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Journal of Applied Mathematics
Volume 2018, Article ID 1582159, 16 pages
https://doi.org/10.1155/2018/1582159
Research Article

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

1Faculdade de Medicina da Universidade de São Paulo, MLS and LIM01-HCFMUSP, São Paulo, SP, Brazil
2Universidade de Campinas, DMA-IMECC, Campinas, SP, Brazil
3School of Applied Mathematics, Fundação Getúlio Vargas, Rio de Janeiro, RJ, Brazil

Correspondence should be addressed to Eduardo Massad; rb.psu.mf.mid@dassamde

Received 7 March 2018; Revised 30 May 2018; Accepted 3 July 2018; Published 1 August 2018

Academic Editor: Zhidong Teng

Copyright © 2018 Silvia Martorano Raimundo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [1315]. 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 by , is then characterized by three classes: susceptible (), incarcerated (), and desisting () 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: , those individuals who have a criminal propensity but have never been incarcerated, not criminally active but susceptible to crime, and , those individuals susceptible to criminal activities who were once incarcerated () and those susceptible who were multiple-time incarcerated () and became criminally active again. Similarly, let the incarcerated population be divided into two categories: , those who are first-time incarcerated, and (i = 2,3,…,n), those who were multiple-time incarcerated at a given time. Finally, 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 (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 crime move to either incarcerated () or desisting () class. The rate of initial participation in crime () at which individuals move from state (not criminally active) to (criminally active and incarcerated) is proportional to intrinsic criminal propensity of individuals. It is also assumed that special intervention programmes () 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 by ; that is, the rate at which inmates move from state to (formerly incarcerated but not criminally active, i.e., ex-offenders) is . In the same way, the average length of the multiple incarceration term is given by , with .

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 including , , and (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 define (i=1,2) as the rate of imprisonment, which captures the return to criminal activity of the individuals released from prison, such that and 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, is the natural mortality rate and (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 Figure 1. The variables and parameters are described in Tables 1 and 2, respectively.

Table 1: Models variables and their biological meaning.
Table 2: Models parameters and their biological meaning.
Figure 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:with generic initial conditions , , , , , , , and . By summing up the above equations, the total population size is variable withThus, in the absence of additional inmate mortalities, that is, , the population size evolves as an immigration model with natural mortality, that is, according to . This equation has a single equilibrium for any initial value of . Thus, in the long run, the population size settles to this constant value. It follows from (2) that .

The differential equation for implies that solutions of (1), starting in the positive orthant , either approach, enter, or remain in the subset defined by

Thus it suffices to consider solutions in the region . Solutions of the initial value problem starting in and defined by (1) exist and are unique on a maximal interval [26]. Since solutions remain bounded in the positively invariant region , 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 .

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

Moreover, from seventh and eighth equations of system (1), we obtain

From the sixth equation of system (1), one gets(i),(ii), which implies

If , model (1) has a low-criminality equilibrium which indicates the existence of offenders who are incarcerated only once in life (), given by

To determine the stability of this equilibrium, the Jacobian of system (1) is computed and evaluated at . Hence, the low-criminality equilibrium is locally asymptotically stable if , where , defined as Criminality Reproduction Number (CRN), is given bywith

Note that, in analogy to the spread of infectious diseases models [29], the CRN, , represents the “average expected number of new offenders originated by a single persisting offender , whilst in a criminal career.” In other words, one person who was incarcerated two times gets into contact with susceptible individuals, just released from first imprisonment, successfully and induces persons to commit crime. In other words, is the average number of individuals who commit crime influenced by one inmate .

For , that is, for , replacing both expressions for given by (4) and given by (5), an expression for is obtained aswhere

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 ) into the expressions in (5), model (1) has a high-criminality equilibrium, , where there is coexistence of both offenders and , given bywith , , and given by (6). Thus, the following result is then established.

Theorem 1. Model (1) has(i)a unique positive equilibrium if ;(ii)a unique positive equilibrium if and ;(iii)two positive equilibria, , if and and ;(iv)no positive equilibrium, otherwise.

Since all model parameters are assumed as nonnegative, it follows from (10) that the coefficient is always positive, for and for . Thus, it is clear from Theorem 1 that model (1) has a unique positive equilibrium, , when , that is, when (case (i)).

Now, for and , the quadratic equation (9) has two positive solutions if and (case (iii)). Hence, assuming that (9) has two positive real solutions, let and be the smaller and higher value of , respectively. Translating it into equilibrium of system (1), the question is to address what means the positive high-criminality equilibria for . It is important to note that, in this case, system (1) can have two equilibria, which are biologically feasible, even though . This idea is explored more deeply below.

It is instructive at this point to explore some qualitative features for and . Firstly, for , it follows that ; the quadratic equation (9) has either a unique positive root (if ) or no positive root (if ). In other words, for and , model (1) has a unique positive high-criminality equilibrium given by (case (ii)).

Moreover, note that when , we have and . Thus, if , thenand model (1) has a unique positive high-criminality equilibrium, , for .

For , we have and with and such thatand if only if . Thus, model (1) has a unique high-criminality equilibrium if .

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 , 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 .

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, (see (7)), and the parameter , 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 and 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 . We fix and explore the system behavior by varying . We also explore what happens when we fix while is varied. These control measures are designed to fight recidivism. In both cases, the recidivism will depend upon the initial sizes of the subpopulation and indicate the possibility of backward bifurcation.

Firstly, it is worth remembering that the quantity measures the average number of new contacts generated by a typical incarcerated individual with those susceptible individuals who are ex-offenders and can become susceptible to the crime again. , in turn, measures the average number of new contacts generated by with those individuals 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 (), it does not generate high criminality rates and, for , 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 could be locally asymptotically stable. On the other hand, the criminality will persist and increase if (worst-case scenario). In this case, the corresponding high-criminality equilibrium 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 loses its stability and a unique appears as increases through one, is known as forward bifurcation in epidemiology.

For models that exhibit this type of bifurcation, the requirement 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 and coexist with the low-criminality equilibrium, ; that is, there are three steady states when is immediately less than one. Thus, the requirement is necessary but is not sufficient to fight the spread of crime.

2.2.1. Analytic Strategy: Fixed with Increasing

The physical implication of backward bifurcation is that the population can engage in a high criminality level even when crosses unity downwards. In other words, in the presence of recidivism, decreasing 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 scale, as . Hence, for , system (1) will present only the locally asymptotically stable low-criminality equilibrium . For , system (1) will present the locally asymptotically stable low-criminality equilibrium plus two positive high-criminality equilibria, and , which will correspond to the solutions of (9): , the higher solution, which corresponds to the stable equilibrium, and , 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 , 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 is chosen as a bifurcation parameter; that is, we fixed , whereas increases.

Figure 2: Backward and forward bifurcations diagram for proportion of reoffenders () for model (1), where is chosen as a bifurcation parameter ( fixed and 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) , , and ; (b) , , and ; and (c) , , and . Forward bifurcation for (a) , , and and (b) , , and . Parameters’ values used are as given in Table 3.
Table 3: Baseline values for model (1).

Figure 2 shows the profile of the proportion of both reoffenders (solid curve) and (dashed curve) as a function of with decreasing values of as increases (i.e., as increases). The solid curve stands for the stable high-criminality equilibrium, , and the dashed curve stands for the unstable high-criminality equilibrium, . For and , one has . Thus, for , model (1) has two positive high-criminality equilibria, while for , model (1) has one positive high-criminality equilibrium. Similarly, for and , one gets . Thus, for , model (1) has two positive high-criminality equilibria, while for , model (1) has one positive high-criminality equilibrium. For and , one gets . Thus, for , model (1) has two positive high-criminality equilibria, while for , model (1) has one positive high-criminality equilibrium. Finally, for and and for and , we have , and the model has one positive high-criminality equilibrium for and no positive equilibrium for . Consequently, model (1) exhibits the forward bifurcation at . As it should be expected, it can be seen in Figure 2 that increases with increasing and a greater reduction in recidivism prevalence is recorded for decreasing values of .

As stated earlier, the physical significance of the phenomenon of backward bifurcation is that the classical requirement of 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 of the model. That is, the presence of backward bifurcation in model (1) suggests that the possibility of avoiding the recidivism event when could be dependent on the initial sizes of the subpopulation . This scenario is illustrated numerically in Figure 3. Thus, if the recidivism is low, then is small, such that the low-criminality equilibrium is stable (Figure 3(a)). In contrast, if the recidivism is higher, then is large, so the high-criminality equilibrium 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 , confirming that model (1) undergoes the phenomenon of backward bifurcation with one stable high-criminality equilibrium (higher, solid curve in Figure 2), one unstable high-criminality equilibrium (lowest dashed curve in Figure 2), and one low-criminality equilibrium .

Figure 3: For , , and . The other parameters’ values are given in Table 3. Profile of the population of reoffenders (). (a) If the recidivism is low (or small), then the equilibrium point is locally asymptotically stable; (b) if the recidivism is higher (or large), then the equilibrium point is locally asymptotically stable.
2.2.2. Analytic Strategy: Fixed with Increasing

Alternatively, from now on we explore the implications of the parameter on the criminal dynamics. It is worth mentioning that, from expression (10), it is easy to verify that if and only ifand thus, in such case, system (1) has two positive high-criminality equilibria (case (iii), see Theorem 1.) for , , and .

A schematic diagram of this bifurcation phenomenon for system (1) is given in Figure 4, where 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 () for model (1), where is chosen as a bifurcation parameter ( fixed with 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) , , and ; (b) , , and ; and (c) , , and ; forward bifurcation for , , and . For and , such that system (1) has a unique positive high-criminality equilibrium . Parameters’ values used are as given in Table 3.

Figure 4 shows the profile of the proportion of both reoffenders (solid curve) and (dashed curve) as a function of with decreasing values of as increases. The solid curve stands for the stable equilibrium and the dashed curve stands for the unstable equilibrium. Note that, for and , 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 Figure 4 that increases with decreasing , such that greater reduction in recidivism prevalence is recorded for decreasing values of . One aspect to differentiate both parameters and 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, for , system (1) presented the locally asymptotically stable low-criminality equilibrium . 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 () and those second-time offenders susceptible to the crime () at a rate , then reducing the number of contacts between and , that is, reducing , could lead to the low-criminality scenario if the proportion of reoffenders is small (see Figure 3). Hence, to reach the low-criminality equilibrium, a community needs to reduce . However, if is large, then the proportion of reoffenders , as well as , increases, which leads to higher crime prevalence, such that the high-criminality scenario emerges.

Figure 5(a) shows the long-term behavior of plotted versus . The threshold is expressed as a minimum value where system (1) has two positive equilibrium points. Thus, for each fixed , there is a corresponding unique threshold . For one such example (see Figure 5(b)), , , and (or ) are the minimum values where system (1) has two positive value. For , we have and . In this case, system (1) has a stable equilibrium point, (the solid curve in Figure 5(b)), and an unstable equilibrium point, (the dashed curve in Figure 5(b)). On the other hand, for , we have , which means that there are no positive equilibrium points for system (1). Moreover, for , we have and (or ) such that, for , one has and . Otherwise, for , one has and (see Figure 6).

Figure 5: (a) The long-term behavior of plotted versus . (b) For , , and , 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 outcome plotted versus . For , , and , 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 () can enter to crime only on their own, so we make . Hence, in the modified model, the flow of the susceptible individuals into the criminal activity depends only on the contact with those individuals who are incarcerated, . We also define (i = 0,1,2) as the rate of imprisonment; it captures the return to criminal activity of those released from prison, such that , , and are the flows into criminal activity which depend on having had contact with those first-time and second-time incarcerated individuals ( and ), respectively. Following the idea of the previous model (1), here (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 Figure 7. The variables and parameters’ values are given in Table 3, except for that is the contact rate between and (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:

2.4. The Existence and Local Stability of Equilibria
2.4.1. Crime-Free Equilibrium

In the absence of crime, that is, for , model (15) has a crime-free equilibrium which is obtained by setting the right-hand sides of system (15) to zero, where

To analyze the local stability of this equilibrium, the Jacobian of system (15) is computed and evaluated at , 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 equilibrium is governed by the Jacobian matrix

where

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

Hence, all the eigenvalues of matrix (17) are negative or have negative real parts if and only if . In summary, the crime-free equilibrium of system (15) is locally asymptotically stable if the Basic Criminality Reproduction Number . Therefore, we have established the following result.

Lemma 2. The crime-free equilibrium of system (15) is locally asymptotically stable if and is unstable if .

2.4.2. Low-Criminality Equilibrium

In the absence of the reoffenders, that is, for , model (15) has a low-criminality equilibrium , where

Note that exists if , that is, if . Now consider the resulting model (15). The local stability of the low-criminality equilibrium , which is examined by linearizing system (15) around , is governed by the Jacobian matrixwith ; ; ; ; and .

The eigenvalues of matrix (22) are , , , and the roots ofwith and .

The eigenvalues and are real negative from definition of the parameters , , and . The eigenvalue is negative if or, equivalently, if

Finally, the eigenvalues and are real negative if and by applying the Routh-Hurwitz criteria [31, 32] on polynomial (23). In this sense, and if and only if .

Thus, we have established the following result.

Lemma 3. The low-criminality equilibrium of system (15) exists and it is locally asymptotically stable if and . Otherwise, is unstable.

It should be mentioned that the consequence of the above result is that when becomes unstable, two scenarios emerge: one for the case where the criminality is eliminated from population and 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: and . In this way, it is instructive to determine the possible interventions on and in order to reduce and 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, if , such that the low-criminality equilibrium is unstable, system (15) has the positive high-criminality equilibrium, , where

Remembering from (21) that , so exists if and only if . Now replacing the expressions for and 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 the following expression:where

Since all the model’s parameters are nonnegative, it follows from (28) that the coefficient is always positive and for . Thus, it is clear that model (15) has a unique positive equilibrium when and . For and , model (15) also has a unique positive equilibrium . Moreover, whenever , if , then and if only if such that model (15) also has a unique positive equilibrium . Finally, for , , and , that is, for and , model (15) has two positive equilibria and . 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 equilibrium if ;(ii)a unique positive equilibrium if and ;(iii)two positive equilibria, and , if and and ;(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 and are chosen as bifurcation parameters.

Figure 8: The phenomena of backward bifurcation and breaking point, when and are chosen as a bifurcation parameter. (a) Backward bifurcation for with and increasing. (b) Breaking point for with increasing, where and and . 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.

Figure 8(a) shows the backward bifurcation for and () with increasing. Figure 8(b) shows the phenomenon of breaking point for , with increasing, where and and . As it should be expected, Figures 8(a) and 8(b) present the same results shown in Figures 2 and 4. In summary, if and , model (15) has the locally asymptotically stable high-criminality equilibrium . Finally, if and and , then there is coexistence of two locally asymptotically stable equilibria, and , such that the stability of these equilibrium points depends on the initial condition of system (15) (see Figure 9).

Figure 9: For , (), , and . The other parameters’ values are given in Table 3. Profile of population of reoffenders (). (a) If the recidivism is low (or small), then the equilibrium point is locally asymptotically stable. (b) If the recidivism is higher (or large), then the equilibrium point is locally asymptotically stable.

In contrast to model (1), when is chosen as a bifurcation parameter, model (15) has the crime-free equilibrium given by (16), which indicates the possibility of the forward bifurcation (Figure 10). If , then the crime-free equilibrium of system (15) is locally asymptotically stable; if , becomes unstable and is locally asymptotically stable if . Thus, the substitution of the assumption that the susceptible individuals () can enter to crime only on their own (i.e., ) by the standard incidence (i.e., ) in model (1) includes the forward bifurcation phenomenon to model (15).

Figure 10: The phenomenon of forward bifurcation when is chosen as a bifurcation parameter. The crime-free equilibrium of system (15) is locally asymptotically stable if and is unstable if . 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., ), 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, (i.e., ), 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 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 and 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 , 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work has been funded by LIM01-HCFMUSP, FAPESP, CNPq and by the Office of Naval Research, USA, Grant Number 2059558. An earlier version of this work has been presented as at EMAp Seminars according to the following link: https://emap.fgv.br/seminarios/contagious-criminal-career-models-showing-backward-bifurcations-implications-crime.

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