Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness

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arXiv:2210.01649v1 [q-bio.PE] 4 Oct 2022

Saddle-node bifurcation of limit cycles in an epidemic model with two

levels of awareness

David Juhera, David Rojasa, Joan Salda˜naa,∗

aDepartament d’Inform`atica, Matem`atica Aplicada i Estad´ıstica

Universitat de Girona, Girona 17003, Catalonia, Spain

Abstract

In this paper we study the appearance of bifurcations of limit cycles in an epidemic model with

two types of aware individuals. All the transition rates are constant except for the alerting

decay rate of the most aware individuals and the rate of creation of the less aware individuals,

which depend on the disease prevalence in a non-linear way. For the ODE model, the numerical

computation of the limit cycles and the study of their stability are made by means of the

Poincar´e map. Moreover, suﬃcient conditions for the existence of an endemic equilibrium are

also obtained. These conditions involve a rather natural relationship between the transmissibility

of the disease and that of awareness. Finally, stochastic simulations of the model under a very

low rate of imported cases are used to conﬁrm the scenarios of bistability (endemic equilibrium

and limit cycle) observed in the solutions of the ODE model.

Keywords: epidemic models, awareness, bifurcations, limit cycles, stochastic simulations.

1. Introduction

The role of human behaviour has been increasingly considered in epidemiological modelling

since the early 2000s [7]. The spread of COVID-19 has highlighted even more its important role

in the progress of infectious diseases. Besides institutional measures as mobility restrictions,

mandatory use of facemasks, or school closings, self-initiated individual behaviours related to

risk aversion are recognized as a driving force in epidemic dynamics [15, 21].

One way to model such behavioural changes in deterministic models is to modify the incidence

term βSI where βdenotes the rate of disease transmission, and Sand Iare the number of

susceptible and infected individuals, respectively. The simplest way to modify it is by assuming

that βis no longer constant but a decreasing function of the prevalence of the disease ([4, 18, 2,

21]). In this mean-ﬁeld formulation of the incidence term, βdepends on the contact rate as well

∗Corresponding author

Email addresses: david.juher@udg.edu (David Juher), david.rojas@udg.edu (David Rojas),

joan.saldana@udg.edu (Joan Salda˜na)

as on the probability of transmission during an infectious contact. So, its reduction can reﬂect

a diminution in the number of social contacts (social distancing), the adoption of measures to

prevent infection while keeping the same contact rate (decrease of the infection probability), or

both.

On the other hand, it is well known that the perception of infection risk is uneven among sus-

ceptible individuals [9]. One way to introduce some heterogeneity in risk-taking propensity has

been to include more types of uninfected individuals characterised by their level of responsive-

ness to risk. For instance, the Susceptible-Aware-Infectious-Susceptible (SAIS) model considers

a new class of non-infected individuals with a higher risk aversion than the susceptible ones, the

so-called aware or alerted individuals, who are characterized by a lower transmission rate [19].

A basic ingredient in such a modelling approach is the transmission of awareness among

individuals [5]. In [11] the authors considered an SAIS model where alerted individuals were

able to transmit awareness by convincing non-aware individuals to take preventive measures

against the infection, which is an example of self-initiated individual behaviour. Moreover, a

new class of aware individuals, the so-called unwilling (U) individuals, is also introduced. They

are characterized by a lower level of alertness which is translated into a lack of willingness

to transmit awareness to susceptible individuals. The existence of this second class of aware

individuals turns out to be necessary to have oscillatory solutions of the SAUIS model with no

births and deaths in the population.

The inﬂow of new susceptible individuals in the population is a key factor in mean-ﬁeld

epidemic models to observe periodic solutions [18]. In dynamic networks models, link dynamics

can also play this role [20]. However, even without demographic processes, behaviourally-induced

epidemic oscillations can also be expected to occur when individuals experience a decline in

awareness as a result of preventive measures taken over long periods of time combined with

low disease prevalence. This fact was, indeed, proved in [11] by analysing the occurrence of a

Hopf bifurcation from an endemic equilibrium of the SAUIS model. Later, the robustness of

such oscillations was conﬁrmed in [10] under the assumption of a low rate εof imported cases

(infections contracted from abroad) by means of stochastic simulations on random networks.

In this paper, we explore an extended version of the SAUIS-εmodel in [10] which considers

that awareness dynamics changes abruptly when disease prevalence crosses a threshold value η.

Precisely, the rate νaof creation of unwilling individuals and the rate of awareness decay δaare

modulated by the following function σm(i) of the fraction iof infected individuals:

σm(i) = 1

1 + (i/η)m, m ≥1.

The sharpness of the reduction of these two rates is controled by the parameter m, while ηis the

half-saturation constant (see Figure 1). In particular, for m≫1, 1 −σm(i) becomes closer to

the unit step function θ(i−η). An extreme example of such abrupt behavioural responses could

be the occurrence of panic waves when new cases of an emerging disease appear in a population.

In contrast to other papers where awareness is considered in terms of non-constant infection

transmission rates (see, for instance, [2, 4, 13, 18, 21]), here we will focus on the awareness dy-

namics themselves and their role in the appearance of periodic solutions (oscillatory epidemics).

In particular, we are interested in how the behaviour of solutions is aﬀected by the reduction of

both the decay of awareness and the creation of unwilling individuals.

2. SAUIS-εmodel with varying coeﬃcients

Each individual in a population can be in one of the following four states: S (susceptible),

A (aware), U (unwilling), and I (infected). The model assumes that aware individuals are

created at alerting rates αiand αafrom susceptible ones after being in contact with infected

and aware individuals, respectively. Aware individuals experience an alerting decay and become

unwilling at a rate δa, while unwilling individuals also appear at rate νafrom contacts between

susceptibles and aware individuals (nodes) and they become susceptible at a rate δu. The

infection transmission rates for susceptible, aware, and unwilling individuals are β,βa, and βu,

respectively, while the recovery rate from infection is δ.

Moreover, following the SAUIS-εmodel introduced in [10], we consider the arrival of imported

cases (infections contracted abroad) at a very low rate ε > 0. This fact prevents stochastic

epidemic oscillations from extinction and, at the same time, the dynamical properties of the

solutions remain close to those of the deterministic ODE model with ε= 0.

Finally, as explained in the introduction, we assume that the rate of awereness decay δaand

the rate νaat which susceptible hosts become unwilling due to a contact with an aware host both

depend on the fraction of infected individuals through a reduction factor given by the function

σm(i). The resulting ODE system governing the epidemic dynamics is then given by:

dt =αis i +αas a −βaa i −δaσm(i)a−εa, βa< β,

dt =δaσm(i)a+νaσm(i)s a −βuu i −δuu−εu, βu< β,

dt = (β s +βaa+βuu−δ)i+ (1 −i)ε, s +a+u+i= 1,

(1)

where s,a,u, and idenote the fractions of hosts in the S,A,U, and Icompartments, respectively.

The diﬀerential equation for shas been omitted because it is redundant.

3. Equilibria

The natural state space of system (1) is Ω := {(a, u, i)∈R3: 0 ≤a+u+i≤1}. The

existence of imported cases from abroad guarantees that the vector ﬁeld deﬁned by this system

on the boundary of Ω points strictly towards its interior. In particular, this implies that Ω is

positively invariant under the ﬂow deﬁned by the solutions of system (1) and, moreover, the

non-existence of disease-free equilibria for this model.

On the other hand, since σm(0) = 1, the same analysis of the bifurcations from the two

disease-free equilibria (DFE) of the model with ε= 0 done in [10] works for our system. For

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Figure 1: Shape of function σm(i) with η= 0.4 for diﬀerent values of m.

instance, taking εas a bifurcation parameter, it follows that one interior equilibrium of (1)

comes from the bifurcation of a DFE of the system with ε= 0 when β < δ. Precisely, either

the DFE e∗

1= (0,0,0) enters Ω for ε > 0 if αa< δa, or the DFE e∗

2= (a∗

0, u∗

0,0) with a∗

δu(1 −δa/αa)/(δa(1 + νa/αa) + δu) and u∗

0=δa/δu(1 + νa/αa)a∗

0enters Ω for ε > 0 if αa> δa.

In both cases, the interior equilibrium of system (1) bifurcates from an asymptotically stable

DFE and is only maintained by the presence of imported cases. So, such an equilibrium is not a

proper endemic equilibrium because it does not result from the disease transmission within the

population.

For β > δ and taking βaas a bifurcation parameter, it follows that e∗

2is still asymptotically

stable if βa< βc

a:= β−(β−δ−(β−βu)u∗

0)/a∗

0. In this case, an interior equilibrium fed by

the imported cases bifurcates from it. So, from now on we will assume that β > δ and βa> βc

to guarantee that no interior equilibrium for ε > 0 arises from a DFE with ε= 0 and, hence,

that any interior equilibrium corresponds to the perturbation of an endemic equilibrium of the

system with ε= 0.

Endemic equilibria are, in general, very diﬃcult to determine analytically. When ε= 0 we

can easily see that any endemic equilibrium lies inside the plane

1−δ

β−1−βa

βa−1−βu

βu−i= 0.(2)

The following result gives suﬃcient conditions for the existence of at least one endemic equilib-

rium point of the model (1) with ε= 0. The proof relies on a version of the Poincar´e-Miranda

theorem in a triangular domain, which we include in the Appendix for completeness.

Lemma 3.1. System (1) with ε= 0 has an endemic equilibrium point in Ωif 0≤βa< βu<

δ < β and αa

δa<β−βa

δ−βa.

Proof. Substituting (2) into the ﬁrst and second equations of (1) we obtain two continuous

functions in the variables (a, u), f1and f2, respectively. We ﬁnd endemic equilibria in the

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arXiv:2210.01649v1[q-bio.PE]4Oct2022Saddle-nodebifurcationoflimitcyclesinanepidemicmodelwithtwolevelsofawarenessDavidJuhera,DavidRojasa,JoanSalda˜naa,∗aDepartamentd’Inform`atica,Matem`aticaAplicadaiEstad´ısticaUniversitatdeGirona,Girona17003,Catalonia,SpainAbstractInthispaperwestudytheappearanceofbi...

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Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness

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