Final Paper
A Writing Guide and the Recommended Outlines for the Final Paper
You may fulfill the requirements for the final paper if you follow the suggestions
below. In the following sections, I discuss the compositions of the expected final paper
elements. At the end, I provide a rough idea about the recommended length and format
of the final paper, how will the paper be evaluated, and tips of journal searching through
Lamar library online sources.
***IMPORTANT:
Articles are attachment titled and underlined for empirical studies to support paper arguments.
1. You may choose your own topic that can be either a type of crime or criminals
TOPIC SERIAL KILLING
Theories: Social learning theory and General strain theory
Your whole purpose here is to make arguments why one theory (or a school of similar
theories) is better than another one (which should be also a good one for the type of
crime or criminals) in explanation why the chosen type of crime/criminals were
committed. In other words, both theories (or schools of theories) are good ones for
explanation of your chosen crime/criminals and you utilize some empirical studies’
findings to support your explanations. Furthermore, you will argue why one theory is
better than the other one due to certain factors (i.e., evidences from the empirical
study findings).
2. Empirical studies used to support your paper arguments should retrieve from Lamar
University Library Quick Search located on the page of http://library.lamar.edu
(more directions are in the end of this guide). Please keep in mind: do NOT use news
media articles to support your arguments, nor use some specific individual cases to
illustrate your points since they are more likely to be biased/edited by the media. To
know if your found article is an academic journal publication, you should find if it has
sections such as literature reviews, data or method session, results/findings of their
studies, and conclusion/discussion. Media reports usually do NOT have these
sessions.
Your paper may contain the following five sections:
1. Introduction. The first part is to discuss some general terms the phenomenon
(crime/criminals) you are intending to study. In other words, you should state what
you intend to do in the paper. For example, you can
a. briefly describe the issue you chose for the paper.
b. briefly describe the opposing views that you are going to present in the paper.
State your position.
c. Or, why have you decided to examine this phenomenon?
2. Descriptive Summary of applied theories. For selected theories that will be applied
to explain your topic of crime/criminals, you should briefly explain how each theory
is applied on factors that lead to the crime/being a criminal (of certain type). You
should especially cover the following points:
a. What are the key elements of each theory offered for explanations on the selected
crime/criminals? Why each theory is a good fit in explaining the crime/criminals?
Your own reasoning can be important here.
b. Among your references, be sure to state how they provide information that is
relevant to your topic? How are theories or reasoning/evidences each provides in
their arguments related to your selected topic? Try to be concise and right up to
the point.
3. Literature Review. You should cite at least Four academic articles (textbook or
novels do not count) which related to the topic you chose (so their empirical findings
could be useful for your arguments). In this part, you should discuss how other
studies have examined or discusses the phenomenon you wish to examine? What
have they found? And how do their findings could be supportive to either theory?
Are their findings strong enough to support your application of the selected theories?
In your review, you may consider to discuss:
a. What does the literature have to say about the dependent variable
(crime/criminals) in your topic?
b. How has it been studied (very briefly)? (e.g., who are the subjects? The research
design and methods has been used?)
c. What have been the key findings that can be related to your topic?
d. Can these findings support your selected theories (both or either one?)?
e. How well theories you selected is applied to your focused crime/delinquency?
** You may also use non academic article (generally, they should be official government
reports). However, you will have to cite the non academic articles (as additional ones) in
your reference list.
4. Discussions and Implications. Next, you should express your own points. You
should make your own arguments as the following suggestions:
a. What is your reaction to the arguments presented by each author? Be sure to
briefly state your opinions on the strength or weakness of the arguments made by
each author from the text.
b. Summarize briefly how do your arguments supported by the literature that you
discussed in the paper? Do not just copy-paste prior sentences.
c. Where do you think future work on this topic should be directed?
d. What remains to be done in this issue?
e. What other empirical evidence may be needed to clarify future arguments
between your chosen theories?
5. Reference List (bibliography) APA
1st article Serial killing and homosexual necrophilia, an exploration
The vast majority of literature regarding sexual homicide is premised on a male perpetrator and female victim, as is the sparse literature regarding acts of necrophilia. Little, therefore, is known of homicide committed by a homosexual male to facilitate necrophilic acts. A case is presented here of such an instance, an instance that escalated into serial homicide and, at least, a dozen victims. The prefacing role of sexual fantasy is explored, as well as the significance of the manner of death when necrophilic intent is the rationale for homicide. Potential indicators of necrophilic interests leading to sexual homicide are noted, and the literature regarding such interests is expanded.
Keywords: Necrophilia; serial killing; sexual paraphilia; homicide; homosexual murder
Introduction
A case is presented of a homosexual serial killer who murdered to secure bodies upon which to inflict his necrophilic fantasies and desires. The study offers an inquiry into the nature of serial killing of men when necrophilic intent is the driving force of those murders. When necrophilia is the rationale for homicide, this case inquires as to the absence or presence of 'overkill', as seen in many murders where the homosexuality of the victim is present or suspected (Bell & Vila, [ 6]; Gojanovic, [17]; Bartlett, [ 4]p583). The role of sexual fantasy, as a preface for offending, is also explored in the context of homosexual homicide. With regard to homosexual necrophilia, the case presented inquires as to the necessity of a distinct taxonomy of the phenomenon, such as that presented by Aggrawal ([ 2]).
Whilst there have been a number of studies of sexualised murder these are, overwhelmingly, accounts premised on heterosexual men murdering women (Bartlett, [ 4]p573). Existing literature regarding homicide involving homosexual males is largely concerned with victims of hate violence (Kelley & Gruenewald, [21]; Mouzos & Thompson, [25]; Tomsen, [29]), there is very limited literature of homosexual perpetrators and victims outside that perspective. As such, studies of homosexual murder carried out for the purposes of committing necrophilic acts is limited to the one account offered by Bartholomew, Milte, and Galbally ([ 3])
Nearly exclusively the sparse academic and professional literature, much like the literature on sexualised homicide, deals with the topic of necrophilia based on the assumption of a male perpetrator imposing his sexual urges on the corpse of a female. That perspective remains constant even when necrophilic acts are not prefaced with a homicide (see for example, Boureghda, Retz, Philipp-Wiegmann, & Rosler, [ 7]; Schlesinger, [27]). The assumption of gender has been consistent across time and even features in longstanding definitions of the paraphilia, such as that offered by Smith and Braun, 'Sexual necrophilia [is] a man's desire to have sexual intercourse or any other kind of sexual contact with a female corpse' (Smith & Braun, [28]p259). Unsurprisingly then, Knafo asserts that almost without exception, necrophiles are males who 'use' females,
The gender disparity reflects that found in many perversions and has been accounted for by postulating castration fears and anxieties and the male objectification of women, obvious in pornography, which highlights the sex act rather than relationships between people (Knafo, [22]p878).
The features which Knafo asserts, the objectification of women, their subjugation for the purposes of offsetting subconscious fears of castration, of owning, consuming, and fetishizing the object of the male sex drive, do not apply neatly in cases of homosexual necrophilia. The explanation for homosexual necrophilic acts cannot lie in the shadow of a patriarchal culture in which the victim is portrayed as the natural subordinate to the perpetrator's dominance and masculinity, the offender is not acting out the gender relations embedded and perpetuated in contemporary culture. The explanation for homosexual necrophilia then requires investigation, which is offered here.
Authorities in the subject area reference homosexual necrophilia as an incidental sidebar, even going so far as to assert that there are no documented cases of 'pure homosexual necrophiles', those who engage in homosexual acts with both the living and the dead (Aggrawal, [ 2]p90). Aggrawal ([ 2]) does, however, offer a preliminary taxonomy of homosexual necrophilia. In the first category he places those who perform homosexual acts with the living, but fantasise about sexual contact with the dead (type I). In the second category he places those who do not engage in homosexual acts with the living but will engage in anal intercourse with the dead (type II). Finally, in the third category, are those who will perform homosexual acts with the living and with the dead (type III), of which he says there are no recorded cases (Aggrawal, [ 2]p90). Overlooked by Aggrawal, it is the last category to which this case belongs.
A case study is employed in the same manner as the study offered by Culhane, Hilstad, Freng, and Gray ([12]); data is sourced here directly from the offender who participated in a three year correspondence, after ethical approval was granted by an institutional ethics board. All quotes are supplied, retrospectively, by the offender himself, information which was easily cross referenced to other sources of data, by studying letters, diaries, an unpublished autobiography, court transcripts, prison records, and witness and confession evidence stored at the National Archives.
The case
The offender was a white male in his early 30s, working in a semi skilled occupation, who lived alone. When eventually apprehended he was found mentally competent to stand trial, although psychiatric assessment revealed him to be suffering from personality disorder not otherwise specified (PD-NOS); psychological evaluations carried out in prison have never deemed him suitable for, or in need of, care in a psychiatric facility. Those reports do, however, consistently judge him as lacking any remorse or empathy for his victims. Describing him as above average intelligence, psychologists who have evaluated him in prison note him as a continual risk to others and have continued to recommend that he should remain in high security conditions for the duration of his life sentence.
At the time of his offending he was a moderately open homosexual; his sexual orientation would not be hidden, but not readily disclosed to work colleagues and family members. He does not report any heterosexual encounters in his personal life. He frequently patronised gay bars and would often engage in 'one night stands', fleeting sexual encounters. Over time though, he came to find 'the gay scene' tiresome and unfulfilling, he would complain of loneliness and displayed signs of depression. His drinking became problematic after the promise of a romantic relationship failed to materialise and he came to be, in his own recollection, 'a full blown alcoholic', using alcohol to cope with rejection. As his feelings of depression and despondency deepened, he would frequently spend time alone, when he would engage in a rich and detailed sexual fantasy life aided by alcohol and music.
As a child, the offender was sexually abused by a close relative. The abuse he suffered was frequent, only ending when his abuser passed away. Struggling with the memories of abuse and with mixed feelings of grief and relief, he became emotionally and physically isolated in his childhood, recalling his abuser as the only person who showed an interest in him in his formative years. He recalls being emotionally detached from his mother and having no feelings whatsoever towards his step father. The family were relatively poor and he blames some of his alienation from other children on his 'shabby clothes'. Generally, however, he was well nourished and remembers suffering from no more than "the usual childhood illnesses. A solitary child with few friends and an acrimonious relationship with his elder half brother – with whom he recalls episodes of sexual experimentation upon his sibling's body as he slept – he was further isolated by his burgeoning homosexuality deemed, as it was, socially unacceptable and, at the time, criminal behaviour.
I now realised I was not like other boys, in either sexual preference or wider emotional aspirations. I was well aware of all the social condemnation that would fall on my head if it were known what my tastes were... There were my early experiences with my [abuser] and now I had inherited a second tier of alienation
To cope with alienation from his peers, and from his family, he withdrew into a fantasy world. His fantasy world was diverse, in some scenarios he would be reunited with his abuser; in others he would come across his own body – injured or dead – and sexually abuse it; he would render other boys unconscious and sexually abuse their passive bodies; and his fantasies would be coupled, and reinforced, by masturbation. Indeed, the sexual exploration of his pubescent years was in the context of this fantasy world,
My feelings for other boys had to be kept hidden because that remained taboo to both peers and to the larger adult world. Therefore, to satisfy the "need" for emotional expression and mutual sustenance, I was obliged to create a friendly, pleasing entity (or friend) within myself. I would hobble along for years, an emotional cripple, with this substitute. He existed as an imaginary person to satisfy my recurring needs
His fantasies centred, and remained so throughout his life, on passivity. Such was the centrality of that theme, and the sexual excitement it provided, unconscious bodies graduated to dead bodies in this sexual fantasy. No doubt this graduation can, in part at least, be attributed to an incident in which he molested an unconscious friend. Although being wracked with fear of discovery, he recalls the excitement of the occasion but also how he mourned its fleeting and short nature. For the corpse is pliable for a longer period and, pivotally, the use of the dead would allow him to permanently 'own' the victim without fear of exposure, as Aggrawal notes, 'the dead cannot refuse, reject or resist; they also do not tell tales or talk back' (Aggrawal, [ 1]:316).
From childhood, through adolescence, and into adulthood, a great deal of time would be spent in this fantasy world, a world which became more intricate as time progressed. Over time, his fantasy life had to become more intricate and detailed to provide continued satisfaction, whilst through masturbation sexual arousal became conditioned to its content. Indeed, so involved in his fantasy world and so determined to maintain its capacity for sexual fulfilment that it came to involve external props (not uncommon in the prefacing time before the commission of sexual violence (Hazelwood & Warren, [18]p135)).
I put talc on my face to erase the living colour. I smear charcoal under my eyes to accentuate a hollow dark look. I put pale blue on my lips. I rub my eyes to make them bloodshot. I have put three holes in my old tee-shirt. I make a mixture of cochineal and saffron to synthesize blood. I soak the 'blood' into the holes and the liquid stains my shirt and runs down my body. I lie, staring-eyed, on the bed in front of the mirror.
Careers in the military and then the police were characterised by isolation owing to his sexual orientation and, at times, repressed romantic feelings towards colleagues. He came to live in a large city, more tolerant of homosexuality than his birth place, but he found no fulfilment in the opportunities for self expression that it offered. Real life experiences did not stimulate him as his fantasy world did, as Hazelwood and Warren note 'fantasy is always perfect' (Hazelwood & Warren, [18]p133). He attempted relationships, as well as 'one night stands', but their lack of fulfilment caused him to socially isolate himself, spending increasing amounts of time at home, alone. Unable to cope with mounting problems, difficulties at work, debt, and without any meaningful friendships or relationship, he further withdrew into a private world, engaging in fantasy with the help of alcohol and music.
...the whole fantasy 'menagerie' [would have] continued if it were not for a crisis of explosive proportions building from accumulating and 'impossible' stresses... The ritual traversed my private boundary and dragged in a real person. I was at my lowest ebb, ever... I had reached a stage where NOTHING at all was going right, and on all fronts. I had never before had all my problems dumped in my lap all at the same time. Something had to give.
It was in this emotional state when, during a night of heavy drinking, he met a man in a bar and invited him home. He declares no premeditation but the following morning, overcome with fear that the man would soon awake and leave, returning him to his state of loneliness, using a ligature he partially strangled the man into unconsciousness and then drowned him in a bucket of water. For some days afterwards, he kept the corpse and used it in acting out his sexual fantasies; a dozen men would come to be killed in this manner, ligature strangulation, sometimes completed with drowning to ensure death. In the deliberate choosing of how he would kill the offender recalls,
I thought of beating him on the head with a blunt instrument but could not do it. I thought about stabbing him to death with a kitchen knife but could not do that either.. I remember feeling it was important not to damage his body in such a way as to mar the purity of the image in the fantasy. I could not bear to think of his smashed head or any stab wounds on him.
After death, in a ritual that would be followed on many occasions the body was washed – although on other occasions he would bathe with the bodies – it would be dressed, undressed, and generally 'cared' for, sometimes for several days. Often he would converse with the corpse as if the person were still alive, eat meals, and watch television together. In terms of sexual exploration, on at least one occasion he attempted anal intercourse with a corpse but found his erection could not be sustained. He dismisses reports of post-mortem signs of anal penetration as natural consequences of strangulation and is adamant that penetrative sex was attempted only once with one victim. However, he openly admits performing fellatio with some, but not all corpses, as well as masturbating over the bodies and performing intercrural sex. More than sexual acts in isolation, however, he would use the bodies as 'props' in his sexual fantasies and role playing,
In a haze of fantastic images the victims had no identity as individual personalities. They did not exist for me as personalities. I cut their words from my ears and from my mind, they would be the images I directed for them. They would have to fit in with my dreams as more props to arrange on the stage of my fantasies. I ingested nothing of them at all, as impersonal as so much raw material. In as much as I ever considered them as people I considered that they would be what their dreams were. For years of practise, I was a champion of preserving my own secret inner world.
Once the corpses displayed outward signs of decay and decomposition they would be stowed underneath the floorboards and covered with air-freshners and salt to disguise the smell. When, after a number of victims had been killed, the lack of storage necessitated disposal of the victims they would be dismembered and burned in suitcases and bags on a bonfire constructed in the backyard of the offender's property.
Discussion
In this case, the murder of actual or perceived homosexual men, with an intent to perpetrate necrophilic acts, was qualitatively different than the majority of reported cases involving such victims. This case contradicts existing literature on homosexual homicide in that there was no extreme violence or 'overkill' (Bell & Vila, [ 6]; Gojanovic, [17]; Bartlett, [ 4]p583). Whilst victims in this case were not subject to 'hate' violence, it is submitted that an important caveat is the necrophilic intent. Destroying the body in the act of homicide subverts the necrophilic intent to, not only have sexual ownership of the corpse, but, in this case, to maintain the body in order to mimic relationship behaviours; eating food, 'conversing', and watching television; behaviour that Hickey would term, 'necrofetishism' (cited in Heasman and Jones 2006p274). The offender explicitly recalls that he did not wish to maim or damage his victims' during the killing process so as to maintain the purity of the image in his fantasy. Others have noted that the choice to strangle is associated with serial killers more generally (Kraemer, Lord, & Heilbrun, [23]p338) and particularly in the sexual killing of women (Chan & Heide, [11]). However, homosexual homicide in men is more commonly committed with an edged weapon (Chan & Beauregard, [10]p84). This case is contrary to that finding; all victims were, at least partially, strangled to death.
Aside from the manner of death, in an additional and important respect this case contradicts another key finding in the literature on homosexual homicide, relating to the role of sexual fantasy. Broadly, the academic literature is replete with accounts, studies, and investigations that document the prefacing and causal link of sexual fantasies to violent sexual crime (Gee & Belofastov, [15]; Gee, Ward, & Eccleston, [16]; Hazelwood and Warren [18]; Jones and Wilson [20]) and particularly lust murder and serial killing, accounts that are premised on a female victim and heterosexual male offender (Geberth, [13]p459; Hickey, [19]p70). The same causal link is found here in contradiction to Bartlett's finding that sexual fantasy imports a gendered component, of a male perpetrator dominating a passive woman or girl, which, he asserts, is not found in gay sexual homicides (Bartlett, [ 4]p592).
Burgess et al have previously reported that in 80% of a sample of violent sex offenders, from childhood and into adulthood, chronic masturbation, daydreaming, and social isolation were found (Burgess, Douglas, D'Agostino, Hartman, & Ressler, [ 8]); that finding is also confirmed here and supported by others (Gee & Belofastov, [15]p52). When fantasy is increasingly rehearsed, it acquires more and more power, strengthening the association between the content of the fantasy and sexual arousal. Stronger associations between fantasy and ejaculation then develop over time (Prentky et al., [26]). Such a finding is not new, however, as relapse prevention theorists have historically proposed that prefacing the committal of sex crimes is arousal to deviant sexual fantasy conditioned through masturbatory activity (Laws, [24]). The fantasy, however, can only satiate for a finite period, masturbatory activity moves a person closer to point of committing a sexual act in reality and, eventually, the fantasy world is so invested by him that it needs to be acted out (Carabellese, Maniglio, Greco, & Catanesi, [ 9] p259). What is new, however, is the finding that homosexual homicide can be prefaced with sexual fantasy, just as in cases of heterosexual sexual homicide.
Additionally, this finding is relevant in the context of necrophilia which is not always prefaced with homicide, homosexual opportunistic necrophiles such as mortuary attendants, for example. There is corroboration of the assertion that necrophilic behaviour in homosexual men is prefaced with a deviant fantasy life, just as has been found in necrophiles who abuse the corpses of women (Aggrawal, [ 2]p76). As this prefacing fantasy life is also found in cases of heterosexual necrophilia, a distinct classification for homosexual men is possibly unnecessary.
It is the necrophilic rationale for murder which appears to be pivotal in this case; serial killing was instrumental only, a means to secure the bodies with which he could act out his fantasies; the offender is adamant that he took no pleasure in the act of killing. The fact that his victims were homosexual, or assumed to be homosexual, merely made it easier for the offender to lure them to his home, on the promise of a sexual encounter. Homosexual men were targeted for their 'ease' as victims, not because their sexual orientation was a necessary component in acting out his necrophilic desires. In this way studies which depict homosexual men at a higher risk of victimisation, because of their risk taking in seeking sexual encounters, is endorsed (Beauregard & Proulx, [ 5]p484; Geberth, [14]p653).
Implications
This is a case which Aggrawal asserts has not been recorded before, type III homosexual necrophilia, in which a perpetrator will engage in homosexual acts with both living and dead males (Aggrawal, [ 2]p90). In a number of ways the case presented here reinforces other accounts of necrophilia, those involving a male perpetrator and female victim, the centrality of low self esteem, sexual fantasy as a precursor to offending, and fear of rejection. Moreover, the offender in this case prefaced the acquiring of a corpse with increasingly intricate sexual fantasies, just as found in cases of heterosexual necrophilia. Seemingly then, if this case is typical and deviant sexual fantasy is present in necrophiles regardless of gender, homosexual necrophilia might not warrant a specific category or distinct taxonomy. If that finding is borne out in other similar cases then there is the possibility of intervention at the fantasy stage, before necrophilic fantasy is acted upon, if potential offenders can be identified. To corroborate that finding, however, the limited literature on the subject of necrophilia would benefit from accounts of female perpetrated necrophilia, concerning both male and female victims.
For clinicians treating those who present thoughts and desires of necrophilia it is important to note the possibility of progression from thought to action. In this case, when fantasies could no longer provide fulfilment, and environmental stressors reduced the inhibition to offend, murder became the means by which a body could be procured to act out those fantasies.
For investigators the manner of death presented here is significant. The offender in this case deliberately targeted strangers and made a conscious decision regarding how he would kill, to complete his sexual fantasy he strangled so as not to inflict unnecessary wounds or damage to his victim. This case then suggests that investigators should consider stranger perpetrated homosexual homicides, without the use of an edged weapon and without evidence of overkill, might have a sexual motive.
2nd article
The Role of Sexual, Sadistic, and Misogynistic Fantasy in Mass and Serial Killing.
A subset of violent criminals fixates on deviant (and often grossly misogynistic) sexual fantasies, or engages in sexually sadistic behavior toward victims. Although the role of psychosexual deviancy is quite common among many notorious serial killers (e.g., Ted Bundy, Gary Ridgway, Dennis Rader), it appears to be a motivating factor for some recent mass killers as well. Evidence indicates that the killer's psychosexual issues were a motivating factor for mass killers (e.g., George Sodini, Jared Lee Loughner, and Elliot Rodger) in their assault. This article will compare and contrast the role and influence of a deviant sexual/misogynistic fantasy formation process for mass and serial killers.
Mass and serial killers share many similarities both prior to their fantasy formation stage, and during the fantasy-conditioning process that spans over a substantial amount of time, and increases in intensity and violence. However, the way in which these fantasies are expressed differs for each type of criminal mindset.2
The common (grossly misogynistic) and sexually sadistic behavior of many serial killers has been studied for well over half a century. This type of deviancy was a prime motivator for notorious serial killers (e.g., Ted Bundy, Gary Ridgeway, Dennis Rader). However, the role of (grossly misogynistic) and psychosexual deviant fantasies fueling the mass killers' behavior is a relatively new phenomena. By conducting ethnographic content analysis on the cases of mass killers (e.g., George Sodini, Jared Lee Loughner, and Elliot Rodger) this research unearths similar fantasy propulsion full of rage, hatred, and sadistic revenge toward women.
The psyche for both serial and mass murderers is marked by low self-esteem; real and perceived rejection; and despair, powerlessness, and self-loathing that is projected onto others. Both mass and serial killers direct blame for their real and perceived shortcomings outwardly. During adolescence and on into adulthood, the potential killer uses fantasy and daydreaming as a means to compensate for deficient social relationships and for some a lack of sexual relationships. Over time, as distorted perceptions become ingrained thought patterns, this potentially violent criminal goes deeper and deeper into a mordant fantasized world. Eventually, it becomes an inefficient coping mechanism; the individual's attempt to re-establish psychological equilibrium must be advanced and the fantasy must be actualized (Proulx et al. [24]:31).
Sadistic serial killers tend to have particularly detailed and elaborate fantasies—"scripts of violence," rich with themes of abuse, control, and dominance (Skrapec [28]). Killing is typically, but not always, a means of sexual arousal and gratification for this offender. This type of serial killer is motivated by themes of absolute power over another human being. The few mass killers who have a sexual motivation for their crimes likewise derive pleasure from fantasies of sadistic violence, control, and domination.
The mass killer, though, is more likely to have generalized feelings of undue superiority over, or unrealistic anger or hatred toward, representative female victims. Sexual sadism, deviancy, and sexual gratification from violence are not the primary motivations for the mass killer. There is an element of selfish, misplaced anger and blame in this killer's fantasy (i.e., "Women are worthless," or "If I can't have intimacy then I will deny it for them [women] and those who care about them").
Defining serial and mass murder
"Mass killings" or "massacres" are defined as the killing of four or more victims, by a lone assailant (or a few assailants), usually in a single location, or in several locations in close proximity; the incident only lasts from a few minutes up to several hours (Federal Bureau of Investigation [FBI] [ 7]:8; Fox and Levin [ 9]). As for "serial murder," the FBI's Behavioral Science Unit's definition requires a minimum of two victims, whom the perpetrator may kill "over a period of months or years" (p. 9). There can be substantial lapses of time (i.e., a cooling off period) between homicides; and during this time, the killer may maintain the appearance of a more or less ordinary, unassuming life (Fox and Levin [ 9]). Thus, the main differences between "mass killing" and "serial killing" lies in the temporal and spatial "relatedness" of the criminal conduct, whereby "relatedness" means how each kind of killer goes about targeting and assaulting victims.
Methodology
The methodology used for this research was ethnographic content analysis and, to a lesser extent, empirical phenomenology and semiotics. Source materials that were analyzed included: artifacts generated by the offenders prior to commission of their crimes (e.g., diaries, manifestos, blogs, drawings, photographs, and videotapes); official findings of governmental review panels; other public documents; survivor, witness or family accounts; news reports; and work conducted previously by other researchers.
This research method systematically dissected the portrait that the killer has painted of "himself and his world" (p. 50; Hickey [13]:353, citing Skrapec [28]). One's thought and behavior always "has context and subjective meaning" (Hickey [13]:354). Thus, "[i]n understanding ... [what] repeated acts of killing mean" to the offender, it is necessary first to inquire about "the motive forces that drive [such] behavior" (Hickey [13]:354). Accordingly, this project had to "identify the principles that organized" the killer's feelings, perceptions, thinking and finally his or her behavior (Hickey [13]:35; citing Kluckhohn and Murray [15]:55). This research, rests on "understanding the stories or narratives of the offenders," even if in reality such stories or narratives are nothing but falsehoods and fantasies (Hickey [13]:354).
With respect to understanding a semiotic approach, Altheide ([ 1]:8), cites researchers Manning and Cullum-Swan ([18]), who describes the objective of semiotics as "to place the [originator's] writing ... into alternative contexts or fields, or to recode the text." "Writing displays reality" as viewed by the originator, and writing is often "a means to an end"—quite literally in the case of mass killers (Plummer [22]:169). Reality, as expressed in these research documents displays a range of emotion behind the texts—neutral and dispassionate observations, angry or rageful tirades, humorous or ironic musings and morose, self-pitying obsessions.
The role of deviant sexual fantasy: Influences on serial murderers
Researchers associated with the Federal Bureau of Investigation's 1979–1983 study of 36 incarcerated sexual murderers in the United States have defined the term fantasy as "an elaborated set of cognitions (or thoughts) characterized by preoccupation (or rehearsal), anchored in emotion and originating in daydreams" (Prentky et al. [23]:889; see also Burgess et al. [ 4]). They additionally put forward fantasy as the principal motivator in serial sexual homicide (Prentky et al. [23]). These researchers further hypothesize that these men are motivated to murder by their way of thinking. Over time, their thinking patterns emerged from, or were influenced by, unrelenting traumatic early life experiences (Burgess et al. [ 4]).
A lust murderer is a serial killer who is influenced by violent deviant sexual impulses. This killer primarily seeks pleasure in his crimes by sexually torturing, raping, and murdering his victims. Although this impetus is frequently observed among serial killers, it is a "comparatively rare typology for mass killers" (Hickey [13]:12). The lust murderer possesses a particular urge not simply to kill, but to sadistically ravage his victim(s) either pre- or post-mortem (Purcell and Arrigo [25]). "The sexualized persecution of the victim is at the core of the assailant's behavior," and violence "is principally inflicted as a means of sustaining arousal and attaining orgasm" (Purcell and Arrigo [25]:26). The sadistic killer is engrossed in a disturbing fantasy world of compulsive fixation on the absolute control over, objectification of, and the infliction of pain onto the victim.
An example of this type of sadistic murderer is serial killer Dennis Rader; also self-named the BTK, which stands for bind, torture and kill. Rader murdered 10 people (mostly women) in Wichita, Kansas from 1974 to 1991. Wanting full credit for his murders and a form of serial killer celebrity status, Rader taunted the police by sending letters describing his torturous killings. A sample of one of his letters espouses his sexual gratification through his sadistic acts of horror:
The victims are tie up-most have been women-phone cut-bring some bondage mater sadistic tendencies-no struggle, outside the death spot-wintness except the Vain's Kids. They were very lucky; a phone call save them. I was go-ng to tape the boys and put plastics bag over there head like I did Joseph, and Shirley. And then hang the girl. God-oh God what a beautiful sexual relief that would been. Josephine, when I hung her really turn me on; her pleading for mercy then the rope took whole, she helpless; staring at me with wide terror filled eyes the rope getting tighter-tighter. (Frater [10])
Rader typically did not rape his victims while alive; rather, what excited him was the power, thrill, and control he had over his victims while he killed them. His favorite modus operandi was to strangle his victims near death, release his hands so they could regain consciousness and awareness of their fate, only to strangle them again. Their suffering empowered and aroused him; once the victim was dead, he would often masturbate into his female victim's underwear.
In a sample of twenty sadistic serial killers analyzed by Hazelwood, Dietz, and Warren ([12]), 80% reported having violent fantasies (Fox and Levin [ 8]:417). The interviews with these killers captured every detail from the offender's perspective—the imagined criminal act, including the capture of a victim, the infliction of extreme pain and suffering, the actual murder, and then disposal of the body (Hazelwood et al. [12]). The conclusion drawn from this and similar research is that repetitive, intense, and detailed "deviant sexual fantasies" are a galvanizing event for the killer's actualization to sadistic violence (Proulx et al. [24]:43; Purcell and Arrigo [25]).
Ted Bundy, who confessed to sadistically killing over 30 women from 1974 to 1978, affirmed in several post-incarceration interviews that his fantasies clearly "accompanie[d] and generate[d] the anticipation that precedes the crime [and that such apprehension] is always more stimulating" than the crime itself (Leyton [17]:125).
The fantasy works as an "internal" driving force to normalize recurring acts of sexual brutality (Purcell and Arrigo [25]:23). Moreover, "the nature and frequency of the deviant sexual fantasies are good predictors of the severity of the offence" (Proulx et al. [24]:43). Through violent sexual abuse and murder, the lust killer literally chases his dreams (Fox and Levin [ 8]:417).
Bundy's modus operandi was to attack his victims with a blunt instrument. Some would die immediately, while others would suffer for "hours or days" until he would strangle them (Hickey [13]:153). Bundy "raped most, if not all, of his victims," and many "were subjected to sodomy and sexual mutilations" (Hickey [13]:153).
The role of deviant sexual fantasy: Influences on mass murderers
Dying for company and intimacy
The sexualized aspect to mass killing, by contrast, tends to involve misogynistic themes of anger or revenge toward women for real or perceived rejection. In some cases, there maybe an element of eroticizing violence and sadistic fantasy, but probably not to the degree as observed with the sadistic serial killer. The mass killer externalizes blame for his own personal psychosexual shortcomings and social deficits onto women.
George Sodini (48 years old) appeared to harbor such odium toward women. On August 5, 2009, Sodini walked into an LA Fitness Center that he had frequented near Pittsburgh, Pennsylvania, turned off the lights, and opened fire on a women's aerobics class. Sodini killed three women, himself and badly injured nine other women (CNN.com/crime[ 6]). Extreme loneliness and the lack of ever having a sexual or close relationship with a woman seemed to fuel Sodini's resentment and vengeance. Prior to Sodini committing these offenses, over a nine-month period, he chronicled his feelings of rejection by women and his deep-seated sexual frustration. On a website registered in Sodini's name, he wrote, "No girlfriend since 1984." "Who knows why? I am not ugly or too weird. No sex since July 1990 either (I was 29)," he writes, "Last time I slept all night with a girlfriend it was 1982. Girls and women don't even give me a second look ANYWHERE." Sodini continued about his problems with women, by writing: "Women just don't like me. There are 30 million desirable women in the U.S. (my estimate) and I cannot find one." In another entry, he writes: "Flying solo for many years is a destroyer" (NYPOST.com[20], Blog Full Text Source).
Sodini was angry and isolated, and craving an intimate relationship that he believed he could never achieve. A violent, misogynistic revenge fantasy appeared to temporarily replace the lack of intimacy, and the inability for Sodini to obtain a female partner. Because of this painful emotional void with the opposite sex, he chose to strike out violently against them. His victims served as representatives, who could be blamed for Sodini's own feelings of anger, inadequacy and humiliation. Here, too, though it is possible that Sodini attained some form of sexual gratification from inflicting a sort of violent revenge against the female gender. It is additionally thought that by killing these women they, along with their partners, would be denied the life and happiness that Sodini desperately craved.
The sadistic mass killer: Elliot Rodger: Flay them alive
More recently, Elliot Rodger (22 years old) wrote a 141-page manifesto titled "My Twisted World," describing his childhood, family problems, frustration over not being able to find a girlfriend, and hatred for women and their boyfriends before going on a rampage through the University of California–Santa Barbara campus. Rodger killed six people, himself, and injured 14 others on May 23, 2014. His onslaught began by stabbing his two roommates and a third man to death in his apartment, he then drove to a sorority house and shot three women outside, killing two of them, next he drove to a deli where he shot to death a male student. Rodger continued to speed through the streets injuring others by both shooting and hitting people with his car. Eventually he crashed his car and shot himself to death.
Rodger, similar to Sodini, wanted to make both women and their male counterparts suffer. He blamed women for depriving him of love and sex, and men because they were the recipients of their affections. In Rodger's words: "I hated all of those obnoxious, boisterous men who were able to enjoy pleasurable sex lives with beautiful girls, but I hated the girls even more, because they were the ones who chose men instead of me" (Rodger [26]:117). Rodger additionally stated, "If I can't have them, no one will" (Rodger [26]:136).
Akin to Sodini, Rodger talks about how he never got to have close intimate experiences with women. Rodger: "I could never have the experience holding hands with a beautiful girl and walking on a moonlit beach, I could never embrace a girlfriend and feel her warmth and love" (Rodger [26]:117). Due to female rejection, Rodger describes women as having sexually starved him and they are "vicious, stupid, cruel animals" (Rodger [26]:117). Like Sodini, Rodger could not understand why women did not like him, and made comments that he was the perfect guy. Instead of taking any blame, Rodger believed that women's sexual attractions were flawed, and that they were attracted to the wrong type of male.
Elliot Rodger writes about his profound frustrations and near explosions toward women and their boyfriends throughout his manifesto, but his recalcitrant and completely out of control indignation did not take a firm hold until he was 19. While at Starbucks, Rodger witnessed a couple kissing passionately and he was in his words, "livid with envious hatred" (Rodger [26]:87). When the couple left Starbucks, Rodger followed them to their car and splashed his coffee all over them. In his manifesto he states: "I had never struck back at my enemies before, and I felt a small sense of spiteful gratification for doing so. I hated them so much" (Rodger [26]:87). Rodger gained a sense of power by physically showing his disdain for the couple. Command over a situation and people had eluded him his entire life; it felt good to be the aggressor.
"Many mass killers desperately try to fit into society and their immediate social groups, be it school or work"; some search for an individuality that will bring them favorable attention from their peers (Murray [19]:237). However, with the inability of achieving "a solid sense of who they are"—other than the label of misfit, loser, or undesirable—"many mass killers give up on conventional means of recognition" (Murray [19]:237). Thus, over time, extremely violent offenders develop what Lonnie Athens ([ 2]) calls a "barbaric individualism." The killer is in fact antagonistic to society, the attitude that eventually emerges.
This was a critical turning point for Elliot Rodger; rather than tapping down his anger, he began to release it on others, and dove deeper into a very graphic sadistic fantasy world completely focused on violent retribution. These thoughts permeated his manifesto from this point on. After throwing the coffee on the couple, Rodger further reflects:
I wanted to do horrible things to that couple. I wanted to inflict pain on all young couples. It was around this point in my life that I realized I was capable of doing such things. I was capable of killing them, and I wanted to. I wanted to kill them slowly, to strip the skins off their flesh. They deserve it. The males deserve it for taking the females away from me, and the females deserve it for choosing those males instead of me. (Rodger [26]:87)
Many more incidents of flinging drinks on couples that openly displayed affection and women that rejected him followed. Rodger always felt slighted by couples that were showing their affections. Statements such as wanting to "flay them alive," and "make them scream in agony as punishment for living a better life" seemed to be at the root of his anger; he so desperately wanted to live their way of life (Rodger [26]:91). At this point, Rodger is not just immersed in a blameful misogynistic fantasy world; he has crossed over into a sadistic fantasy world of both torture and murder, not unlike sexual sadist serial killers.
In his manifesto, Rodger continued to vilify women because they controlled who got sex and who did not. Rodger believes "there is no creature more evil and depraved than a human female. Women are like the plague. They don't deserve to have any rights. Women are vicious, evil, barbaric animals, and they need to be treated as such" (Rodger [26]:136). He further decrees that all women should be quarantined and starved to death. He additionally states that: "A few women would be spared, however for the sake of reproduction. These women would be kept and bred in secret labs" (Rodger [26]:136). Like a mad scientist, he wants to create a woman he can control and that will love him.
At the very end of his manifesto, Rodger becomes even more detailed in his vengeful plans. He describes how on the day of retribution he will begin by luring people one by one into his personal torture and killing chamber, knock them out with a hammer, slit their throats, behead them, and stash their heads in a bag to use later for further shock value when he parades down the streets further massacring everyone (Rodger [26]).
His detailed description of his plans rivals actual killing methods used by numerous sadistic serial killers combined. In his twisted sadistic fantasy world, he wants to use violence to stop the abuse of rejection, gain power and respect, and frighten others.
Misogyny, psychosis, and revenge: Jared Lee Loughner—women enjoy rape
Jared Lee Loughner (22 years old) presents an example of a mass killer with a misogynistic sexual revenge fantasy infused with some form of psychotic paranoia. On January 8, 2011, Jared Lee Loughner went on a rampage outside a Safeway grocery store in Tucson, Arizona. Loughner killed 6 people and injured 14 others, including severely injuring U.S. Representative Gabrielle Giffords (NYTimes.com[21]). The shooting took place at a constituency gathering or "town hall meeting" that was purposefully held at a commercial location with little or no security, so that Representative Giffords could meet and talk with ordinary citizens going about their weekend errands. Based on evidence gathered in the police investigation, it appears that there are several underlying motivations to Loughner's behavior: sexual fixation and anger directed at women; revenge for the rejection that Loughner experienced in social, academic, and employment circles; and the effect of untreated mental illness or psychosis.
Prior to the massacre Loughner posted 131 messages between April and June 2010 in a private forum associated with the online game Earth Empires (CBSNews.com[ 5]; WSJ.com[30]). Many of the postings touched on various themes including: misogynistic, angry rants about women. On April 24, 2010, one particularly disturbing rant titled "Why Rape" explained how women in college actually enjoyed being raped (WSJ.com[30]). In this posting, Loughner incoherently wrote: "There are Rape victims that are under the influence of substance. The drinking is leading them to rape. The loneliness will bring you to depression. Being alone for a very long time will inevitably lead you to rape" (WSJ.com[30]). Loughner followed up on May 5, 2010 with an entry titled "Talk, Talk, Talking about Rejection"—the posting appeared to be eliciting other forum members for their stories of rejection by the opposite sex (WSJ.com[30]). The next day, Loughner wrote, "It's funny ... when ... they say lets go on a date about 3 times ... and they don't ... go..." (WSJ.com [30]). Through this forum, Loughner could vent his frustration, commiserate with others similar to him, and build justification for his beliefs and fantasized revenge. The forum appeared to become a fruitful fantasy formation tool.
Comrades in arms
Elliot Rodger fed and justified his misogynistic fantasies in a similar fashion by joining the online chat club "Pua-Hate.com," that in Rodger's own words is a "forum full of men who are starved of sex, just like me" (Rodger [26]:117). He goes on to describe what the club provided for him:
Many of them have their own theories of what women are attracted to, and many of them share my hatred of women, though unlike me they would be too cowardly to act on it. Reading the posts on that website only confirmed many of the theories I had about how wicked and degenerate women really are. (Rodger [26]:117–118)
The overall theme of this chat room is to find comradery with other males rejected by women. The members of this organization often refer to themselves as "incels," which stands for involuntary celibacy (Glasstetter [11]). One of Rodger's comments in this forum was, "If we can't solve our problems we must DESTROY our problems" (NYTimes.com 2014). In part of another reference about "incels," Rodger states "Start envisioning a world where WOMEN FEAR YOU" (Glasstetter [11]). Throughout more of Rodger's statements about the participants in this forum, it is evident that he feels a connection with these men, yet he considers himself as supreme and stronger than they are. Even in an environment most suited for him to connect with others, he still cannot do so completely. Over time, in this forum, Rodger becomes too extreme in his hatred even for their taste.
Bruised manhood
The search for, and destruction of, their idealized woman
Serial killer Ted Bundy's "main motivation was power and control"; however there was an obvious sexual element of torture in his killings in order to achieve this feeling of control (Holmes and Holmes [14]:49). Bundy wanted to totally possess his victims, and he gained immense pleasure knowing the victim's life or death depended on him solely (Holmes and Holmes [14]).
This pleasure and arousal for the killer is entrenched in hatred and loathing, as reported by Sears' ([27]) study on the etiology of the serial killer (Holmes and Holmes [14]:60). For Bundy, this hatred was focused onto an archetype victim. Unlike most serial killers who seek their victims from the transient fringes of society, Bundy sought out the idealized woman. His victims typically were attractive, upper-middle-class brunettes (Leyton [17]:125). It is supposed that Bundy's choice of victims stemmed from a turbulent relationship and rejection by his fiancée Stephanie Brooks, who came from a genteel, well-connected background to which Bundy always aspired. Bundy came from a chaotic lower-middle-class family; a circumstance he despised. When the relationship with Stephanie ended, Bundy perceived the loss as also destroying his chance to climb above his origins. Bundy felt this loss deeply; therefore, he indulged his feelings of animosity and revenge by seeking representative female victims who physically resembled Stephanie and appeared to come from a similar social class from which he so acutely felt rejection.
Mass killer Elliot Rodger was enamored with attractive blond women, and felt that sorority girls epitomized this type of female. These unattainable women had shunned him his entire life. In his attempt to gain a sense of power and control, after violently killing three men in his apartment during the start of his actual massacre, Rodger's next stop was a sorority house. George Sodini thought the women that worked out at the L.A. Fitness Center were a perfect representation of all the attractive women that had always eluded him. Bundy, Rodger, and Sodini all sought out whom they believed to be the unattainable perfect woman.
Internalized feminization of man
With 48 serial murder convictions, Gary Leon Ridgway (52 years old) at the time of his arrest in 2001 is perhaps the most prolific serial killer in U.S. history. Ridgway's victims were primarily prostitutes and runaways that were easy to pick up while also remaining undetected, and they were typically not reported as missing right away, if at all. Ridgway—also known as the Green River Killer—strangled many of his victims during sex, and left the bodies in clusters along banks of the Seattle, Washington Green River. Ridgway has stated that he enjoyed choking his victims, and that killing prostitutes was a "career" (Hickey [13]:25).
It is believed that with each of Ridgway's murders, "he was reinforcing his own male privilege that he could never express with his overbearing mother whom he was unable to please; the punishment of his victims compensated for his bruised manhood" (Levi-Minzi and Shields [16]:82).
The impact of sexually motivated revenge killing is documented among many theorists. In the book Lustmord: Sexual Murder in Welmar Germany, Maria Tatar ([29]) states that, "Women are punished and blamed for the feminization of men, their loss of control, impotence and even castration. In short, to dismember woman allows man to remember himself." This violent form of murder—whether it involves the physical sexual dismemberment of a victim by a sadistic serial killer, or whether it involves a large-scale assault on many lives by a mass killer— can, at least in part for some, stem from sexual inadequacy turned into rage.
Conclusion
Psychosexual deviant fantasies fueling the mass killer's behavior is a relatively new phenomena. A content analysis of three recent mass killers unearths similar fantasy propulsion to that of a sadistic serial killer, full of misogynistic rage, hatred, and sadistic revenge toward women.
Some common psychological characteristics that develop for both serial and mass killers are feelings of inadequacy, self-doubt, and worthlessness. The potential killer externalizes blame and seeks solace in fantasies of revenge as a substitute for their lack of social connections. Over time their obsessive rumination both conditions and provides justification, in the murderer's mind, to actualize their violent plans.
The murderer that craves ultimate power and control "kills because he is gratified by his ability to hold the fate of another in his hands" (Holmes and Holmes [14]:41). This appears to be true with both mass and serial murderers. However, with sexually sadistic killers, it is the need for "power, dominance, and control," to the degree that "the victim is demeaned and humiliated," thus the "killer is able to feel superior, exalting in the victim's suffering" (Borgeson and Kuehnle [ 3]:24). For some sadistic lust murderers a sexual gratification is achieved either pre- or post-mortem.
For a few mass killers, a sexual aspect influences their motivation—who, how, where, when—to kill. Severe psychosexual issues are a theme underlying the behavior of some mass killers. The themes, though, tend to involve deeply misogynistic feelings, anger, or hatred for women—including indignation over having romantic overtures repeatedly rejected. The mass killer typically lacks an intense need for sexual gratification from the imagined and actual homicides. Mass killers George Sodini and Jared Lee Loughner had a deep-seated hatred for and were continually rejected by women. Their misogynistic fantasies of homicidal payback were directed toward representative female victims.
However, Elliot Rodger had deep-seated misogynistic and sadistic fantasies unerringly like a hedonistic serial killer. He was a fantasy hybrid killer that yearned for enacting systematic torture and murder on a massive scale. He wanted to feel the power and control over a person hands on like a sadistic serial killer, as well as guns blazing like a mass killer and ostensibly wanted to accomplish all of this without a cooling off period.
These sexual fantasy patterns do not just spring up spontaneously out of nowhere; they develop over years and years, and this most likely means that people close to the killer (parents, siblings, former girlfriends/boyfriends, school-mates, teachers) potentially could have seen this pattern developing at earlier stages. This is an important point with policy/response implications and the need to educate the appropriate people about catching these patterns as early as possible. The earlier, the more likely a successful intervention, the later, the more likely a full-blown tragedy and many lost lives.
3rd article
Fractional Order Mathematical Model of Serial Killing with Different Choices of Control Strategy
Abstract
The current manuscript describes the dynamics of a fractional mathematical model of serial killing under the Mittag–Leffler kernel. Using the fixed point theory approach, we present a qualitative analysis of the problem and establish a result that ensures the existence of at least one solution. Ulam’s stability of the given model is presented by using nonlinear concepts. The iterative fractional-order Adams–Bashforth approach is being used to find the approximate solution. The suggested method is numerically simulated at various fractional orders. The simulation is carried out for various control strategies. Over time, all of the compartments demonstrate convergence and stability. Different fractional orders have produced an excellent comparison outcome, with low fractional orders achieving stability sooner.
Keywords: Adams–Bashforth method; fixed point theory; serial killing; Mittag–Leffler kernel
1. Introduction
All governments throughout the globe have focused their attention on crimes, specifically serial killings. All the crime is a dignified sociological problem that has been broadly studied in the scientific literature [1]. Every year, billions of dollars are spent around the world to control crime. Different imprisonment and restoration centers are made to handle the crimes of addicted persons [2]. Crime has five types, violent crime, white-collar crime, property crime, organized crime, and consensual crime [3]. One type of violent crime is a serial killer. Serial killer refers to one who commits the crime (murder) three times or more with a cooling-off period between the murders [4,5]. These killers murder their targets and create a very negative impact on society. The killers may or may not be strangers to the target. The killers prefer the areas where they have a recognizable base and the area is being circumscribed by their offenses [6,7].
Gangs cannot be ignored in the history of crimes. Gangs are expanding worldwide and are involved in different types of crimes in rural and urban areas, especially violent crimes. Globally, 48 percent of violent crimes are reportedly due to gangs. Further, 27 percent of crimes are general in small cities, while 66 percent of general crimes in big cities due to gangs. In order to securely carry out their perverted urges and bloodlust, a serial killer joins a criminal organization. As such, the existence of gangs has emerged as the most sensitive public matter [8,9].
The mathematical model provides an examination of the development of criminality and its different effects, including sociological and economic factors. Such types of research suggest possible strategies for decreasing and controlling crimes [10,11,12,13]. Recently, several researchers have studied different models for crimes by using ordinary and partial differential equations [2,14,15,16,17]. The model used is expressed in the form of the following DEs:
dShdt=Λ−b(β1+β2)ShGhN−πSh+β4u4Jh,dWhdt=bβ1ShGhN−(u1+π)Wh,dChdt=u1Wh+bβ2ShGhN−(u2+u3+π)Ch,dGhdt=u2Ch−(π+β3)Gh+β4(1−u4)Jh,dJhdt=u3Ch+β3Gh−(β4+π)Jh,
(1)
In Model (1), the total population is divided into five classes based on stages of addiction to crimes (serial killing). The first class is the susceptible humans Sh in the population that may or may not have contact with serial killers. Most individuals are in the age of 14 years or greater. The Wh is the weaponized class, Ch is the active class of serial killers, Gh is the class of gang and Jh is the serial killers detained in jail. The parameters in Model (1) are given with complete descriptions below (Table 1).
Table 1. Meaning of parameters of the model (1).
Table
Researchers have given fractional calculus a lot of attention, and it has been applied in a variety of disciplines. Researchers have developed mathematical models for a variety of diseases, such as in [18,19,20].The majority of mathematical models are based on integer-order differential and integral equations. Fractional differential equations (FDEs) have been extensively utilized for the last twenty years to construct models of real processes with a higher degree of precision and accuracy [21,22]. Many scholars have utilized a variety of approaches to analyze fractional order (FO) mathematical models qualitatively; see, for example, [23,24]. Non-linear FDEs are notoriously difficult to solve. To deal with this problem, many mathematicians have constructed a variety of approaches for computing approximate solutions for nonlinear systems [25,26,27,28].
To address the shortcomings of the ordinary operator, a variety of fractional order derivatives have been designed [29]. Riemann–Liouville constructed the definition of the fractional derivative (FD). Later on, Caputo subsequently redefined and enhanced the definition of FD. The definition of Caputo FD is based on the singular power-law kernel. The study of real problems using FDs frequently results in singularities that are unsatisfactory for mathematical model dynamics. After many decades, a new FD known as the Caputo–Fabrizio (CF) operator was defined through a non-singular kernel to avoid such a problem [30]. In this operator, there is the kernel’s locality problem. To address these limitations, Atangana and Baleanu (AB) [31] introduced a novel type of FD through the nonsingular and nonlocal kernel, which we call the Mittag–Leffler kernel. The new derivative operator was also employed, ensuring that the kernel has neither singularity nor localization. The AB operator has many applications in the applied sciences. For instance, Rahman et al. used the AB operator to analyze the TB disease with incomplete treatment [32]. Ahmad et al. investigated the tumor-immune-vitamins model using the AB fractional operator [33]. The Kawahara equation has been studied under the AB operator by Rahman et al. in [34].
In this paper, we examine a novel fractional mathematical model of serial killing. This model, reported in [35], has not been studied for particular crimes in the sense of ABC FO. This new work provides qualitative and quantitative results regarding the dynamics of serial killing in terms of different strategies.
5. Numerical Simulations and Discussion
In this section, we establish the approximate solution of our considered Model (2) using various parameters given in Table 2 for verification of the proposed scheme. The initial values for all cases of the given system are Sh(0)=1000,Wh(0)=20,Ch(0)=20, Gh(0)=10 and Jh(0)=10. We have taken four different sets of parameter, one without control and the remaining three simulated by applying some control strategies for all of the compartments in problem (2) and at a different fractional order of ℘. Figure 1a–d represents the the dynamics of susceptible humans (free of crimes) Sh(t) at a different fractional order of ℘ before and after control strategies. Figure 1a is the plot before control parameters in which the susceptible people are decreasing and transferring to the serial killers. While Figure 1b–d are the plots after we applied the control strategies by changing the values of the most affected parameter, as given in Table 2. In these three figures, the susceptible class is controlled and increasing. Figure 1e is the combined graph. Figure 2a–d show the dynamics of weaponization in humans Wh(t) at different fractional-orders of ℘ before and after control strategies. Figure 2a is the plot before the control parameters in which the weaponized people are increasing and, thus, the number of serial killers also increases. While Figure 2a–d are the graphs after applying the control strategies by changing the values of most affected parameters as given in Table 2. The weapons class is controlled, reduced, and approaching zero in these three figures. Figure 2e is the combined graph. In Figure 3a–d, the dynamics of serial killer humans Ch(t) have been shown at different fractional-orders of ℘ before and after control strategies. Figure 1a is the plot before the control parameters in which the serial killers are increasing and converging. While Figure 3b–d are the simulations after applying control strategies by changing the values of most affected parameters, as given in Table 2. In these three figures, the serial killers’ class is controlled, decreasing, and tends to zero. Figure 3e is the combined graph. In Figure 4a–d, the dynamics of gang member humans Gh(t) have been shown at different fractional-orders of ℘ before and after control strategies. Figure 4a is the plot before the control parameters in which the gang members are increasing and diverging. While Figure 4b–d are the simulations after applying the control strategies by changing the values of the most affected parameters, as given in Table 2. In these three figures, the gang member class is controlled, decreased, and tends to zero. Figure 4e is the combined graph. In the first four subfigures (a–d) of Figure 5, the behavior of serial killers detained in jail Jh(t) have been shown at different fractional-orders of ℘ before and after control strategies. Figure 5a is the plot before control parameters in which the jailed members are extremely increasing and diverging. The Figure 5b–d are the representations of jailed members after applying the control strategies by changing the values of the most influencing parameters, as given in Table 2. In these three figures, the gang member class is controlled, decreased, and tends to zero. Figure 5e is the combined graph.
Fractalfract 06 00162 g001 550Figure 1. (a–e) Plots of Sh(t) having four different sets, before and after control strategies in the problem under analysis (2) at various arbitrary orders.
Fractalfract 06 00162 g002 550Figure 2. (a–e) Plots of Wh(t) having four different sets, before and after control strategies in the problem under analysis (2) at various arbitrary orders.
Fractalfract 06 00162 g003a 550Fractalfract 06 00162 g003b 550Figure 3. (a–e) Plots of serial killers Ch(t) having four different sets, before and after control strategies in the problem under analysis (2) at various arbitrary orders.
Fractalfract 06 00162 g004a 550Fractalfract 06 00162 g004b 550Figure 4. (a–e) Plots of gang members Gh(t) having four different sets, before and after control strategies in the problem under analysis (2) at various arbitrary orders.
Fractalfract 06 00162 g005 550Figure 5. (a–e) Plots of Jh(t) having four different sets, before and after control strategies in the problem under analysis (2) at various arbitrary orders.
Table 2. Values of the parameters used in Model (2) with different control strategies.
Table
6. Conclusions
In the current paper, we have studied a fractional-order mathematical model of serial killing under the ABC operator. We have proved the existence of the solution to the problem under examination by using the fixed point theorem. We used the Adams–Basforth method to attain an approximate solution to the given model. We showed the stability of the given model through the nonlinear concepts of Ulam–Hyers. We performed numerical simulations for two different cases to support our analytical findings. We discussed the given model for control with and without control strategies. All components of the presented model have attained stability and convergence. The stability of the decay process adjusts quickly to small fractional orders, whereas the stability of the growth process adjusts quickly to higher orders. The given model provided global information due to fractional analysis of an extra degree of freedom. Thus, the fractional-order Model (2) is superior to the integer-order Model (1). In the future, more global and generalized operators might be used to investigate the given model.
4th article
A Literature Review on Female Serial Killing: Examining Gendered Features of the Crime
https://escholarship.org/uc/item/6qk0h8xv