# Contagious Criminal Career Models Showing Backward Bifurcations: Implications for Crime Control Policies

Authors: Silvia Martorano Raimundo; Hyun Mo Yang; Eduardo Massad
Journal: Journal of Applied Mathematics
 (2018)

---

## Abstract

We provide a theoretical framework to study how criminal behaviors can be treated as an infectious phenomenon. There are two infectious diseases like models that mimic the role of convicted criminals in contaminating individuals not yet engaged in the criminal career. Equilibrium analyses of each model are studied in detail. The models proposed in this work include the social, economic, personal, and pressure from peers aspects that can, theoretically, determine the probability with which a susceptible individual with criminal propensity engages in a criminal career. These crime-inducing parameters are treated mathematically and their inclusion in the model aims to help policy-makers design crime control strategies. We propose, to the best of our knowledge by the first time in quantitative criminology, the existence of thresholds for the stability of crime-endemic equilibrium which are the equivalent to the “basic reproduction number” widely used in the mathematical epidemiology literature. Both models presented the phenomena of backward bifurcation and breaking-point when the contact rates are chosen as bifurcation parameters. The finding of backward bifurcation in both models implies that there is an endemic equilibrium of criminality even when the threshold parameter for contagion is below unit, which, in turn, implies that control strategies are more difficult to achieve considerable impact on crime control.


---

## Body

## 1. Introduction

There is now substantial support in the specialized literature on economics, sociology, criminology, and social psychology to the attempts to explain how and why an individual’s propensity to engage in criminal behavior is influenced by his/her social context [1]. One interesting metaphor is the one that states that criminal behavior is contagious or that individuals can be susceptible to what economists call endogenous effects [2]. According to this effect, the social milieu in which individuals live may change the individual’s propensity to engage in that same criminal behavior as their peers [3, 4]. In addition, the individual’s criminal behavior can be affected by other attributes of his/her neighbors, like in the “role model” theory [5] or the peer pressure to maintain local order [6]. Moreover, the institutional or other characteristics of neighborhoods, including crime prevalence, may induce criminal behavior in susceptible individuals [7].However, in spite of the large theoretical literature on whether or not crime is contagious, the empirical support for this hypothesis is still limited. Some authors (see [8, 9]) reported an excess in the variation in crime rates across areas which cannot be explained only by the variation in standard sociodemographic determinants of criminal behavior. This suggests that social interactions are more important for less serious than more serious crimes. In a famous study, Crane (1991) (see [10]) showed that in the presence of endogenous residential sorting such reports may be biased by the causal effects of environmental and some individual or family characteristics on the selection of the neighborhood. As pointed out by Kling and Ludwig (see [1]), however, even in the absence of the biased selection problem, it would be very difficult to determine which of the theoretical perspectives above are responsible for any observed neighborhood effects on criminal behavior.The influence of others’ behavior on criminal offences of susceptible individuals is called “behavioral contagion” [11, 12]. Behavioral contagion has been defined as the spread of any attitude or behavior from one individual or group of individuals to another individual or group of individuals throughout a social network of varying structures [11].Social contagion arises among people interacting in social structures of diverse nature [13–15]. In such interactions information, behavioral innovation, belief, or meme is transmitted in a similar way to infectious diseases spread in groups of susceptible individuals [16]. Contagion occurs when susceptible individuals interact with contagious people in such a way that this interaction results in a new case [17]. In social networks, contact is defined by the communication and/or imitation of influential processes that make transmission potentially effective (see [13] pp. 1288-1289).The social contagion theory of violence describes the spread of criminal behavior as similar to the spread of infectious diseases [18]. In such a context, the contagious nature of criminal behavior can be understood, described, and analyzed with the tools developed for studying infectious diseases spreading, in particular the use of mathematical models [19]. However, in contrast with directly transmitted diseases, the infectivity of violence does not require direct contact between susceptible and infected individuals.In this paper, we provide a different approach to the contagious effect of criminal behavior in susceptible individuals. Rather than considering direct contagious effect by the social environment, we consider a contagious effect by criminals already convicted and who control criminal activities from inside prison, a common effect of some countries like Brazil.The advent of organized crime in Brazilian prisons, especially in the state of São Paulo, and its role in the contagion of criminality to susceptible individuals outside the prisons constitute the object of this article. The gang leaders the Capital’s First Command (PCC, Primeiro Comando da Capital) (see [20]) unleashed a series of attacks in May 2006, resulting in deaths, brought cities to a halt, and cornered authorities in charge preventing them from applying law and order and these are the starting as well as reference points taken. In addition, the gang leaders were sustained by an organization maintained by a hierarchical structure of disciplined and obedient employees capable of executing orders without questioning them. Operating from a base of support networks disseminated in distinct mobile points throughout the state, they revealed that they had an able and agile communication system among leaders, followers, and those who took orders, through protected channels barely permeated by external interference by means of cell phones, telephone exchanges, and carrier pigeons [21].Criminal contagion from inside the prison system of Brazil to outside susceptible individuals with criminal propensity will be considered as an infectious event in a dynamical system context. The models proposed in this work include the social, economic, personal, and pressure from peers aspects that can, theoretically, determine the probability with which a susceptible individual with criminal propensity engage in a criminal career. These crime-inducing parameters are treated mathematically and their inclusion in the model aims to help policy-makers design crime control strategies.


## 2. The Models

The models presented here are based on the criminal activity within a population in an effort to understand how the criminal careers change and evolve over time. We assume a population-based approach, similar to those models of the spreading of infections that confer temporary/permanent immunity. By introducing the key epidemiological concept of a threshold from mass action law [22, 23], we illustrate the fundamental relationship between incarceration and recidivism within a population and use it to show how the criminal activity could be controlled to reduce the likelihood of an individual to engage in a criminal career. We compute and analyze this threshold considering the spread of the crime and the dynamics of incarceration and recidivism. Keeping in mind both the threshold and the perspective of criminal dynamics, it is possible to evaluate what programs of rehabilitation or prevention contribute to the reduction of the recidivism and the number of contacts among individuals susceptible to crime, offenders, and ex-offenders.In epidemiology theory, the core groups are conceptualized as being the individuals in a population who will infect more than one person over the duration of infection. Core groups are recognized as playing a central role in sustaining the infection in a population and interventions targeting these groups are central to an effective prevention response. Building on this, we extend the concept of core groups to the inmate population.Especially in inmate population, most criminality is found among core groups and the criminality will only become more generalized if the contact spreads throughout other networks. Hence, it is very likely that incarcerated individuals will contact those susceptible individuals who have never been incarcerated but have an intrinsic criminal propensity. The basic problem is to find out when this contact occurs, who regulates contact, what types of contact are feasible and desirable, and what are the effects of contact (or lack thereof) on susceptible individuals.In this way, the total population size, denoted byN(t), is then characterized by three classes: susceptible (S), incarcerated (C), and desisting (D) individuals. We specify first-time (i = 1) and multiple-time (i = 2, 3,…,n) incarceration by adding subscripts to model variables and parameters.


### 2.1. The Partially Contagious Criminality Model (PCCM) Formulation

Let the susceptible in core group population be divided into two categories:S0, those individuals who have a criminal propensity but have never been incarcerated, not criminally active but susceptible to crime, and Si, those individuals susceptible to criminal activities who were once incarcerated (S1) and those susceptible who were multiple-time incarcerated (Si,i=2,3,…,n) and became criminally active again. Similarly, let the incarcerated population be divided into two categories: C1, those who are first-time incarcerated, and Ci (i = 2,3,…,n), those who were multiple-time incarcerated at a given time. Finally, D0 represents those individuals who have a criminal propensity but desisted from criminal life either by their own or as a result of early interventions when discharged out of prison (desisting offenders/criminal desisters), and Di (i = 1,2,3,…,n) represents those individuals who were either first-time or multiple-time incarcerated but desisted from criminal behavior (desisting offenders/criminal desisters) due to variety of reasons.It is assumed that individuals susceptible to the crimeS0 move to either incarcerated (C1) or desisting (D0) class. The rate of initial participation in crime (δ0) at which individuals move from state S0 (not criminally active) to C1 (criminally active and incarcerated) is proportional to intrinsic criminal propensity of individuals. It is also assumed that special intervention programmes (γ1) may change their basic propensity traits and affect the decisions to engage in crime [24]. As a consequence, we should expect that certain interventions discourage participation in crime, resulting in desistance from criminal activity and reintegration back into society. Primary prevention is an attempt to reduce the risk of behaviors that potentially lead to incarceration. This point is one of the great interests here.The average length of the primary incarceration term is given by1/τ1; that is, the rate at which inmates move from state C1 to S1 (formerly incarcerated but not criminally active, i.e., ex-offenders) is τ1. In the same way, the average length of the multiple incarceration term is given by 1/τ2, with 1/τ2>1/τ1.It is worth mentioning that the representation of the cycles of criminal dynamics, with a focus on modeling the recidivism process (criminal careers), could be extended includingSi, Ci, and Di (i = 2,3,…,n). However, we do not intend to develop complex models that account for these cycles and we will explore two cycles only (i = 1,2).In addition, it is assumed that the recidivism prevention takes place during incarceration and after release back into society. Its purpose is to reduce the risk of an individual reoffending and eventually returning to the prison system. However, some individuals may resume criminal activity very soon after being released from prisons depending on having contact with those individuals still incarcerated and criminally active. It should be mentioned that Walsh and Graig (see [25]) indicated that recidivism is also called falling back into a previous criminal behavior.We also defineβi (i=1,2) as the rate of imprisonment, which captures the return to criminal activity of the individuals released from prison, such that β1 and β2 describe the rates with which individuals engage into criminal activity, depending on having had any contagious contact with those incarcerated criminals. They are analogous to the effective contact rate in infectious diseases models.Finally,μ is the natural mortality rate and ai (i=1,2) is the incarceration-related additional deaths rate (inmate mortality rate can be caused by illness, such as AIDS-related, suicide, accidental self-injury, execution, or any other unspecified cause). Moreover, since the model monitors human populations, all parameters are assumed as nonnegative. We also assumed homogeneous population without any differences in age or in crime type occurrences. In addition, we used the number of incarcerations as a proxy for reoffending.The flow diagram of the PCCM for two stages is depicted in Figure1. The variables and parameters are described in Tables 1 and 2, respectively.Table 1
Models variables and their biological meaning.
Variables Description S0 Individuals not criminally active but susceptible to crime (core group) C1 First-time incarcerated individuals S1 First-time ex-offenders individuals who are again susceptible to crime Di Individuals who desist from criminal behavior (i = 0,1,2) (desisting offenders/criminal desisters) C2 Individuals who were two or multiple times incarcerated (recidivists, reoffenders) S2 Second-time ex-offenders individuals and susceptible to crime (at least two incarcerations)Table 2
Models parameters and their biological meaning.
Parameters Description Λ Rate of recruitment of individuals into the core group δ0 Basic flow to the criminality (time-1) (criminal propensity) β1 Rate of cooptation of first-time ex-offenders (contact rate betweenC2 and S1) β2 Rate of cooptation of second-time ex-offenders (contact rate betweenC2 and S2) γ0 Rate of early desistance from crime γ1 Rate of desistance from crime when in the first cycle γ2 Rate of desistance from crime when in at least the second cycle μ Natural inmate mortality rate a1 Additional inmate mortality rate when inC1 a2 Additional inmate mortality rate when inC2 τ1 Release rate from incarceration of first-time offenders τ2 Release rate from incarceration of at least second-time offendersFigure 1
The flow diagram for the low-high criminality model (1).Combining the above derivations and assumptions, it follows that the model for transmission dynamics of criminality for two stages is given by the following nonlinear system of differential equations:(1)dS0dt=Λ-δ0+γ0+μS0dD0dt=γ0S0-μD0dC1dt=δ0S0-τ1+μ+a1C1dS1dt=τ1C1-γ1+μS1-β1C2S1dD1dt=γ1S1-μD1dC2dt=β1S1+β2S2C2-τ2+μ+a2C2dS2dt=τ2C2-β2C2S2-γ2+μS2dD2dt=γ2S2-μD2with generic initial conditions S0(0)≥0, D0(0)≥0, C1(0)≥0, S1(0)≥0, D1(0)≥0, C2(0)≥0, S2(0)≥0, and D2(0)≥0. By summing up the above equations, the total population size N(t) is variable with(2)dNdt=Λ-μN-a1C1-a2C2Thus, in the absence of additional inmate mortalities, that is, a1=a2=0, the population size evolves as an immigration model with natural mortality, that is, according to dN/dt=Λ-μN. This equation has a single equilibrium N=N(0)=Λ/μ for any initial value of N(0). Thus, in the long run, the population size settles to this constant value. It follows from (2) that limt→∞N(t)≤Λ/μ=N(0).The differential equation forN implies that solutions of (1), starting in the positive orthant R+8, either approach, enter, or remain in the subset R+8 defined by(3)Ω=S0,D0,C1,S1,D1,C2,S2,D2∈R+8:S0+D0+C1+S1+D1+C2+S2+D2≤N0.Thus it suffices to consider solutions in the regionΩ. Solutions of the initial value problem starting in Ω and defined by (1) exist and are unique on a maximal interval [26]. Since solutions remain bounded in the positively invariant region Ω, the initial value problem is then both mathematically and epidemiologically well posed [27]. Hence, it is sufficient to consider the dynamics of the flow generated by model (1) in Ω.


### 2.2. Analysis of the PCCM Model

In this section, system (1) is qualitatively analyzed to investigate the existence of its equilibria [28] and the control strategies of its dynamical behavior.From system (1), with the right-hand size equal to zero, it can be seen from the first five equations that the coordinates of the equilibrium point are given, respectively, by(4)S0=Λδ0+γ0+μ,D0=γ0μS0,C1=δ0τ1+μ+a1S0,S1=τ1δ0τ1+μ+a1μ+γ1+β1C2S0,D1=γ1μS1.Moreover, from seventh and eighth equations of system (1), we obtain(5)S2=τ2C2μ+γ2+β2C2,D2=γ2μS2.From the sixth equation of system (1), one gets(i)
C2=0,(ii)
C2≠0, which implies β1S1+β2S2-(τ2+μ+a2)=0.IfC2=0, model (1) has a low-criminality equilibrium Pl=(S0,D0,C1,S1,D1,0,0,0) which indicates the existence of offenders who are incarcerated only once in life (C1), given by(6)S0=Λδ0+γ0+μ,D0=γ0μS0,C1=δ0τ1+μ+a1S0,S1=τ1δ0τ1+μ+a1μ+γ1S0,D1=γ1μS1.To determine the stability of this equilibrium, the Jacobian of system (1) is computed and evaluated at Pl. Hence, the low-criminality equilibrium Pl=(S0,D0,C1,S1,D1,0,0,0) is locally asymptotically stable if R1∗<1, where R1∗, defined as Criminality Reproduction Number (CRN), is given by(7)R1∗=β1β1∗with(8)β1∗=τ1+μ+a1μ+γ1τ2+μ+a2τ1δ0S0.Note that, in analogy to the spread of infectious diseases models [29], the CRN, R1∗, represents the “average expected number of new offenders originated by a single persisting offender C2∗, whilst in a criminal career.” In other words, one person C2∗ who was incarcerated two times gets into contact with S1 susceptible individuals, just released from first imprisonment, successfully and induces R1∗ persons to commit crime. In other words, R1∗ is the average number of individuals who commit crime influenced by one inmate C2∗.ForC2≠0, that is, for β1S1+β2S2-(τ2+μ+a2)=0, replacing both expressions for S1 given by (4) and S2 given by (5), an expression for C2=C2∗>0 is obtained as(9)b2C2∗2+b1C2∗+b0=0,where(10)b2=β1β2μ+a2.b1=μ+γ1τ2+μ+a2β2β1μ+γ2β2μ+γ1+μ+a2τ2+μ+a2-R1∗,b0=μ+γ1μ+γ2τ2+μ+a21-R1∗,Let us now determine the conditions under which the quadratic equation (9) has positive real roots; that is, we search for the existence of multiple equilibria of system (1). However, the expression for the discriminant of the quadratic equation (9) is very complex, so we will analyze the signs of its coefficients to ensure the existence of real solutions. Thus, the conditions under which this equation has either one or two positive real roots can be determined, and these results translated into nontrivial equilibrium of system (1) which is biologically feasible (i.e., positive).Hence, substituting the positive real solutions of the quadratic equation (9) (i.e., positive values of C2∗) into the expressions in (5), model (1) has a high-criminality equilibrium, Ph=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗), where there is coexistence of both offenders C1 and C2∗, given by(11)S1∗=τ1δ0τ1+μ+a1+μ+γ1+β1C2∗S0D1∗=γ1μS1∗S2∗=τ2C2∗μ+γ2+β2C2∗D2∗=γ2μS2∗,with S0, D0, and C1 given by (6). Thus, the following result is then established.Theorem 1.
Model (1) has(i)
a unique positive equilibriumPh if b0<0⇔R1∗>1;(ii)
a unique positive equilibriumPh if b0=0 and b1<0;(iii)
two positive equilibria,Ph, if b0>0 and b1<0 and b12-4b2b0>0;(iv)
no positive equilibrium, otherwise.Since all model parameters are assumed as nonnegative, it follows from (10) that the coefficient b2 is always positive, b0<0 for R1∗>1 and b0>0 for R1∗<1. Thus, it is clear from Theorem 1 that model (1) has a unique positive equilibrium, Ph, when b0<0, that is, when R1∗>1 (case (i)).Now, forb0>0 and R1∗<1, the quadratic equation (9) has two positive solutions if b1<0 and b12-4b2b0>0 (case (iii)). Hence, assuming that (9) has two positive real solutions, let C2- and C2+ be the smaller and higher value of C2∗, respectively. Translating it into equilibrium of system (1), the question is to address what means the positive high-criminality equilibria Ph for R1∗<1. It is important to note that, in this case, system (1) can have two equilibria, which are biologically feasible, even though R1∗<1. This idea is explored more deeply below.It is instructive at this point to explore some qualitative features forR1∗=1 and R1∗>1. Firstly, for R1∗=1, it follows that b0=0; the quadratic equation (9) has either a unique positive root (if b1<0) or no positive root (if b1>0). In other words, for R1∗=1 and b1<0, model (1) has a unique positive high-criminality equilibrium given by Ph (case (ii)).Moreover, note that whenβ2=0, we have b2=0 and b1=(μ+γ2)(τ2+μ+a2)β1. Thus, if R1∗>1, then(12)C2∗β2=0=μ+γ1β1R1∗-1,and model (1) has a unique positive high-criminality equilibrium, Ph, for β2=0.Forβ2→∞, we have b1=(μ+γ1)(τ2+μ+a2)β2(μ+a2)/(τ2+μ+a2)-R1∗ and b2=β1β2(μ+a2) with b0≪b1 and b0≪b2 such that(13)C2∗β2→∞=μ+γ1τ2+μ+a2β1μ+a2R1∗-μ+a2τ2+μ+a2,and C2∗>0 if only if R1∗>(μ+a2)/(τ2+μ+a2). Thus, model (1) has a unique high-criminality equilibrium Ph if β2→∞.Unfortunately, this high-criminality equilibrium cannot be studied from its closed form, so we carried out its local stability using numerical methods. The results are provided next.Finally, if (i), (ii), and (iii) do not occur, then there are no endemic equilibria for system (1).In what follows, model (1) admits two realistic scenarios: the best-case scenario, where offenders were incarcerated once and it is still possible to fight crime and recidivism is given by the low-criminality equilibrium Pl=(S0,D0,C1,S1,D1,0,0,0), and the worst-case scenario, where the offenders were incarcerated at least once (recidivism), which can potentially lead to increased criminal activities, given by the high-criminality equilibrium Ph=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗).Having found the scenarios in which there exist the equilibria for system (1), it is instructive to analyze whether or not these equilibria are stable under any of these scenarios. Moreover, together with the CRN, R1∗ (see (7)), and the parameter β2, we see that each scenario can be used as a check for the existence and the stability of the equilibria. Another crucial question is which of the two incidence rates β1 and β2 is more likely to affect the criminal dynamics in general and criminal contacts in particular. These points are of great interest here and we then explore how our model behaves under control strategies when R1∗<1. We fix β2 and explore the system behavior by varying β1. We also explore what happens when we fix β1 while β2 is varied. These control measures are designed to fight recidivism. In both cases, the recidivism will depend upon the initial sizes of the subpopulation C2 and indicate the possibility of backward bifurcation.Firstly, it is worth remembering that the quantityβ1 measures the average number of new contacts generated by a typical incarcerated individual C2 with those susceptible individuals S1 who are ex-offenders and can become susceptible to the crime again. β2, in turn, measures the average number of new contacts generated by C2 with those individuals S2 who were reoffenders (recidivist behavior) and are susceptible to the crime once again.In this way, there are two groups of offenders: those who are under much lower risk, most of whom will go to prison once and not come back (ex-offenders), and those who repeatedly do crimes and come back multiple times to the prison (reoffenders). The strong implication of the findings is that individuals who are incarcerated are extremely likely to reoffend once they are free.Note that when system (1) has a small influx of reoffenders (C2), it does not generate high criminality rates and, for R1∗<1, it is still possible to minimize the spreading of crime (best-case scenario). We will see that, in this case, the corresponding low-criminality equilibrium Pl could be locally asymptotically stable. On the other hand, the criminality will persist and increase if R1∗>1 (worst-case scenario). In this case, the corresponding high-criminality equilibrium Ph could be locally asymptotically stable. This phenomenon, where the possibility of fighting the spread of the crime is lost and the criminality is potentially active, that is, where Pl loses its stability and a unique Ph appears as R1∗ increases through one, is known as forward bifurcation in epidemiology.For models that exhibit this type of bifurcation, the requirementR1∗<1 is necessary and sufficient for the high-criminality elimination. In contrast, other models undergo another type of bifurcation, known as “backward bifurcation” in epidemiology, where two equilibria Ph+ and Ph- coexist with the low-criminality equilibrium, Pl; that is, there are three steady states when R1∗ is immediately less than one. Thus, the requirement R1∗<1 is necessary but is not sufficient to fight the spread of crime.


#### 2.2.1. Analytic Strategy:β2 Fixed with β1 Increasing

The physical implication of backward bifurcation is that theC2 population can engage in a high criminality level even when R1∗ crosses unity downwards. In other words, in the presence of recidivism, decreasing R1∗ below one is not a sufficient condition to make the criminality level decrease. This has very important consequence for crime control, as will be discussed later in this paper.To check the possibility of backward bifurcation in model (1), it is necessary to know other subthreshold, and we will refer to this limit point, as expressed on the R1∗ scale, as R1thr. Hence, for R1∗<R1thr, system (1) will present only the locally asymptotically stable low-criminality equilibrium Pl. For Rthr∗<R1∗<1, system (1) will present the locally asymptotically stable low-criminality equilibrium Pl plus two positive high-criminality equilibria, Ph+ and Ph-, which will correspond to the solutions of (9): C2+, the higher solution, which corresponds to the stable equilibrium, and C2-, the smaller solution, which corresponds to the unstable equilibrium.Although the critical value of the bifurcation could not be found analytically due to the high dimension of system (1), this task can be performed numerically. In this way, our simulations show that there exists a critical value Rthr∗<R1∗=1, where model (1) undergoes backward bifurcation.A schematic diagram of the backward and the forward bifurcations for system (1) is given in Figure 2, where β1 is chosen as a bifurcation parameter; that is, we fixed β2, whereas β1 increases.Figure 2
Backward and forward bifurcations diagram for proportion of reoffenders (C2∗) for model (1), where β1 is chosen as a bifurcation parameter (β2 fixed and β1 increasing). The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). Backward bifurcation for (a) β2=1000, β1=5.371, and R1thr=0.347; (b) β2=500, β1=5.845, and R1thr=0.378; and (c) β2=50, β1=11, and R1thr=0.712. Forward bifurcation for (a) β2=17.7, β1=15.45, and R1thr=R1∗=1 and (b) β2=10.0, β1=15.45, and R1thr<R1∗=1. Parameters’ values used are as given in Table 3.Table 3
Baseline values for model (1).
Variable Description Λ 0.015  (years-1) δ0 0.02  (years-1) β1 v a r i a b l e  (years-1) β2 v a r i a b l e  (years-1) γ0 0.2  (years-1) γ1 0.1  (years-1) γ2 0.08  (years-1) μ0 0.015  (years-1) a1 0.03  (years-1) a2 0.025  (years-1) τ1 0.2  (years-1) τ2 0.1  (years-1)Figure2 shows the profile of the proportion of both reoffenders C2+ (solid curve) and C2- (dashed curve) as a function of R1∗ with decreasing values of β2=1000;500;50;17.7;10(year-1) as R1∗ increases (i.e., as β1 increases). The solid curve stands for the stable high-criminality equilibrium, C2+, and the dashed curve stands for the unstable high-criminality equilibrium, C2-. For β2=1000 and β1=5.371, one has R1thr=0.347. Thus, for 0.347<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. Similarly, for β2=500 and β1=5.845, one gets R1thr=0.378. Thus, for 0.378<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. For β2=50 and β1=11, one gets R1thr=0.712. Thus, for 0.712<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. Finally, for β2=17.7 and β1=15.45 and for β2=10.0 and β1=15.45, we have R1thr≤R1∗=1, and the model has one positive high-criminality equilibrium for R1∗>1 and no positive equilibrium for R1∗<1. Consequently, model (1) exhibits the forward bifurcation at R1∗=1. As it should be expected, it can be seen in Figure 2 that R1thr increases with increasing β1 and a greater reduction in recidivism prevalence is recorded for decreasing values of β2.As stated earlier, the physical significance of the phenomenon of backward bifurcation is that the classical requirement ofR1∗<1 is no longer sufficient for avoiding the recidivism prevalence, as it is for the forward bifurcation. In such a scenario, the recidivism would depend on the initial sizes of the subpopulation C2 of the model. That is, the presence of backward bifurcation in model (1) suggests that the possibility of avoiding the recidivism event when R1∗<1 could be dependent on the initial sizes of the subpopulation C2. This scenario is illustrated numerically in Figure 3. Thus, if the recidivism is low, then C2(0) is small, such that the low-criminality equilibrium Pl is stable (Figure 3(a)). In contrast, if the recidivism is higher, then C2(0) is large, so the high-criminality equilibrium Ph is stable (Figure 3(b)). Therefore, the stability of these equilibrium points depends on the initial condition of system (1). This clearly indicates the coexistence of two locally asymptotically stable equilibria when R1∗<1, confirming that model (1) undergoes the phenomenon of backward bifurcation with one stable high-criminality equilibrium Ph+ (higher, solid curve in Figure 2), one unstable high-criminality equilibrium Ph- (lowest dashed curve in Figure 2), and one low-criminality equilibrium Pl.Figure 3
Forβ1=11, β2=50, and R1thr<R1∗<1. The other parameters’ values are given in Table 3. Profile of the population of reoffenders (C2). (a) If the recidivism is low (or C2(0) small), then the equilibrium point Pl is locally asymptotically stable; (b) if the recidivism is higher (or C2(0) large), then the equilibrium point Ph is locally asymptotically stable.
(a) (b)


#### 2.2.2. Analytic Strategy:β1 Fixed with β2 Increasing

Alternatively, from now on we explore the implications of the parameterβ2 on the criminal dynamics. It is worth mentioning that, from expression (10), it is easy to verify that b1<0 if and only if(14)β2>β1μ+γ2μ+γ1R1∗-μ+a2/τ2+μ+a2>0,and thus, in such case, system (1) has two positive high-criminality equilibria (case (iii), see Theorem 1.) for b0>0, b12-4b2b0>0, and R1∗>(μ+a2)/(τ2+μ+a2).A schematic diagram of this bifurcation phenomenon for system (1) is given in Figure 4, where β2 is chosen as a bifurcation parameter. As mentioned previously, this bifurcation phenomenon of model (1) is only illustrated numerically.Figure 4
Breaking point and forward bifurcation diagrams for proportion of reoffenders (C2∗) for model (1), where β2 is chosen as a bifurcation parameter (β1 fixed with β2 increasing). The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). The phenomenon of the breaking point for (a) β1=11, R1∗=0.71, and β2thr=50; (b) β1=7.5, R1∗=0.486, and β2thr=146.8; and (c) β1=5.845, R1∗=0.378, and β2thr=499; forward bifurcation for β1=15.449, R1∗=1, and β2thr=17.7. For β1=30 and R1∗=1.94, β2=0 such that system (1) has a unique positive high-criminality equilibrium Ph. Parameters’ values used are as given in Table 3.Figure4 shows the profile of the proportion of both reoffenders C2+ (solid curve) and C2- (dashed curve) as a function of β2 with decreasing values of β1=30;15.449;11;7.5;5.845(year-1) as β2 increases. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. Note that, for β1=30 and R1∗>1, β2=0 such that system (1) has a unique positive high-criminality equilibrium (see case (i) of Theorem 1 and (12)).It should be also noticed in Figure4 that R1∗ increases with decreasing β2, such that greater reduction in recidivism prevalence is recorded for decreasing values of β1. One aspect to differentiate both parameters β1 and β2 is the way that they account for the transition to criminal activity.In the low-criminality scenario, the only way someone becomes criminally active is on his/her own, without interacting with another person who is already criminally active. This restriction is necessary in order to illustrate the dynamics of a system which would lead to low-criminality equilibrium. If new criminal could only emerge on his/her own, then reducing the number of individuals who are not criminally active (core group) through either intervention or spontaneous desistance could lead to the lowest criminal scenario. For this, we calculated the threshold between the low-criminality equilibrium and the high-criminality equilibrium. Therefore, forR1∗<R1thr, system (1) presented the locally asymptotically stable low-criminality equilibrium Pl. Although not a realistic possibility, understanding the intrinsic criminal propensity that leads to such kind of deviant behavior is a necessary step for understanding the high-criminality prevalence in the model, which will be presented in the next section.In addition, in the high-criminality scenario, the way someone can become criminally active is to interact with another person who is already criminally active. Thus, if new reoffenders could only emerge by interaction between reoffenders (C2∗) and those second-time offenders susceptible to the crime (S2∗) at a rate β2, then reducing the number of contacts between C2∗ and S1, that is, reducing β1, could lead to the low-criminality scenario if the proportion of reoffenders C2∗ is small (see Figure 3). Hence, to reach the low-criminality equilibrium, a community needs to reduce β1. However, if β1 is large, then the proportion of reoffenders C2∗, as well as S2∗, increases, which leads to higher crime prevalence, such that the high-criminality scenario emerges.Figure5(a) shows the long-term behavior of ln⁡β2thr plotted versus β1. The threshold β2thr is expressed as a minimum value where system (1) has two positive equilibrium points. Thus, for each fixed β1, there is a corresponding unique threshold β2thr. For one such example (see Figure 5(b)), β1=11, R1∗=0.71, and β2thr=50 (or ln⁡50=3.91) are the minimum values where system (1) has two positive value. For β2>β2thr, we have C2+>0 and C2->0. In this case, system (1) has a stable equilibrium point, C2+ (the solid curve in Figure 5(b)), and an unstable equilibrium point, C2- (the dashed curve in Figure 5(b)). On the other hand, for β2<β2thr, we have C2+=C2-=0, which means that there are no positive equilibrium points for system (1). Moreover, for β1thr=15.4492917, we have R1∗=1 and β2thr=17.867 (or ln⁡17.867=2.883) such that, for β2<β2thr, one has C2+=0 and C2-<0. Otherwise, for β2>β2thr, one has C2+>0 and C2-=0 (see Figure 6).Figure 5
(a) The long-term behavior ofln⁡β2thr plotted versus β1. (b) For β1=11, R1∗=0.71, and ln⁡β2=3.91, system (1) has two positive equilibrium points. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. The other parameters’ values are given in Table 3.Figure 6
Long-term physical outcomeC2∗ plotted versus ln⁡β2. For β1thr=15.449, R1∗=1, and β2>β2thr, system (1) has one positive equilibrium point. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. The other parameters’ values are given in Table 3.


### 2.3. The Full Contagion Criminal Model (FCCM) Formulation

In this section, we relax the assumption made in the PCCM model (1) that the susceptible individuals (S0) can enter to crime only on their own, so we make δ0=0. Hence, in the modified model, the flow of the susceptible individuals S0 into the criminal activity depends only on the contact with those individuals who are incarcerated, C1. We also define βi (i = 0,1,2) as the rate of imprisonment; it captures the return to criminal activity of those released from prison, such that β0, β1, and β2 are the flows into criminal activity which depend on having had contact with those first-time and second-time incarcerated individuals (C1 and C2), respectively. Following the idea of the previous model (1), here βi (i = 0,1,2) is analogous to the effective contact rate in infectious diseases model. This derivation adopts a standard incidence formulation, where the contact rate is assumed to be constant, unlike the case of the mass action formulation, where the contact rate depends on the size of the total population (see, e.g., [27, 30], for detailed derivation of these incidences’ functions).The flow diagram of the FCCM is depicted in Figure7. The variables and parameters’ values are given in Table 3, except for β0 that is the contact rate between C1 and S0 (cooptation rate).Figure 7
The flow diagram for the free-low-high criminality model (15).The model is represented by the following nonlinear system of differential equations:(15)dS0dt=Λ-β0C1S0-γ0+μS0dD0dt=γ0S0-μD0dC1dt=β0C1S0-τ1+μ+a1C1dS1dt=τ1C1-γ1+μS1-β1C2S1dD1dt=γ1S1-μD1dC2dt=β1S1+β2S2C2-τ2+μ+a2C2dS2dt=τ2C2-β2C2S2-γ2+μS2dD2dt=γ2S2-μD2


### 2.4. The Existence and Local Stability of Equilibria

#### 2.4.1. Crime-Free Equilibrium

In the absence of crime, that is, forC1=C2=0, model (15) has a crime-free equilibrium P0=(S00,D00,0,0,0,0,0,0) which is obtained by setting the right-hand sides of system (15) to zero, where(16)S00=Λμ+γ0,D0=γ0μS00.To analyze the local stability of this equilibrium, the Jacobian of system (15) is computed and evaluated at P0, which is locally asymptotically stable if the real parts of the eigenvalues of the Jacobian matrix are all negative.Thus, the local stability of the crime-free equilibriumP0 is governed by the Jacobian matrix(17)MP0=AP000BP0,where(18)AP0=-μ+γ0-β0S000β0S00-τ1+μ+a1,(19)BP0=-μ+γ1000-τ2+μ+a200τ2-μ+γ2.It is easy to verify that the three eigenvalues of matrix (19) are always negative. In the same way, it is also straightforward to verify that one of the eigenvalues of matrix (18) is always negative, while the other is negative whenever(20)R0=β0τ1+μ+a1S00<1.Hence, all the eigenvalues of matrix (17) are negative or have negative real parts if and only if R0<1. In summary, the crime-free equilibrium P0 of system (15) is locally asymptotically stable if the Basic Criminality Reproduction Number R0<1. Therefore, we have established the following result.Lemma 2.
The crime-free equilibriumP0 of system (15) is locally asymptotically stable if R0<1 and is unstable if R0>1.


#### 2.4.2. Low-Criminality Equilibrium

In the absence of the reoffenders, that is, forC2=0, model (15) has a low-criminality equilibrium Pl∗=(S0,D0,C1,S1,D1,0,0,0), where(21)S0=τ1+μ+a1β0D0=γ0μS0C1=Λβ0-μ+γ0τ1+μ+a1τ1+μ+a1β0=μ+γ0β0R0-1S1=τ1C1μ+γ1D1=γ1μS1.Note thatPl∗ exists if C1>0, that is, if R0>1. Now consider the resulting model (15). The local stability of the low-criminality equilibrium Pl, which is examined by linearizing system (15) around Pl, is governed by the Jacobian matrix(22)MPl=M11-β0S0000β0C1M220000τ1M33-β1S10000M440000τ2M55,with M11=-β0C1-(μ+γ0); M22=β0S0-(τ1+μ+a1); M33=-(μ+γ1); M44=β1S1-(τ2+μ+a2); and M55=-(μ+γ2).The eigenvalues of matrix (22) are λ1=M55, λ2=M44, λ3=M33, and the roots of(23)ϕ2+z1ϕ+z0=0,with z1=Λβ0/(τ1+μ+a1)>0 and z0=Λβ0-(μ+γ0)(τ1+μ+a1)>0(⇔R0>1).The eigenvaluesλ1 and λ3 are real negative from definition of the parameters μ, γ1, and γ2. The eigenvalue λ2 is negative if(24)β1<τ2+μ+a2S1=β0τ2+μ+a2μ+γ1τ1μ+γ0R0-1 or, equivalently, if(25)R1=β1τ1μ+γ0R0-1τ2+μ+a2μ+γ1β0<1Finally, the eigenvaluesλ4 and λ5 are real negative if z0>0 and z1>0 by applying the Routh-Hurwitz criteria [31, 32] on polynomial (23). In this sense, z0>0 and z1>0 if and only if R0>1.Thus, we have established the following result.Lemma 3.
The low-criminality equilibriumPl∗ of system (15) exists and it is locally asymptotically stable if R0>1 and R1<1. Otherwise, Pl∗ is unstable.It should be mentioned that the consequence of the above result is that whenPl∗ becomes unstable, two scenarios emerge: one for the case where the criminality is eliminated from population and P0 is stable and the other where new criminals could emerge and there is high-criminality prevalence in the population such that model (15) has a high-criminality equilibrium with coexistence of both offenders: C1≠0 and C2≠0. In this way, it is instructive to determine the possible interventions on β0 and β1 in order to reduce R0 and R1 below one, that is, to guarantee the conditions under which the criminality is eliminated, or at least having its incidence reduced.


#### 2.4.3. High-Criminality Equilibrium

In what follows, ifR1>1, such that the low-criminality equilibrium Pl∗ is unstable, system (15) has the positive high-criminality equilibrium, Ph∗=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗), where(26)S1∗=τ1C1μ+γ1+β1C2∗,S2∗=τ2C2∗μ+γ2+β2C2∗,D1∗=γ1μS1∗,D2∗=γ2μS2∗.Remembering from (21) that C1>0⇔R0>1, so Ph∗ exists if and only if R0>1. Now replacing the expressions for S1∗ and S2∗ in the sixth equation of system (15), the positive high-criminality equilibrium, which cannot be expressed cleanly in closed form, can then be obtained by solving for C2∗ the following expression:(27)η2C2∗2+η1C2∗+η0=0,where(28)η2=β1β2μ+a2,η1=μ+γ1τ2+μ+a2β2β1μ+γ2β2μ+γ1+μ+a2τ2+μ+a2-R1η0=μ+γ1μ+γ2μ+γ2+a21-R1.Since all the model’s parameters are nonnegative, it follows from (28) that the coefficient η2 is always positive and η0<0 for R1>1. Thus, it is clear that model (15) has a unique positive equilibrium Ph+ when R1>1 and R0>1. For η0=0 and η1<0, model (15) also has a unique positive equilibrium Ph+. Moreover, whenever R0>1, if R1=1, then η0=0 and η1<0 if only if (29)β2>τ2+μ+a2μ+γ2β1τ2μ+γ1,such that model (15) also has a unique positive equilibrium Ph∗. Finally, for η0>0, η1<0, and η12-4η2η0>0, that is, for R0>1 and R1<1, model (15) has two positive equilibria Ph+ and Ph-. Note that these equilibrium points cannot be studied in its closed form, so we carried out its local stability using numerical methods.Thus, the following result is then established.Theorem 4.
Model (15) has (i)
a unique positive equilibriumPh+ if η0<0⇔R1>1;(ii)
a unique positive equilibriumPh+ if η0=0 and η1<0;(iii)
two positive equilibria,Ph+andPh-,ifη0>0 and η1<0andη12-4η2η0>0;(iv)
no positive equilibrium, otherwise.As explored in model (1), model (15) also exhibits the phenomenon of backward bifurcation (Figure 8(a)) and breaking point (Figure 8(b)) when β1 and β2 are chosen as bifurcation parameters.Figure 8
The phenomena of backward bifurcation and breaking point, whenβ1 and β2 are chosen as a bifurcation parameter. (a) Backward bifurcation for β2=5 with β0 and β1 increasing. (b) Breaking point for β0=β1=5 with β2 increasing, where β2thr=121.2 and R0=1.139 and R1=0.464. The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). Parameters’ values used are as given in Table 3.
(a) (b)Figure8(a) shows the backward bifurcation for β2=5 and β0=8.4 (R0=2.39) with β1 increasing. Figure 8(b) shows the phenomenon of breaking point for β1=β0=5, with β2 increasing, where β2∗=121.2 and R0=1.139 and R1=0.464. As it should be expected, Figures 8(a) and 8(b) present the same results shown in Figures 2 and 4. In summary, if R0>1 and R1>1, model (15) has the locally asymptotically stable high-criminality equilibrium Ph+. Finally, if R0>1 and R1<1 and β2>β2thr, then there is coexistence of two locally asymptotically stable equilibria, Pl∗ and Ph+, such that the stability of these equilibrium points depends on the initial condition of system (15) (see Figure 9).Figure 9
Forβ1=2.0, β2=4.4 (β2>β2thr), R0=2.39, and R1=0.89. The other parameters’ values are given in Table 3. Profile of population of reoffenders (C2). (a) If the recidivism is low (or C2(0) small), then the equilibrium point Pl∗ is locally asymptotically stable. (b) If the recidivism is higher (or C2(0) large), then the equilibrium point Ph+ is locally asymptotically stable.
(a) (b)In contrast to model (1), when β0 is chosen as a bifurcation parameter, model (15) has the crime-free equilibrium P0 given by (16), which indicates the possibility of the forward bifurcation (Figure 10). If R0<1, then the crime-free equilibrium P0 of system (15) is locally asymptotically stable; if R0>1, P0 becomes unstable and Pl∗ is locally asymptotically stable if R1<1. Thus, the substitution of the assumption that the susceptible individuals (S0) can enter to crime only on their own (i.e., δ0) by the standard incidence (i.e., β1C1S0) in model (1) includes the forward bifurcation phenomenon to model (15).Figure 10
The phenomenon of forward bifurcation whenβ0 is chosen as a bifurcation parameter. The crime-free equilibrium P0 of system (15) is locally asymptotically stable if R0<1 and is unstable if R0>1. Parameters’ values used are as given in Table 3.


## 2.1. The Partially Contagious Criminality Model (PCCM) Formulation

Let the susceptible in core group population be divided into two categories:S0, those individuals who have a criminal propensity but have never been incarcerated, not criminally active but susceptible to crime, and Si, those individuals susceptible to criminal activities who were once incarcerated (S1) and those susceptible who were multiple-time incarcerated (Si,i=2,3,…,n) and became criminally active again. Similarly, let the incarcerated population be divided into two categories: C1, those who are first-time incarcerated, and Ci (i = 2,3,…,n), those who were multiple-time incarcerated at a given time. Finally, D0 represents those individuals who have a criminal propensity but desisted from criminal life either by their own or as a result of early interventions when discharged out of prison (desisting offenders/criminal desisters), and Di (i = 1,2,3,…,n) represents those individuals who were either first-time or multiple-time incarcerated but desisted from criminal behavior (desisting offenders/criminal desisters) due to variety of reasons.It is assumed that individuals susceptible to the crimeS0 move to either incarcerated (C1) or desisting (D0) class. The rate of initial participation in crime (δ0) at which individuals move from state S0 (not criminally active) to C1 (criminally active and incarcerated) is proportional to intrinsic criminal propensity of individuals. It is also assumed that special intervention programmes (γ1) may change their basic propensity traits and affect the decisions to engage in crime [24]. As a consequence, we should expect that certain interventions discourage participation in crime, resulting in desistance from criminal activity and reintegration back into society. Primary prevention is an attempt to reduce the risk of behaviors that potentially lead to incarceration. This point is one of the great interests here.The average length of the primary incarceration term is given by1/τ1; that is, the rate at which inmates move from state C1 to S1 (formerly incarcerated but not criminally active, i.e., ex-offenders) is τ1. In the same way, the average length of the multiple incarceration term is given by 1/τ2, with 1/τ2>1/τ1.It is worth mentioning that the representation of the cycles of criminal dynamics, with a focus on modeling the recidivism process (criminal careers), could be extended includingSi, Ci, and Di (i = 2,3,…,n). However, we do not intend to develop complex models that account for these cycles and we will explore two cycles only (i = 1,2).In addition, it is assumed that the recidivism prevention takes place during incarceration and after release back into society. Its purpose is to reduce the risk of an individual reoffending and eventually returning to the prison system. However, some individuals may resume criminal activity very soon after being released from prisons depending on having contact with those individuals still incarcerated and criminally active. It should be mentioned that Walsh and Graig (see [25]) indicated that recidivism is also called falling back into a previous criminal behavior.We also defineβi (i=1,2) as the rate of imprisonment, which captures the return to criminal activity of the individuals released from prison, such that β1 and β2 describe the rates with which individuals engage into criminal activity, depending on having had any contagious contact with those incarcerated criminals. They are analogous to the effective contact rate in infectious diseases models.Finally,μ is the natural mortality rate and ai (i=1,2) is the incarceration-related additional deaths rate (inmate mortality rate can be caused by illness, such as AIDS-related, suicide, accidental self-injury, execution, or any other unspecified cause). Moreover, since the model monitors human populations, all parameters are assumed as nonnegative. We also assumed homogeneous population without any differences in age or in crime type occurrences. In addition, we used the number of incarcerations as a proxy for reoffending.The flow diagram of the PCCM for two stages is depicted in Figure1. The variables and parameters are described in Tables 1 and 2, respectively.Table 1
Models variables and their biological meaning.
Variables Description S0 Individuals not criminally active but susceptible to crime (core group) C1 First-time incarcerated individuals S1 First-time ex-offenders individuals who are again susceptible to crime Di Individuals who desist from criminal behavior (i = 0,1,2) (desisting offenders/criminal desisters) C2 Individuals who were two or multiple times incarcerated (recidivists, reoffenders) S2 Second-time ex-offenders individuals and susceptible to crime (at least two incarcerations)Table 2
Models parameters and their biological meaning.
Parameters Description Λ Rate of recruitment of individuals into the core group δ0 Basic flow to the criminality (time-1) (criminal propensity) β1 Rate of cooptation of first-time ex-offenders (contact rate betweenC2 and S1) β2 Rate of cooptation of second-time ex-offenders (contact rate betweenC2 and S2) γ0 Rate of early desistance from crime γ1 Rate of desistance from crime when in the first cycle γ2 Rate of desistance from crime when in at least the second cycle μ Natural inmate mortality rate a1 Additional inmate mortality rate when inC1 a2 Additional inmate mortality rate when inC2 τ1 Release rate from incarceration of first-time offenders τ2 Release rate from incarceration of at least second-time offendersFigure 1
The flow diagram for the low-high criminality model (1).Combining the above derivations and assumptions, it follows that the model for transmission dynamics of criminality for two stages is given by the following nonlinear system of differential equations:(1)dS0dt=Λ-δ0+γ0+μS0dD0dt=γ0S0-μD0dC1dt=δ0S0-τ1+μ+a1C1dS1dt=τ1C1-γ1+μS1-β1C2S1dD1dt=γ1S1-μD1dC2dt=β1S1+β2S2C2-τ2+μ+a2C2dS2dt=τ2C2-β2C2S2-γ2+μS2dD2dt=γ2S2-μD2with generic initial conditions S0(0)≥0, D0(0)≥0, C1(0)≥0, S1(0)≥0, D1(0)≥0, C2(0)≥0, S2(0)≥0, and D2(0)≥0. By summing up the above equations, the total population size N(t) is variable with(2)dNdt=Λ-μN-a1C1-a2C2Thus, in the absence of additional inmate mortalities, that is, a1=a2=0, the population size evolves as an immigration model with natural mortality, that is, according to dN/dt=Λ-μN. This equation has a single equilibrium N=N(0)=Λ/μ for any initial value of N(0). Thus, in the long run, the population size settles to this constant value. It follows from (2) that limt→∞N(t)≤Λ/μ=N(0).The differential equation forN implies that solutions of (1), starting in the positive orthant R+8, either approach, enter, or remain in the subset R+8 defined by(3)Ω=S0,D0,C1,S1,D1,C2,S2,D2∈R+8:S0+D0+C1+S1+D1+C2+S2+D2≤N0.Thus it suffices to consider solutions in the regionΩ. Solutions of the initial value problem starting in Ω and defined by (1) exist and are unique on a maximal interval [26]. Since solutions remain bounded in the positively invariant region Ω, the initial value problem is then both mathematically and epidemiologically well posed [27]. Hence, it is sufficient to consider the dynamics of the flow generated by model (1) in Ω.


## 2.2. Analysis of the PCCM Model

In this section, system (1) is qualitatively analyzed to investigate the existence of its equilibria [28] and the control strategies of its dynamical behavior.From system (1), with the right-hand size equal to zero, it can be seen from the first five equations that the coordinates of the equilibrium point are given, respectively, by(4)S0=Λδ0+γ0+μ,D0=γ0μS0,C1=δ0τ1+μ+a1S0,S1=τ1δ0τ1+μ+a1μ+γ1+β1C2S0,D1=γ1μS1.Moreover, from seventh and eighth equations of system (1), we obtain(5)S2=τ2C2μ+γ2+β2C2,D2=γ2μS2.From the sixth equation of system (1), one gets(i)
C2=0,(ii)
C2≠0, which implies β1S1+β2S2-(τ2+μ+a2)=0.IfC2=0, model (1) has a low-criminality equilibrium Pl=(S0,D0,C1,S1,D1,0,0,0) which indicates the existence of offenders who are incarcerated only once in life (C1), given by(6)S0=Λδ0+γ0+μ,D0=γ0μS0,C1=δ0τ1+μ+a1S0,S1=τ1δ0τ1+μ+a1μ+γ1S0,D1=γ1μS1.To determine the stability of this equilibrium, the Jacobian of system (1) is computed and evaluated at Pl. Hence, the low-criminality equilibrium Pl=(S0,D0,C1,S1,D1,0,0,0) is locally asymptotically stable if R1∗<1, where R1∗, defined as Criminality Reproduction Number (CRN), is given by(7)R1∗=β1β1∗with(8)β1∗=τ1+μ+a1μ+γ1τ2+μ+a2τ1δ0S0.Note that, in analogy to the spread of infectious diseases models [29], the CRN, R1∗, represents the “average expected number of new offenders originated by a single persisting offender C2∗, whilst in a criminal career.” In other words, one person C2∗ who was incarcerated two times gets into contact with S1 susceptible individuals, just released from first imprisonment, successfully and induces R1∗ persons to commit crime. In other words, R1∗ is the average number of individuals who commit crime influenced by one inmate C2∗.ForC2≠0, that is, for β1S1+β2S2-(τ2+μ+a2)=0, replacing both expressions for S1 given by (4) and S2 given by (5), an expression for C2=C2∗>0 is obtained as(9)b2C2∗2+b1C2∗+b0=0,where(10)b2=β1β2μ+a2.b1=μ+γ1τ2+μ+a2β2β1μ+γ2β2μ+γ1+μ+a2τ2+μ+a2-R1∗,b0=μ+γ1μ+γ2τ2+μ+a21-R1∗,Let us now determine the conditions under which the quadratic equation (9) has positive real roots; that is, we search for the existence of multiple equilibria of system (1). However, the expression for the discriminant of the quadratic equation (9) is very complex, so we will analyze the signs of its coefficients to ensure the existence of real solutions. Thus, the conditions under which this equation has either one or two positive real roots can be determined, and these results translated into nontrivial equilibrium of system (1) which is biologically feasible (i.e., positive).Hence, substituting the positive real solutions of the quadratic equation (9) (i.e., positive values of C2∗) into the expressions in (5), model (1) has a high-criminality equilibrium, Ph=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗), where there is coexistence of both offenders C1 and C2∗, given by(11)S1∗=τ1δ0τ1+μ+a1+μ+γ1+β1C2∗S0D1∗=γ1μS1∗S2∗=τ2C2∗μ+γ2+β2C2∗D2∗=γ2μS2∗,with S0, D0, and C1 given by (6). Thus, the following result is then established.Theorem 1.
Model (1) has(i)
a unique positive equilibriumPh if b0<0⇔R1∗>1;(ii)
a unique positive equilibriumPh if b0=0 and b1<0;(iii)
two positive equilibria,Ph, if b0>0 and b1<0 and b12-4b2b0>0;(iv)
no positive equilibrium, otherwise.Since all model parameters are assumed as nonnegative, it follows from (10) that the coefficient b2 is always positive, b0<0 for R1∗>1 and b0>0 for R1∗<1. Thus, it is clear from Theorem 1 that model (1) has a unique positive equilibrium, Ph, when b0<0, that is, when R1∗>1 (case (i)).Now, forb0>0 and R1∗<1, the quadratic equation (9) has two positive solutions if b1<0 and b12-4b2b0>0 (case (iii)). Hence, assuming that (9) has two positive real solutions, let C2- and C2+ be the smaller and higher value of C2∗, respectively. Translating it into equilibrium of system (1), the question is to address what means the positive high-criminality equilibria Ph for R1∗<1. It is important to note that, in this case, system (1) can have two equilibria, which are biologically feasible, even though R1∗<1. This idea is explored more deeply below.It is instructive at this point to explore some qualitative features forR1∗=1 and R1∗>1. Firstly, for R1∗=1, it follows that b0=0; the quadratic equation (9) has either a unique positive root (if b1<0) or no positive root (if b1>0). In other words, for R1∗=1 and b1<0, model (1) has a unique positive high-criminality equilibrium given by Ph (case (ii)).Moreover, note that whenβ2=0, we have b2=0 and b1=(μ+γ2)(τ2+μ+a2)β1. Thus, if R1∗>1, then(12)C2∗β2=0=μ+γ1β1R1∗-1,and model (1) has a unique positive high-criminality equilibrium, Ph, for β2=0.Forβ2→∞, we have b1=(μ+γ1)(τ2+μ+a2)β2(μ+a2)/(τ2+μ+a2)-R1∗ and b2=β1β2(μ+a2) with b0≪b1 and b0≪b2 such that(13)C2∗β2→∞=μ+γ1τ2+μ+a2β1μ+a2R1∗-μ+a2τ2+μ+a2,and C2∗>0 if only if R1∗>(μ+a2)/(τ2+μ+a2). Thus, model (1) has a unique high-criminality equilibrium Ph if β2→∞.Unfortunately, this high-criminality equilibrium cannot be studied from its closed form, so we carried out its local stability using numerical methods. The results are provided next.Finally, if (i), (ii), and (iii) do not occur, then there are no endemic equilibria for system (1).In what follows, model (1) admits two realistic scenarios: the best-case scenario, where offenders were incarcerated once and it is still possible to fight crime and recidivism is given by the low-criminality equilibrium Pl=(S0,D0,C1,S1,D1,0,0,0), and the worst-case scenario, where the offenders were incarcerated at least once (recidivism), which can potentially lead to increased criminal activities, given by the high-criminality equilibrium Ph=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗).Having found the scenarios in which there exist the equilibria for system (1), it is instructive to analyze whether or not these equilibria are stable under any of these scenarios. Moreover, together with the CRN, R1∗ (see (7)), and the parameter β2, we see that each scenario can be used as a check for the existence and the stability of the equilibria. Another crucial question is which of the two incidence rates β1 and β2 is more likely to affect the criminal dynamics in general and criminal contacts in particular. These points are of great interest here and we then explore how our model behaves under control strategies when R1∗<1. We fix β2 and explore the system behavior by varying β1. We also explore what happens when we fix β1 while β2 is varied. These control measures are designed to fight recidivism. In both cases, the recidivism will depend upon the initial sizes of the subpopulation C2 and indicate the possibility of backward bifurcation.Firstly, it is worth remembering that the quantityβ1 measures the average number of new contacts generated by a typical incarcerated individual C2 with those susceptible individuals S1 who are ex-offenders and can become susceptible to the crime again. β2, in turn, measures the average number of new contacts generated by C2 with those individuals S2 who were reoffenders (recidivist behavior) and are susceptible to the crime once again.In this way, there are two groups of offenders: those who are under much lower risk, most of whom will go to prison once and not come back (ex-offenders), and those who repeatedly do crimes and come back multiple times to the prison (reoffenders). The strong implication of the findings is that individuals who are incarcerated are extremely likely to reoffend once they are free.Note that when system (1) has a small influx of reoffenders (C2), it does not generate high criminality rates and, for R1∗<1, it is still possible to minimize the spreading of crime (best-case scenario). We will see that, in this case, the corresponding low-criminality equilibrium Pl could be locally asymptotically stable. On the other hand, the criminality will persist and increase if R1∗>1 (worst-case scenario). In this case, the corresponding high-criminality equilibrium Ph could be locally asymptotically stable. This phenomenon, where the possibility of fighting the spread of the crime is lost and the criminality is potentially active, that is, where Pl loses its stability and a unique Ph appears as R1∗ increases through one, is known as forward bifurcation in epidemiology.For models that exhibit this type of bifurcation, the requirementR1∗<1 is necessary and sufficient for the high-criminality elimination. In contrast, other models undergo another type of bifurcation, known as “backward bifurcation” in epidemiology, where two equilibria Ph+ and Ph- coexist with the low-criminality equilibrium, Pl; that is, there are three steady states when R1∗ is immediately less than one. Thus, the requirement R1∗<1 is necessary but is not sufficient to fight the spread of crime.


### 2.2.1. Analytic Strategy:β2 Fixed with β1 Increasing

The physical implication of backward bifurcation is that theC2 population can engage in a high criminality level even when R1∗ crosses unity downwards. In other words, in the presence of recidivism, decreasing R1∗ below one is not a sufficient condition to make the criminality level decrease. This has very important consequence for crime control, as will be discussed later in this paper.To check the possibility of backward bifurcation in model (1), it is necessary to know other subthreshold, and we will refer to this limit point, as expressed on the R1∗ scale, as R1thr. Hence, for R1∗<R1thr, system (1) will present only the locally asymptotically stable low-criminality equilibrium Pl. For Rthr∗<R1∗<1, system (1) will present the locally asymptotically stable low-criminality equilibrium Pl plus two positive high-criminality equilibria, Ph+ and Ph-, which will correspond to the solutions of (9): C2+, the higher solution, which corresponds to the stable equilibrium, and C2-, the smaller solution, which corresponds to the unstable equilibrium.Although the critical value of the bifurcation could not be found analytically due to the high dimension of system (1), this task can be performed numerically. In this way, our simulations show that there exists a critical value Rthr∗<R1∗=1, where model (1) undergoes backward bifurcation.A schematic diagram of the backward and the forward bifurcations for system (1) is given in Figure 2, where β1 is chosen as a bifurcation parameter; that is, we fixed β2, whereas β1 increases.Figure 2
Backward and forward bifurcations diagram for proportion of reoffenders (C2∗) for model (1), where β1 is chosen as a bifurcation parameter (β2 fixed and β1 increasing). The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). Backward bifurcation for (a) β2=1000, β1=5.371, and R1thr=0.347; (b) β2=500, β1=5.845, and R1thr=0.378; and (c) β2=50, β1=11, and R1thr=0.712. Forward bifurcation for (a) β2=17.7, β1=15.45, and R1thr=R1∗=1 and (b) β2=10.0, β1=15.45, and R1thr<R1∗=1. Parameters’ values used are as given in Table 3.Table 3
Baseline values for model (1).
Variable Description Λ 0.015  (years-1) δ0 0.02  (years-1) β1 v a r i a b l e  (years-1) β2 v a r i a b l e  (years-1) γ0 0.2  (years-1) γ1 0.1  (years-1) γ2 0.08  (years-1) μ0 0.015  (years-1) a1 0.03  (years-1) a2 0.025  (years-1) τ1 0.2  (years-1) τ2 0.1  (years-1)Figure2 shows the profile of the proportion of both reoffenders C2+ (solid curve) and C2- (dashed curve) as a function of R1∗ with decreasing values of β2=1000;500;50;17.7;10(year-1) as R1∗ increases (i.e., as β1 increases). The solid curve stands for the stable high-criminality equilibrium, C2+, and the dashed curve stands for the unstable high-criminality equilibrium, C2-. For β2=1000 and β1=5.371, one has R1thr=0.347. Thus, for 0.347<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. Similarly, for β2=500 and β1=5.845, one gets R1thr=0.378. Thus, for 0.378<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. For β2=50 and β1=11, one gets R1thr=0.712. Thus, for 0.712<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. Finally, for β2=17.7 and β1=15.45 and for β2=10.0 and β1=15.45, we have R1thr≤R1∗=1, and the model has one positive high-criminality equilibrium for R1∗>1 and no positive equilibrium for R1∗<1. Consequently, model (1) exhibits the forward bifurcation at R1∗=1. As it should be expected, it can be seen in Figure 2 that R1thr increases with increasing β1 and a greater reduction in recidivism prevalence is recorded for decreasing values of β2.As stated earlier, the physical significance of the phenomenon of backward bifurcation is that the classical requirement ofR1∗<1 is no longer sufficient for avoiding the recidivism prevalence, as it is for the forward bifurcation. In such a scenario, the recidivism would depend on the initial sizes of the subpopulation C2 of the model. That is, the presence of backward bifurcation in model (1) suggests that the possibility of avoiding the recidivism event when R1∗<1 could be dependent on the initial sizes of the subpopulation C2. This scenario is illustrated numerically in Figure 3. Thus, if the recidivism is low, then C2(0) is small, such that the low-criminality equilibrium Pl is stable (Figure 3(a)). In contrast, if the recidivism is higher, then C2(0) is large, so the high-criminality equilibrium Ph is stable (Figure 3(b)). Therefore, the stability of these equilibrium points depends on the initial condition of system (1). This clearly indicates the coexistence of two locally asymptotically stable equilibria when R1∗<1, confirming that model (1) undergoes the phenomenon of backward bifurcation with one stable high-criminality equilibrium Ph+ (higher, solid curve in Figure 2), one unstable high-criminality equilibrium Ph- (lowest dashed curve in Figure 2), and one low-criminality equilibrium Pl.Figure 3
Forβ1=11, β2=50, and R1thr<R1∗<1. The other parameters’ values are given in Table 3. Profile of the population of reoffenders (C2). (a) If the recidivism is low (or C2(0) small), then the equilibrium point Pl is locally asymptotically stable; (b) if the recidivism is higher (or C2(0) large), then the equilibrium point Ph is locally asymptotically stable.
(a) (b)


### 2.2.2. Analytic Strategy:β1 Fixed with β2 Increasing

Alternatively, from now on we explore the implications of the parameterβ2 on the criminal dynamics. It is worth mentioning that, from expression (10), it is easy to verify that b1<0 if and only if(14)β2>β1μ+γ2μ+γ1R1∗-μ+a2/τ2+μ+a2>0,and thus, in such case, system (1) has two positive high-criminality equilibria (case (iii), see Theorem 1.) for b0>0, b12-4b2b0>0, and R1∗>(μ+a2)/(τ2+μ+a2).A schematic diagram of this bifurcation phenomenon for system (1) is given in Figure 4, where β2 is chosen as a bifurcation parameter. As mentioned previously, this bifurcation phenomenon of model (1) is only illustrated numerically.Figure 4
Breaking point and forward bifurcation diagrams for proportion of reoffenders (C2∗) for model (1), where β2 is chosen as a bifurcation parameter (β1 fixed with β2 increasing). The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). The phenomenon of the breaking point for (a) β1=11, R1∗=0.71, and β2thr=50; (b) β1=7.5, R1∗=0.486, and β2thr=146.8; and (c) β1=5.845, R1∗=0.378, and β2thr=499; forward bifurcation for β1=15.449, R1∗=1, and β2thr=17.7. For β1=30 and R1∗=1.94, β2=0 such that system (1) has a unique positive high-criminality equilibrium Ph. Parameters’ values used are as given in Table 3.Figure4 shows the profile of the proportion of both reoffenders C2+ (solid curve) and C2- (dashed curve) as a function of β2 with decreasing values of β1=30;15.449;11;7.5;5.845(year-1) as β2 increases. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. Note that, for β1=30 and R1∗>1, β2=0 such that system (1) has a unique positive high-criminality equilibrium (see case (i) of Theorem 1 and (12)).It should be also noticed in Figure4 that R1∗ increases with decreasing β2, such that greater reduction in recidivism prevalence is recorded for decreasing values of β1. One aspect to differentiate both parameters β1 and β2 is the way that they account for the transition to criminal activity.In the low-criminality scenario, the only way someone becomes criminally active is on his/her own, without interacting with another person who is already criminally active. This restriction is necessary in order to illustrate the dynamics of a system which would lead to low-criminality equilibrium. If new criminal could only emerge on his/her own, then reducing the number of individuals who are not criminally active (core group) through either intervention or spontaneous desistance could lead to the lowest criminal scenario. For this, we calculated the threshold between the low-criminality equilibrium and the high-criminality equilibrium. Therefore, forR1∗<R1thr, system (1) presented the locally asymptotically stable low-criminality equilibrium Pl. Although not a realistic possibility, understanding the intrinsic criminal propensity that leads to such kind of deviant behavior is a necessary step for understanding the high-criminality prevalence in the model, which will be presented in the next section.In addition, in the high-criminality scenario, the way someone can become criminally active is to interact with another person who is already criminally active. Thus, if new reoffenders could only emerge by interaction between reoffenders (C2∗) and those second-time offenders susceptible to the crime (S2∗) at a rate β2, then reducing the number of contacts between C2∗ and S1, that is, reducing β1, could lead to the low-criminality scenario if the proportion of reoffenders C2∗ is small (see Figure 3). Hence, to reach the low-criminality equilibrium, a community needs to reduce β1. However, if β1 is large, then the proportion of reoffenders C2∗, as well as S2∗, increases, which leads to higher crime prevalence, such that the high-criminality scenario emerges.Figure5(a) shows the long-term behavior of ln⁡β2thr plotted versus β1. The threshold β2thr is expressed as a minimum value where system (1) has two positive equilibrium points. Thus, for each fixed β1, there is a corresponding unique threshold β2thr. For one such example (see Figure 5(b)), β1=11, R1∗=0.71, and β2thr=50 (or ln⁡50=3.91) are the minimum values where system (1) has two positive value. For β2>β2thr, we have C2+>0 and C2->0. In this case, system (1) has a stable equilibrium point, C2+ (the solid curve in Figure 5(b)), and an unstable equilibrium point, C2- (the dashed curve in Figure 5(b)). On the other hand, for β2<β2thr, we have C2+=C2-=0, which means that there are no positive equilibrium points for system (1). Moreover, for β1thr=15.4492917, we have R1∗=1 and β2thr=17.867 (or ln⁡17.867=2.883) such that, for β2<β2thr, one has C2+=0 and C2-<0. Otherwise, for β2>β2thr, one has C2+>0 and C2-=0 (see Figure 6).Figure 5
(a) The long-term behavior ofln⁡β2thr plotted versus β1. (b) For β1=11, R1∗=0.71, and ln⁡β2=3.91, system (1) has two positive equilibrium points. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. The other parameters’ values are given in Table 3.Figure 6
Long-term physical outcomeC2∗ plotted versus ln⁡β2. For β1thr=15.449, R1∗=1, and β2>β2thr, system (1) has one positive equilibrium point. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. The other parameters’ values are given in Table 3.


## 2.2.1. Analytic Strategy:β2 Fixed with β1 Increasing

The physical implication of backward bifurcation is that theC2 population can engage in a high criminality level even when R1∗ crosses unity downwards. In other words, in the presence of recidivism, decreasing R1∗ below one is not a sufficient condition to make the criminality level decrease. This has very important consequence for crime control, as will be discussed later in this paper.To check the possibility of backward bifurcation in model (1), it is necessary to know other subthreshold, and we will refer to this limit point, as expressed on the R1∗ scale, as R1thr. Hence, for R1∗<R1thr, system (1) will present only the locally asymptotically stable low-criminality equilibrium Pl. For Rthr∗<R1∗<1, system (1) will present the locally asymptotically stable low-criminality equilibrium Pl plus two positive high-criminality equilibria, Ph+ and Ph-, which will correspond to the solutions of (9): C2+, the higher solution, which corresponds to the stable equilibrium, and C2-, the smaller solution, which corresponds to the unstable equilibrium.Although the critical value of the bifurcation could not be found analytically due to the high dimension of system (1), this task can be performed numerically. In this way, our simulations show that there exists a critical value Rthr∗<R1∗=1, where model (1) undergoes backward bifurcation.A schematic diagram of the backward and the forward bifurcations for system (1) is given in Figure 2, where β1 is chosen as a bifurcation parameter; that is, we fixed β2, whereas β1 increases.Figure 2
Backward and forward bifurcations diagram for proportion of reoffenders (C2∗) for model (1), where β1 is chosen as a bifurcation parameter (β2 fixed and β1 increasing). The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). Backward bifurcation for (a) β2=1000, β1=5.371, and R1thr=0.347; (b) β2=500, β1=5.845, and R1thr=0.378; and (c) β2=50, β1=11, and R1thr=0.712. Forward bifurcation for (a) β2=17.7, β1=15.45, and R1thr=R1∗=1 and (b) β2=10.0, β1=15.45, and R1thr<R1∗=1. Parameters’ values used are as given in Table 3.Table 3
Baseline values for model (1).
Variable Description Λ 0.015  (years-1) δ0 0.02  (years-1) β1 v a r i a b l e  (years-1) β2 v a r i a b l e  (years-1) γ0 0.2  (years-1) γ1 0.1  (years-1) γ2 0.08  (years-1) μ0 0.015  (years-1) a1 0.03  (years-1) a2 0.025  (years-1) τ1 0.2  (years-1) τ2 0.1  (years-1)Figure2 shows the profile of the proportion of both reoffenders C2+ (solid curve) and C2- (dashed curve) as a function of R1∗ with decreasing values of β2=1000;500;50;17.7;10(year-1) as R1∗ increases (i.e., as β1 increases). The solid curve stands for the stable high-criminality equilibrium, C2+, and the dashed curve stands for the unstable high-criminality equilibrium, C2-. For β2=1000 and β1=5.371, one has R1thr=0.347. Thus, for 0.347<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. Similarly, for β2=500 and β1=5.845, one gets R1thr=0.378. Thus, for 0.378<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. For β2=50 and β1=11, one gets R1thr=0.712. Thus, for 0.712<R1∗<1, model (1) has two positive high-criminality equilibria, while for R1∗>1, model (1) has one positive high-criminality equilibrium. Finally, for β2=17.7 and β1=15.45 and for β2=10.0 and β1=15.45, we have R1thr≤R1∗=1, and the model has one positive high-criminality equilibrium for R1∗>1 and no positive equilibrium for R1∗<1. Consequently, model (1) exhibits the forward bifurcation at R1∗=1. As it should be expected, it can be seen in Figure 2 that R1thr increases with increasing β1 and a greater reduction in recidivism prevalence is recorded for decreasing values of β2.As stated earlier, the physical significance of the phenomenon of backward bifurcation is that the classical requirement ofR1∗<1 is no longer sufficient for avoiding the recidivism prevalence, as it is for the forward bifurcation. In such a scenario, the recidivism would depend on the initial sizes of the subpopulation C2 of the model. That is, the presence of backward bifurcation in model (1) suggests that the possibility of avoiding the recidivism event when R1∗<1 could be dependent on the initial sizes of the subpopulation C2. This scenario is illustrated numerically in Figure 3. Thus, if the recidivism is low, then C2(0) is small, such that the low-criminality equilibrium Pl is stable (Figure 3(a)). In contrast, if the recidivism is higher, then C2(0) is large, so the high-criminality equilibrium Ph is stable (Figure 3(b)). Therefore, the stability of these equilibrium points depends on the initial condition of system (1). This clearly indicates the coexistence of two locally asymptotically stable equilibria when R1∗<1, confirming that model (1) undergoes the phenomenon of backward bifurcation with one stable high-criminality equilibrium Ph+ (higher, solid curve in Figure 2), one unstable high-criminality equilibrium Ph- (lowest dashed curve in Figure 2), and one low-criminality equilibrium Pl.Figure 3
Forβ1=11, β2=50, and R1thr<R1∗<1. The other parameters’ values are given in Table 3. Profile of the population of reoffenders (C2). (a) If the recidivism is low (or C2(0) small), then the equilibrium point Pl is locally asymptotically stable; (b) if the recidivism is higher (or C2(0) large), then the equilibrium point Ph is locally asymptotically stable.
(a) (b)


## 2.2.2. Analytic Strategy:β1 Fixed with β2 Increasing

Alternatively, from now on we explore the implications of the parameterβ2 on the criminal dynamics. It is worth mentioning that, from expression (10), it is easy to verify that b1<0 if and only if(14)β2>β1μ+γ2μ+γ1R1∗-μ+a2/τ2+μ+a2>0,and thus, in such case, system (1) has two positive high-criminality equilibria (case (iii), see Theorem 1.) for b0>0, b12-4b2b0>0, and R1∗>(μ+a2)/(τ2+μ+a2).A schematic diagram of this bifurcation phenomenon for system (1) is given in Figure 4, where β2 is chosen as a bifurcation parameter. As mentioned previously, this bifurcation phenomenon of model (1) is only illustrated numerically.Figure 4
Breaking point and forward bifurcation diagrams for proportion of reoffenders (C2∗) for model (1), where β2 is chosen as a bifurcation parameter (β1 fixed with β2 increasing). The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). The phenomenon of the breaking point for (a) β1=11, R1∗=0.71, and β2thr=50; (b) β1=7.5, R1∗=0.486, and β2thr=146.8; and (c) β1=5.845, R1∗=0.378, and β2thr=499; forward bifurcation for β1=15.449, R1∗=1, and β2thr=17.7. For β1=30 and R1∗=1.94, β2=0 such that system (1) has a unique positive high-criminality equilibrium Ph. Parameters’ values used are as given in Table 3.Figure4 shows the profile of the proportion of both reoffenders C2+ (solid curve) and C2- (dashed curve) as a function of β2 with decreasing values of β1=30;15.449;11;7.5;5.845(year-1) as β2 increases. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. Note that, for β1=30 and R1∗>1, β2=0 such that system (1) has a unique positive high-criminality equilibrium (see case (i) of Theorem 1 and (12)).It should be also noticed in Figure4 that R1∗ increases with decreasing β2, such that greater reduction in recidivism prevalence is recorded for decreasing values of β1. One aspect to differentiate both parameters β1 and β2 is the way that they account for the transition to criminal activity.In the low-criminality scenario, the only way someone becomes criminally active is on his/her own, without interacting with another person who is already criminally active. This restriction is necessary in order to illustrate the dynamics of a system which would lead to low-criminality equilibrium. If new criminal could only emerge on his/her own, then reducing the number of individuals who are not criminally active (core group) through either intervention or spontaneous desistance could lead to the lowest criminal scenario. For this, we calculated the threshold between the low-criminality equilibrium and the high-criminality equilibrium. Therefore, forR1∗<R1thr, system (1) presented the locally asymptotically stable low-criminality equilibrium Pl. Although not a realistic possibility, understanding the intrinsic criminal propensity that leads to such kind of deviant behavior is a necessary step for understanding the high-criminality prevalence in the model, which will be presented in the next section.In addition, in the high-criminality scenario, the way someone can become criminally active is to interact with another person who is already criminally active. Thus, if new reoffenders could only emerge by interaction between reoffenders (C2∗) and those second-time offenders susceptible to the crime (S2∗) at a rate β2, then reducing the number of contacts between C2∗ and S1, that is, reducing β1, could lead to the low-criminality scenario if the proportion of reoffenders C2∗ is small (see Figure 3). Hence, to reach the low-criminality equilibrium, a community needs to reduce β1. However, if β1 is large, then the proportion of reoffenders C2∗, as well as S2∗, increases, which leads to higher crime prevalence, such that the high-criminality scenario emerges.Figure5(a) shows the long-term behavior of ln⁡β2thr plotted versus β1. The threshold β2thr is expressed as a minimum value where system (1) has two positive equilibrium points. Thus, for each fixed β1, there is a corresponding unique threshold β2thr. For one such example (see Figure 5(b)), β1=11, R1∗=0.71, and β2thr=50 (or ln⁡50=3.91) are the minimum values where system (1) has two positive value. For β2>β2thr, we have C2+>0 and C2->0. In this case, system (1) has a stable equilibrium point, C2+ (the solid curve in Figure 5(b)), and an unstable equilibrium point, C2- (the dashed curve in Figure 5(b)). On the other hand, for β2<β2thr, we have C2+=C2-=0, which means that there are no positive equilibrium points for system (1). Moreover, for β1thr=15.4492917, we have R1∗=1 and β2thr=17.867 (or ln⁡17.867=2.883) such that, for β2<β2thr, one has C2+=0 and C2-<0. Otherwise, for β2>β2thr, one has C2+>0 and C2-=0 (see Figure 6).Figure 5
(a) The long-term behavior ofln⁡β2thr plotted versus β1. (b) For β1=11, R1∗=0.71, and ln⁡β2=3.91, system (1) has two positive equilibrium points. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. The other parameters’ values are given in Table 3.Figure 6
Long-term physical outcomeC2∗ plotted versus ln⁡β2. For β1thr=15.449, R1∗=1, and β2>β2thr, system (1) has one positive equilibrium point. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. The other parameters’ values are given in Table 3.


## 2.3. The Full Contagion Criminal Model (FCCM) Formulation

In this section, we relax the assumption made in the PCCM model (1) that the susceptible individuals (S0) can enter to crime only on their own, so we make δ0=0. Hence, in the modified model, the flow of the susceptible individuals S0 into the criminal activity depends only on the contact with those individuals who are incarcerated, C1. We also define βi (i = 0,1,2) as the rate of imprisonment; it captures the return to criminal activity of those released from prison, such that β0, β1, and β2 are the flows into criminal activity which depend on having had contact with those first-time and second-time incarcerated individuals (C1 and C2), respectively. Following the idea of the previous model (1), here βi (i = 0,1,2) is analogous to the effective contact rate in infectious diseases model. This derivation adopts a standard incidence formulation, where the contact rate is assumed to be constant, unlike the case of the mass action formulation, where the contact rate depends on the size of the total population (see, e.g., [27, 30], for detailed derivation of these incidences’ functions).The flow diagram of the FCCM is depicted in Figure7. The variables and parameters’ values are given in Table 3, except for β0 that is the contact rate between C1 and S0 (cooptation rate).Figure 7
The flow diagram for the free-low-high criminality model (15).The model is represented by the following nonlinear system of differential equations:(15)dS0dt=Λ-β0C1S0-γ0+μS0dD0dt=γ0S0-μD0dC1dt=β0C1S0-τ1+μ+a1C1dS1dt=τ1C1-γ1+μS1-β1C2S1dD1dt=γ1S1-μD1dC2dt=β1S1+β2S2C2-τ2+μ+a2C2dS2dt=τ2C2-β2C2S2-γ2+μS2dD2dt=γ2S2-μD2


## 2.4. The Existence and Local Stability of Equilibria

### 2.4.1. Crime-Free Equilibrium

In the absence of crime, that is, forC1=C2=0, model (15) has a crime-free equilibrium P0=(S00,D00,0,0,0,0,0,0) which is obtained by setting the right-hand sides of system (15) to zero, where(16)S00=Λμ+γ0,D0=γ0μS00.To analyze the local stability of this equilibrium, the Jacobian of system (15) is computed and evaluated at P0, which is locally asymptotically stable if the real parts of the eigenvalues of the Jacobian matrix are all negative.Thus, the local stability of the crime-free equilibriumP0 is governed by the Jacobian matrix(17)MP0=AP000BP0,where(18)AP0=-μ+γ0-β0S000β0S00-τ1+μ+a1,(19)BP0=-μ+γ1000-τ2+μ+a200τ2-μ+γ2.It is easy to verify that the three eigenvalues of matrix (19) are always negative. In the same way, it is also straightforward to verify that one of the eigenvalues of matrix (18) is always negative, while the other is negative whenever(20)R0=β0τ1+μ+a1S00<1.Hence, all the eigenvalues of matrix (17) are negative or have negative real parts if and only if R0<1. In summary, the crime-free equilibrium P0 of system (15) is locally asymptotically stable if the Basic Criminality Reproduction Number R0<1. Therefore, we have established the following result.Lemma 2.
The crime-free equilibriumP0 of system (15) is locally asymptotically stable if R0<1 and is unstable if R0>1.


### 2.4.2. Low-Criminality Equilibrium

In the absence of the reoffenders, that is, forC2=0, model (15) has a low-criminality equilibrium Pl∗=(S0,D0,C1,S1,D1,0,0,0), where(21)S0=τ1+μ+a1β0D0=γ0μS0C1=Λβ0-μ+γ0τ1+μ+a1τ1+μ+a1β0=μ+γ0β0R0-1S1=τ1C1μ+γ1D1=γ1μS1.Note thatPl∗ exists if C1>0, that is, if R0>1. Now consider the resulting model (15). The local stability of the low-criminality equilibrium Pl, which is examined by linearizing system (15) around Pl, is governed by the Jacobian matrix(22)MPl=M11-β0S0000β0C1M220000τ1M33-β1S10000M440000τ2M55,with M11=-β0C1-(μ+γ0); M22=β0S0-(τ1+μ+a1); M33=-(μ+γ1); M44=β1S1-(τ2+μ+a2); and M55=-(μ+γ2).The eigenvalues of matrix (22) are λ1=M55, λ2=M44, λ3=M33, and the roots of(23)ϕ2+z1ϕ+z0=0,with z1=Λβ0/(τ1+μ+a1)>0 and z0=Λβ0-(μ+γ0)(τ1+μ+a1)>0(⇔R0>1).The eigenvaluesλ1 and λ3 are real negative from definition of the parameters μ, γ1, and γ2. The eigenvalue λ2 is negative if(24)β1<τ2+μ+a2S1=β0τ2+μ+a2μ+γ1τ1μ+γ0R0-1 or, equivalently, if(25)R1=β1τ1μ+γ0R0-1τ2+μ+a2μ+γ1β0<1Finally, the eigenvaluesλ4 and λ5 are real negative if z0>0 and z1>0 by applying the Routh-Hurwitz criteria [31, 32] on polynomial (23). In this sense, z0>0 and z1>0 if and only if R0>1.Thus, we have established the following result.Lemma 3.
The low-criminality equilibriumPl∗ of system (15) exists and it is locally asymptotically stable if R0>1 and R1<1. Otherwise, Pl∗ is unstable.It should be mentioned that the consequence of the above result is that whenPl∗ becomes unstable, two scenarios emerge: one for the case where the criminality is eliminated from population and P0 is stable and the other where new criminals could emerge and there is high-criminality prevalence in the population such that model (15) has a high-criminality equilibrium with coexistence of both offenders: C1≠0 and C2≠0. In this way, it is instructive to determine the possible interventions on β0 and β1 in order to reduce R0 and R1 below one, that is, to guarantee the conditions under which the criminality is eliminated, or at least having its incidence reduced.


### 2.4.3. High-Criminality Equilibrium

In what follows, ifR1>1, such that the low-criminality equilibrium Pl∗ is unstable, system (15) has the positive high-criminality equilibrium, Ph∗=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗), where(26)S1∗=τ1C1μ+γ1+β1C2∗,S2∗=τ2C2∗μ+γ2+β2C2∗,D1∗=γ1μS1∗,D2∗=γ2μS2∗.Remembering from (21) that C1>0⇔R0>1, so Ph∗ exists if and only if R0>1. Now replacing the expressions for S1∗ and S2∗ in the sixth equation of system (15), the positive high-criminality equilibrium, which cannot be expressed cleanly in closed form, can then be obtained by solving for C2∗ the following expression:(27)η2C2∗2+η1C2∗+η0=0,where(28)η2=β1β2μ+a2,η1=μ+γ1τ2+μ+a2β2β1μ+γ2β2μ+γ1+μ+a2τ2+μ+a2-R1η0=μ+γ1μ+γ2μ+γ2+a21-R1.Since all the model’s parameters are nonnegative, it follows from (28) that the coefficient η2 is always positive and η0<0 for R1>1. Thus, it is clear that model (15) has a unique positive equilibrium Ph+ when R1>1 and R0>1. For η0=0 and η1<0, model (15) also has a unique positive equilibrium Ph+. Moreover, whenever R0>1, if R1=1, then η0=0 and η1<0 if only if (29)β2>τ2+μ+a2μ+γ2β1τ2μ+γ1,such that model (15) also has a unique positive equilibrium Ph∗. Finally, for η0>0, η1<0, and η12-4η2η0>0, that is, for R0>1 and R1<1, model (15) has two positive equilibria Ph+ and Ph-. Note that these equilibrium points cannot be studied in its closed form, so we carried out its local stability using numerical methods.Thus, the following result is then established.Theorem 4.
Model (15) has (i)
a unique positive equilibriumPh+ if η0<0⇔R1>1;(ii)
a unique positive equilibriumPh+ if η0=0 and η1<0;(iii)
two positive equilibria,Ph+andPh-,ifη0>0 and η1<0andη12-4η2η0>0;(iv)
no positive equilibrium, otherwise.As explored in model (1), model (15) also exhibits the phenomenon of backward bifurcation (Figure 8(a)) and breaking point (Figure 8(b)) when β1 and β2 are chosen as bifurcation parameters.Figure 8
The phenomena of backward bifurcation and breaking point, whenβ1 and β2 are chosen as a bifurcation parameter. (a) Backward bifurcation for β2=5 with β0 and β1 increasing. (b) Breaking point for β0=β1=5 with β2 increasing, where β2thr=121.2 and R0=1.139 and R1=0.464. The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). Parameters’ values used are as given in Table 3.
(a) (b)Figure8(a) shows the backward bifurcation for β2=5 and β0=8.4 (R0=2.39) with β1 increasing. Figure 8(b) shows the phenomenon of breaking point for β1=β0=5, with β2 increasing, where β2∗=121.2 and R0=1.139 and R1=0.464. As it should be expected, Figures 8(a) and 8(b) present the same results shown in Figures 2 and 4. In summary, if R0>1 and R1>1, model (15) has the locally asymptotically stable high-criminality equilibrium Ph+. Finally, if R0>1 and R1<1 and β2>β2thr, then there is coexistence of two locally asymptotically stable equilibria, Pl∗ and Ph+, such that the stability of these equilibrium points depends on the initial condition of system (15) (see Figure 9).Figure 9
Forβ1=2.0, β2=4.4 (β2>β2thr), R0=2.39, and R1=0.89. The other parameters’ values are given in Table 3. Profile of population of reoffenders (C2). (a) If the recidivism is low (or C2(0) small), then the equilibrium point Pl∗ is locally asymptotically stable. (b) If the recidivism is higher (or C2(0) large), then the equilibrium point Ph+ is locally asymptotically stable.
(a) (b)In contrast to model (1), when β0 is chosen as a bifurcation parameter, model (15) has the crime-free equilibrium P0 given by (16), which indicates the possibility of the forward bifurcation (Figure 10). If R0<1, then the crime-free equilibrium P0 of system (15) is locally asymptotically stable; if R0>1, P0 becomes unstable and Pl∗ is locally asymptotically stable if R1<1. Thus, the substitution of the assumption that the susceptible individuals (S0) can enter to crime only on their own (i.e., δ0) by the standard incidence (i.e., β1C1S0) in model (1) includes the forward bifurcation phenomenon to model (15).Figure 10
The phenomenon of forward bifurcation whenβ0 is chosen as a bifurcation parameter. The crime-free equilibrium P0 of system (15) is locally asymptotically stable if R0<1 and is unstable if R0>1. Parameters’ values used are as given in Table 3.


## 2.4.1. Crime-Free Equilibrium

In the absence of crime, that is, forC1=C2=0, model (15) has a crime-free equilibrium P0=(S00,D00,0,0,0,0,0,0) which is obtained by setting the right-hand sides of system (15) to zero, where(16)S00=Λμ+γ0,D0=γ0μS00.To analyze the local stability of this equilibrium, the Jacobian of system (15) is computed and evaluated at P0, which is locally asymptotically stable if the real parts of the eigenvalues of the Jacobian matrix are all negative.Thus, the local stability of the crime-free equilibriumP0 is governed by the Jacobian matrix(17)MP0=AP000BP0,where(18)AP0=-μ+γ0-β0S000β0S00-τ1+μ+a1,(19)BP0=-μ+γ1000-τ2+μ+a200τ2-μ+γ2.It is easy to verify that the three eigenvalues of matrix (19) are always negative. In the same way, it is also straightforward to verify that one of the eigenvalues of matrix (18) is always negative, while the other is negative whenever(20)R0=β0τ1+μ+a1S00<1.Hence, all the eigenvalues of matrix (17) are negative or have negative real parts if and only if R0<1. In summary, the crime-free equilibrium P0 of system (15) is locally asymptotically stable if the Basic Criminality Reproduction Number R0<1. Therefore, we have established the following result.Lemma 2.
The crime-free equilibriumP0 of system (15) is locally asymptotically stable if R0<1 and is unstable if R0>1.


## 2.4.2. Low-Criminality Equilibrium

In the absence of the reoffenders, that is, forC2=0, model (15) has a low-criminality equilibrium Pl∗=(S0,D0,C1,S1,D1,0,0,0), where(21)S0=τ1+μ+a1β0D0=γ0μS0C1=Λβ0-μ+γ0τ1+μ+a1τ1+μ+a1β0=μ+γ0β0R0-1S1=τ1C1μ+γ1D1=γ1μS1.Note thatPl∗ exists if C1>0, that is, if R0>1. Now consider the resulting model (15). The local stability of the low-criminality equilibrium Pl, which is examined by linearizing system (15) around Pl, is governed by the Jacobian matrix(22)MPl=M11-β0S0000β0C1M220000τ1M33-β1S10000M440000τ2M55,with M11=-β0C1-(μ+γ0); M22=β0S0-(τ1+μ+a1); M33=-(μ+γ1); M44=β1S1-(τ2+μ+a2); and M55=-(μ+γ2).The eigenvalues of matrix (22) are λ1=M55, λ2=M44, λ3=M33, and the roots of(23)ϕ2+z1ϕ+z0=0,with z1=Λβ0/(τ1+μ+a1)>0 and z0=Λβ0-(μ+γ0)(τ1+μ+a1)>0(⇔R0>1).The eigenvaluesλ1 and λ3 are real negative from definition of the parameters μ, γ1, and γ2. The eigenvalue λ2 is negative if(24)β1<τ2+μ+a2S1=β0τ2+μ+a2μ+γ1τ1μ+γ0R0-1 or, equivalently, if(25)R1=β1τ1μ+γ0R0-1τ2+μ+a2μ+γ1β0<1Finally, the eigenvaluesλ4 and λ5 are real negative if z0>0 and z1>0 by applying the Routh-Hurwitz criteria [31, 32] on polynomial (23). In this sense, z0>0 and z1>0 if and only if R0>1.Thus, we have established the following result.Lemma 3.
The low-criminality equilibriumPl∗ of system (15) exists and it is locally asymptotically stable if R0>1 and R1<1. Otherwise, Pl∗ is unstable.It should be mentioned that the consequence of the above result is that whenPl∗ becomes unstable, two scenarios emerge: one for the case where the criminality is eliminated from population and P0 is stable and the other where new criminals could emerge and there is high-criminality prevalence in the population such that model (15) has a high-criminality equilibrium with coexistence of both offenders: C1≠0 and C2≠0. In this way, it is instructive to determine the possible interventions on β0 and β1 in order to reduce R0 and R1 below one, that is, to guarantee the conditions under which the criminality is eliminated, or at least having its incidence reduced.


## 2.4.3. High-Criminality Equilibrium

In what follows, ifR1>1, such that the low-criminality equilibrium Pl∗ is unstable, system (15) has the positive high-criminality equilibrium, Ph∗=(S0,D0,C1,S1∗,D1∗,C2∗,S2∗,D2∗), where(26)S1∗=τ1C1μ+γ1+β1C2∗,S2∗=τ2C2∗μ+γ2+β2C2∗,D1∗=γ1μS1∗,D2∗=γ2μS2∗.Remembering from (21) that C1>0⇔R0>1, so Ph∗ exists if and only if R0>1. Now replacing the expressions for S1∗ and S2∗ in the sixth equation of system (15), the positive high-criminality equilibrium, which cannot be expressed cleanly in closed form, can then be obtained by solving for C2∗ the following expression:(27)η2C2∗2+η1C2∗+η0=0,where(28)η2=β1β2μ+a2,η1=μ+γ1τ2+μ+a2β2β1μ+γ2β2μ+γ1+μ+a2τ2+μ+a2-R1η0=μ+γ1μ+γ2μ+γ2+a21-R1.Since all the model’s parameters are nonnegative, it follows from (28) that the coefficient η2 is always positive and η0<0 for R1>1. Thus, it is clear that model (15) has a unique positive equilibrium Ph+ when R1>1 and R0>1. For η0=0 and η1<0, model (15) also has a unique positive equilibrium Ph+. Moreover, whenever R0>1, if R1=1, then η0=0 and η1<0 if only if (29)β2>τ2+μ+a2μ+γ2β1τ2μ+γ1,such that model (15) also has a unique positive equilibrium Ph∗. Finally, for η0>0, η1<0, and η12-4η2η0>0, that is, for R0>1 and R1<1, model (15) has two positive equilibria Ph+ and Ph-. Note that these equilibrium points cannot be studied in its closed form, so we carried out its local stability using numerical methods.Thus, the following result is then established.Theorem 4.
Model (15) has (i)
a unique positive equilibriumPh+ if η0<0⇔R1>1;(ii)
a unique positive equilibriumPh+ if η0=0 and η1<0;(iii)
two positive equilibria,Ph+andPh-,ifη0>0 and η1<0andη12-4η2η0>0;(iv)
no positive equilibrium, otherwise.As explored in model (1), model (15) also exhibits the phenomenon of backward bifurcation (Figure 8(a)) and breaking point (Figure 8(b)) when β1 and β2 are chosen as bifurcation parameters.Figure 8
The phenomena of backward bifurcation and breaking point, whenβ1 and β2 are chosen as a bifurcation parameter. (a) Backward bifurcation for β2=5 with β0 and β1 increasing. (b) Breaking point for β0=β1=5 with β2 increasing, where β2thr=121.2 and R0=1.139 and R1=0.464. The higher solution corresponds to the stable equilibrium (solid curve); the smaller solution corresponds to the unstable equilibrium (dashed curve). Parameters’ values used are as given in Table 3.
(a) (b)Figure8(a) shows the backward bifurcation for β2=5 and β0=8.4 (R0=2.39) with β1 increasing. Figure 8(b) shows the phenomenon of breaking point for β1=β0=5, with β2 increasing, where β2∗=121.2 and R0=1.139 and R1=0.464. As it should be expected, Figures 8(a) and 8(b) present the same results shown in Figures 2 and 4. In summary, if R0>1 and R1>1, model (15) has the locally asymptotically stable high-criminality equilibrium Ph+. Finally, if R0>1 and R1<1 and β2>β2thr, then there is coexistence of two locally asymptotically stable equilibria, Pl∗ and Ph+, such that the stability of these equilibrium points depends on the initial condition of system (15) (see Figure 9).Figure 9
Forβ1=2.0, β2=4.4 (β2>β2thr), R0=2.39, and R1=0.89. The other parameters’ values are given in Table 3. Profile of population of reoffenders (C2). (a) If the recidivism is low (or C2(0) small), then the equilibrium point Pl∗ is locally asymptotically stable. (b) If the recidivism is higher (or C2(0) large), then the equilibrium point Ph+ is locally asymptotically stable.
(a) (b)In contrast to model (1), when β0 is chosen as a bifurcation parameter, model (15) has the crime-free equilibrium P0 given by (16), which indicates the possibility of the forward bifurcation (Figure 10). If R0<1, then the crime-free equilibrium P0 of system (15) is locally asymptotically stable; if R0>1, P0 becomes unstable and Pl∗ is locally asymptotically stable if R1<1. Thus, the substitution of the assumption that the susceptible individuals (S0) can enter to crime only on their own (i.e., δ0) by the standard incidence (i.e., β1C1S0) in model (1) includes the forward bifurcation phenomenon to model (15).Figure 10
The phenomenon of forward bifurcation whenβ0 is chosen as a bifurcation parameter. The crime-free equilibrium P0 of system (15) is locally asymptotically stable if R0<1 and is unstable if R0>1. Parameters’ values used are as given in Table 3.


## 3. Results

To illustrate the theoretical results contained in this paper, models (1) and (15) are simulated using baseline parameters values/ranges summarized in Table 3 (unless otherwise stated). The parameters are chosen for simulations purposes only, so we could illustrate our qualitative results. Moreover, it is worth mentioning that if the assumption made in the PCCM model (1), that is, susceptible individuals can enter to crime only on their own (i.e., δ0≠0), and the assumptions made in FCCM model (15), that is, susceptible individuals get into criminal behavior depending only on the contact with those individuals who are incarcerated, C1 (i.e., β0C1S0), were considered in a single model, this new model would be more realistic. However, the analytical results and the phenomena of backward bifurcation and breaking point of this new model would be similar to model (1), except for the fact that R0 would no longer exist. For this reason, we did not study this more realistic mixed model.


## 4. Discussion

In this paper, we attempt to present theoretical models of criminal careers using the dynamical system approach traditionally used in the study of infectious diseases spreading in a homogeneous population. The models consider crime dynamics as contagious phenomena in which a susceptible cohort of individuals with criminal propensity is “infected” by criminal individuals who have been convicted for their crimes and are arrested in a prison. The contagion occurs by several ways, in particular through the sending of messages by mobile phones (a very widespread habit in Brazilian prisons) and through “carrier pigeons” represented by relatives and lawyers.We propose, to the best of our knowledge by the first time in quantitative criminology, the existence of thresholds for the stability of crime-endemic equilibrium which are the equivalent to the “basic reproduction number” widely used in the mathematical epidemiology literature [22], as shown in Lemmas 2 and 3. Both model (1) and model (15), however, exhibit the phenomena of backward bifurcation and breaking point when the contact rates β1 and β2 are chosen as bifurcation parameters. Since data strongly suggests that standard incidence formulation is more suited for modeling human diseases [22, 23, 27] and we adopted this formulation in the current study, the above results show that the phenomena of the backward bifurcation and the breaking point could be important properties of the criminality model.As in other criminal career models, our models seek to provide a theoretical framework to analyse the longitudinal behavior of individuals who commit criminal offenses [33]. In addition, we centrered our analysis, although in an implicit way, in the parameter considered to be the most important for the analysis of criminal career, namely, the rate at which offenders commit crime, denoted in the specialized literature by the Greek letter λ, coincidently, the same symbol used to denote the force-of-infection in epidemiology of transmissible diseases.Other dynamical system models proposed the analysis of criminal dynamics including differential equations, like the works by Farrington synthesized in Refs. [16, 34]. However, these models are either linear (the former) or related to other kinds of infectious contagion (the latter).Finally, it is noteworthy that the models proposed in the paper are intended only to provide a theoretical framework upon which other works can provide empirical support for the assumptions and values for the parameters determinant of the dynamical behavior of the systems here studied. The present work, therefore, is intended to provide the first step in the study of criminal careers as determined by contagious events like the ones related to the phenomenon of incarcerated criminals influencing the behavior of susceptible juveniles outside prisons with a criminal propensity. The finding of backward bifurcation in both models, however, implies that there is an endemic equilibrium of criminality even when the threshold parameter for contagion is below unit which, in turn, implies that control strategies are more difficult to achieve any considerable impact on crime control in situations similar to the ones here analyzed.



---

*Source: 1582159-2018-08-01.xml*