii) a saddle if Φ(−1)<0; or
(iv) a non-hyperbolic if Φ(−1) = 0and ξ=0, 2; or if ξ2<4ζand ζ=1.
Panigoro et al. |189
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
η
7.7
7.8
7.9
8.0
S
E0−unstable
E0−stable
ˆ
E−stable
η≈0.5714 (R0= 1)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
η
0.0
0.1
0.2
0.3
I
E0−unstable
E0−stable
ˆ
E−stable
η≈0.5714 (R0= 1)
(a) Bifurcation diagram driven by η
(b) Phase portrait for different values of η
Figure 5. Bifurcation diagrams and phase portraits of model
(5)
driven by
η
using parameter values:
Λ=0
.
8
,
β=0.05, ω=0.6, δ1=0.1, δ2=0.1, and h=0.1
3 Numerical results
To explore the complexity of the dynamical behaviors, some numerical simulations are demon-
strated. Since no one specific epidemiological case is related to the model, we use hypothetical
parameter values for the simulations. We first set the parameter values as follows.
Λ=0.8, β=0.01, η=0.6, δ1=0.1, δ2=0.1, ω=0.6, h=0.5. (9)
By using the parameter values
(9)
, we give the following subsections to show the global sensitivity
analysis, forward, and period-doubling bifurcations.
Global sensitivity analysis
To investigate the most influential parameter of model
(5)
, we perform the global sensitivity
analysis [
43
,
44
]. The Partial Rank Correlation Coefficient (PRCC) [
45
] is employed for parameter
ranking along with Saltelli sampling [
46
] to generate the sample data around the parameter values
given by
(9)
. We consider the basic reproduction number and the population densities for the
190 |Bulletin of Biomathematics, 2024, Vol. 2, No. 2, 182–197
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ω
7.0
7.2
7.4
7.6
7.8
8.0
S
E0−unstable
E0−stable
ˆ
E−stable
ω≈0.5667 (R0= 1)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ω
0.0
0.2
0.4
0.6
0.8
1.0
I
E0−unstable
E0−stable
ˆ
E−stable
ω≈0.5667 (R0= 1)
(a) Bifurcation diagram driven by η
(b) Phase portrait for different values of ω
Figure 6. Bifurcation diagrams and phase portraits of model
(5)
driven by
ω
using parameter values:
Λ=0
.
8
,
β=0.05, η=0.6, δ1=0.1, δ2=0.1, and h=0.1
constraint function and the rank of the parameter as the objective function. Since the birth rate
(Λ)
and the natural death rate
δi
,
i=1
,
2
,
3
can be obtained directly if the real data exists, we
only focus on the impact of the infection rate
(β)
, the recovery rate
(ω)
, and the half-saturation
constant (η).
We first investigate the most influential parameter of model
(5)
concerning the value of the basic
reproduction number
(R0)
. As a result, we have the infection rate
(β)
become the most influential
parameter with PRCC
0
.
583
while
η
and
ω
are respectively at the second and the third rank with
PRCC
−0
.
583
and
−0
.
426
. See the bar chart of PRCC results in Figure 1. We also confirm that
β
has a positive relationship with
R0
while
η
and
ω
have a negative relationship with
R0
by
observing the sign of the PRCC results. This means that if the value of
β
increases, then the value
of
R0
increases. If the value of
η
or
ω
increases, the value of
R0
will decrease. We give the contour
plot of these conditions in Figure 2.
Now, we investigate the most influential parameter concerning the density of the susceptible
compartment
(S)
and the infected compartment
(I)
. Again, by applying PRCC and Saltelli
sampling as well as computing the PRCC value for
n
in range
[0
,
50]
, we find
β
still becomes
the most influential parameter to the density of
S
and
I
. See the numerical results in Figure 3.
Panigoro et al. |191
(a) Bifurcation diagram
(b) Some periodic solutions
Figure 7. Bifurcation diagram and periodic solutions of model
(5)
driven by
h
using parameter values:
Λ=0
.
8
,
β=0.05, η=0.6, ω=0.6, δ1=0.1, δ2=0.1, and h=0.1
From the sign of the PRCC values, we also verify that
β
has a negative relationship with
S
and a positive relationship with
I
, while
η
and
ω
have a positive relationship with
S
and a
negative relationship with
I
. This means that when the infection rate increases, the density of
the susceptible compartment decreases while the density of the infected compartment increases.
When the recovery rate and the half-saturation constant increase, the density of the susceptible
compartment increases, and the density of the infected compartment decreases.
Forward bifurcations
We then investigate the impact of the infection rate
(β)
, the half-saturation constant
(η)
, and the
recovery rate
(ω)
on the dynamics of model
(5)
. Let the parameter values as in
(9)
. By varying
192 |Bulletin of Biomathematics, 2024, Vol. 2, No. 2, 182–197
the value of
β
in the interval
[0
,
0
.
12]
, we investigate the existence and stability condition for each
fixed point. As a result, we have the bifurcation diagram as given in Figure 4(a) and phase portrait
in Figure 4(b). When
β<β∗
where
β∗≈0
.
0525
(or
R0≈1
), the nearby solution converges to
DFF
E0= (8
,
0)
which indicate the DFF is a sink. When
β
crosses
β∗
, the DFF loses its stability
followed by the occurrence of EFP
ˆ
E
where the DFF becomes a saddle and EFP is a sink. This
phenomenon is called forward bifurcation where
β
is the bifurcation parameter and
β∗
is the
bifurcation point. Similar dynamical behaviors are presented when the half saturation parameter
(η)
and the recovery rate
(ω)
is varied. Using
(9)
and varying
η
in interval
[0
.
2
,
1]
, we have
a forward bifurcation where
η
and
η∗≈0
.
5714
(or
R0≈1
) are respectively the bifurcation
parameter and bifurcation point. The forward bifurcation also occurs when
ω
crosses
ω∗≈0
.
5667
which confirms that
ω
and
ω∗
respectively become the bifurcation parameter and bifurcation
point. See Figure 5 and Figure 6 for the numerical simulations of the bifurcation diagrams and
their corresponding phase portraits. At these phenomena, we conclude that
β
,
η
, and
ω
have
impacts on the existence and stability of DFE and FEP. The disease in the population will become
extinct or endemic when the infection rate, half saturation constant, and the recovery rate are
varied.
Period-doubling bifurcation
In this subsection, we present the occurrence of the sequence of period-doubling bifurcation as
well as the example of the period of the solutions when the step-size
(h)
. The parameter values
given by
(9)
are set and
h
is varied in
[2
.
5
,
3
.
3]
. As a result, we have Figure 7(a) as the bifurcation
diagram. We confirm that the sink EFP becomes unstable when crosses
h≈2
.
61
and a period-2
solution occurs. Each branch of the periodic solution also split into the other period-2 solution
and so forth. This indicates the existence of the sequence of period-doubling bifurcation. We give
Figure 7(b) to show some of the periodic solutions such as a sink for
h=2
.
5
, period-2 for
h=2
.
9
,
period-4 for
h=3
.
1
, period-8 for
h=3
.
12
, period-6 for
h=3
.
175
, and period-5 for
h=3
.
276
.
This phenomenon shows that the endemic point may lose its stability when the step-size becomes
larger. Therefore, if we have less data for some interval of time, the dynamical behaviors may
change via period-doubling bifurcation and the forecasting will be wrong.
4 Conclusion
The discrete-time SIS-epidemic model with a saturated infection rate has been studied. Some
analytical and numerical results have been investigated. Two fixed points have been identified
namely disease-free and endemic fixed points as well as the basic reproduction number. We
have shown that the existence and stability of each fixed point depend on the basic reproduction
number. More information about the dynamics of the model has been explored numerically. The
PRCC along with Saltelli sampling has been used to investigate the most influential parameter
concerning the value of the basic reproduction number and the density of each compartment which
shows that the infection rate becomes the most influenced one. The existence of some bifurcations
is also demonstrated namely forward bifurcations and period-doubling bifurcation. We conclude
that the infection rate, the half-saturation constant, and the recovery rate have an impact not only
on the stability of the fixed points but also on the occurrence of forward bifurcation. Although the
model is mathematically explored such as the stability condition, sensitivity analysis, and some
bifurcation phenomena, this work has less epidemiological interpretation since we do not apply
this model to any epidemiological cases. Moreover, the model also studies two compartments only,
which means we can explore more by adding some compartments based on the real phenomena
in nature. This limitation will become interesting to study further.
Panigoro et al. |193
Declarations
Use of AI tools
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this
article.
Data availability statement
All data generated or analyzed during this study are included in this article.
Ethical approval
The authors state that this research complies with ethical standards. This research does not involve
either human participants or animals.
Consent for publication
Not applicable
Conflicts of interest
The authors declare that they have no conflict of interest.
Funding
This research was funded by Lembaga Penelitian dan Pengabdian Kepada Masyarakat-Universitas
Negeri Gorontalo via PNBP-UNG with DIPA-UNG No. SP DIPA-023.17.2.677521/2023/2023,
under contract No. B/704/UN47.D1/PT.01.03/2023.
Author’s contributions
H.S.P.: Conceptualization, Methodology, Validation, Writing - Review & Editing, Supervision,
Project Administration, Funding Acquisition. E.R.: Conceptualization, Methodology, Validation,
Resources, Writing - Original Draft, Visualization, Supervision, Project Administration, Funding
Acquisition. S.K.N.: Conceptualization, Formal Analysis, Resources, Writing - Original Draft
N.P.H.G.: Software, Formal Analysis, Investigation, Data Curation, Writing - Original Draft,
Visualization O.J.P.: Software, Validation, Investigation, Writing - Review & Editing. All authors
discussed the results and contributed to the final manuscript.
Acknowledgements
The authors express their profound gratitude to the specialists who willingly imparted their
knowledge and insights, which greatly aided in the creation of this manuscript.
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How to cite this article: Panigoro, H.S., Rahmi, E., Nasib, S.K., Gawa, N.P.H. & Peter, O.J. (2024).
Bifurcations on a discrete-time SIS-epidemic model with saturated infection rate. Bulletin of
Biomathematics, 2(2), 182-197. https://doi.org/10.59292/bulletinbiomath.2024008
