A mathematical model with nonlinear relapse conditions for a forward-backward bifurcation Fabio Sanchez1 Jorge Arroyo-Esquivel2 and Juan G. Calvo1

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A mathematical model with nonlinear relapse:

conditions for a forward-backward bifurcation

Fabio Sanchez1,*,+, Jorge Arroyo-Esquivel2,+, and Juan G. Calvo1,+

1Universidad de Costa Rica, Centro de Investigaci´on en Matem´atica Pura y Aplicada - Escuela de Matem´atica,

San Jos´e, Costa Rica

2Department of Mathematics, University of California Davis, CA, USA

*fabio.sanchez@ucr.ac.cr

+these authors contributed equally to this work

Abstract

We constructed a Susceptible-Addicted-Reformed model and explored the dynam-

ics of nonlinear relapse in the Reformed population. The transition from susceptible

considered at-risk is modeled using a strictly decreasing general function, mimicking

an inﬂuential factor that reduces the ﬂow into the addicted class. The basic repro-

ductive number is computed. Furthermore, R0determines the local asymptotically

stability of the addicted-free equilibrium. Conditions for a forward-backward bifur-

cation were established using R0and other threshold quantities. A stochastic version

of the model is presented, and some numerical examples are shown. Results showed

that the inﬂuence of the temporarily reformed individuals is highly sensitive to the

initial addicted population.

Keywords: Nonlinear relapse; backward bifurcation; epidemic models; social determinants;

addiction

1 Introduction

Infectious diseases have been a burden to public health for some time. The transmission mecha-

nisms of pathogens are mainly close contact with an infectious host, airborne, via a vector, and

in some cases via contact with an infected area [2]. However, more recently, health authorities

worldwide have been vigilant with the high number of mental health incidents, highlighted pri-

marily by the Covid-19 pandemic [21]. Social factors provide a unique challenge to construct

mathematical models that include social aspects not typically included in epidemic models. How-

ever, the incorporation of social determinants in these models is inherently diﬃcult. In previous

work, social determinants were introduced as “epidemics” where transmission happened through

social interactions, similar to infectious diseases. For example, a drinking dynamics model using a

nonlinear system of diﬀerential equations [17], where the “infectious” class was the drinking pop-

ulation and the interaction between nondrinkers and drinkers simulated an epidemic process. We

based our theoretical framework on the latter. Other models simulating social dynamics include:

a bulimia model [9], drug models [18, 1], and a sex worker industry model [5], among others.

Mathematical models applied to infectious diseases have become common; more recently, an

insurmountable number of models arose during the Covid-19 pandemic [4, 16, 8] (and references

arXiv:2210.15481v1 [math.DS] 17 Oct 2022

therein). In general terms, when studying infectious diseases, mathematical models help under-

stand disease transmission dynamics. Furthermore, mathematical models, in some cases, can

provide insight to health authorities to construct and develop eﬃcient public health policies [8].

Modeling social interactions as epidemic processes can provide a helpful understanding of the

phenomenon studied. Here, we model addiction as an infectious disease where the interactions

between the non-addicted and addicted individuals can cause an “epidemic” process and confer

an “infection”. Drug addiction has been a problem worldwide for many decades [14, 19, 3]. In

particular, when the crack “epidemic” of the 1980s was in full force, the derivative of cocaine, a

more pure and more expensive narcotic, led to a faster addiction and deterioration of individuals

that consumed the drug [10, 7].

Furthermore, relapse rates of addicted individuals, especially those that used potent narcotics

such as crack cocaine, methamphetamine, fentanyl, and heroin, among others, are very high [15,

13]. In the model constructed here, we looked at nonlinear relapse rates and the inﬂuence of those

who recovered and want to provide support for non-addicted presumed susceptible individuals.

This is done via a general function that depends on the temporarily recovered population and

other parameters.

Epidemic models have helped describe transmission dynamics of infectious pathogens and

derive strategies for their control, prevention, and reduction of incidence, among others. Here, we

provide a theoretical framework to study social phenomena studied via an epidemic model and

highlight the sensitivity of initial conditions.

The article is organized as follows: in Section 2, we give details of the mathematical model.

In Section 3, we present the mathematical analysis. In Section 4, we provide a stochastic version

of the model and provide some numerical examples. Finally, in Section 5, we provide a discussion

based on our results.

2 Mathematical Model

The model we consider is based on [17], where authors explored the impact of nonlinear inﬂuence

on drinking behavior dynamics. In our model, we consider three compartments: susceptible

individuals (S), addicted individuals (A), and temporarily reformed individuals ( ˜

S). The model

transitions follow the typical SIR model [2, 12].

The recruitment rate, β, represents the strength of social inﬂuence on susceptible (at-risk)

individuals. In this context, transmission is a collective behavior rather than an individual con-

sequence; i.e., recruitment is not typically the work of a single individual, but instead is a result

of the collective inﬂuence of a group of individuals as a whole [6]. Moreover, κ∈[0,1] denotes

the cost of addiction, and ν∈[0,1] is the willingness of reformed individuals to deter at-risk

individuals from addiction. We then consider a positive, strictly decreasing smooth function g

deﬁned by:

gκ,ν (˜

S) = κ

1 + ν˜

,(1)

which is a reducing factor that impacts transitions from Sto A. The function grepresents the

impact of reformed individuals in the at-risk population. Here, high values of νimply that a large

proportion of the reformed class is helping the susceptible population, considered at-risk.

The relapse of the reformed population is possible through interactions with individuals in

the addicted class considered infectious, which refers to conditions that possibly spread through

a strong collective social component. In our model, individuals can temporarily recover at rate γ

and transition into the susceptible (at-risk) class ( ˜

S). Rehabilitation programs have the potential

to use the social inﬂuence of reformed individuals to deter at-risk individuals from relapse. How-

ever, reformed individuals typically encounter environmental pressures that may lead to relapse.

Reformed individuals can once again become addicted via interaction with individuals in the ad-

dicted class A, with relapse rate φ, that denotes the “social inﬂuence” of temporarily reformed

individuals. Finally, individuals leave the system at rate µ, typically considered the natural exit

rate.

The model we just described corresponds to the system of nonlinear diﬀerential equations

given by:

dt =µN −βg(˜

S)SA

N−µS,

dt =βg(˜

S)SA

N+φ˜

N−(µ+γ)A, (2)

d˜

dt =γA −φ˜

N−µ˜

where N=S+A+˜

Sis the total (presumed constant) population. We re-scale system (2) by

substituting s=S

N,a=A

N, ˜s=˜

N, obtaining the equivalent system of equations:

dt =µ−βg(˜s)sa −µs, (3a)

dt =βg(˜s)sa +φ˜sa −(µ+γ)a, (3b)

d˜s

dt =γa −φ˜sa −µ˜s. (3c)

It is clear that s+a+ ˜s= 1 and the reducing factor (1) is therefore given by

g(˜s) = κ

1 + ν˜s∈[0,1].(4)

3 Mathematical analysis

We ﬁrst analyze the addiction-free equilibrium, (s∗

0, a∗

0,˜s∗

0) = (1,0,0), that can be used to de-

termine the basic reproductive number, R0. In epidemiology, the basic reproductive number

represents the number of secondary infections produced by an average infected individual; when

this number is less than one, the disease typically dies out, while when it is greater than one, there

will be an epidemic [2]. In this context, we consider R0to be a measure of the strength of the

social inﬂuence of addicted individuals to recruit individuals into a vice. As we will demonstrate,

R0>1 implies the establishment of an infectious agent and R0<1 typically implies that the

number of addicted individuals decreases and goes to zero. Albeit, our model can sustain an

addiction when R0<1 under particular initial conditions.

3.1 Basic reproductive number and addiction-free equilibrium

We use the next generation operator method [12] to compute R0. Let

F=βg(˜s)sa +φ˜sa and V= (µ+γ)a,

where Fand Vcontains all terms ﬂowing into aand ﬂowing out of a, respectively. It holds that

F=∂F

∂a 



(s∗

0,a∗

0,˜s∗

=βg(0) and V=∂V

∂a 



(s∗

0,a∗

0,˜s∗

=µ+γ.

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Amathematicalmodelwithnonlinearrelapse:conditionsforaforward-backwardbifurcationFabioSanchez1,*,+,JorgeArroyo-Esquivel2,+,andJuanG.Calvo1,+1UniversidaddeCostaRica,CentrodeInvestigacionenMatematicaPurayAplicada-EscueladeMatematica,SanJose,CostaRica2DepartmentofMathematics,UniversityofCaliforniaDa...

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A mathematical model with nonlinear relapse conditions for a forward-backward bifurcation Fabio Sanchez1 Jorge Arroyo-Esquivel2 and Juan G. Calvo1

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