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International
Journal of
Mathematical
Analysis and
Modelling
(Formerly Journal of the Nigerian Society
for Mathematical Biology)
Volume 5, Issue 2 (Sept.), 2022
ISSN (Print): 2682 - 5694
ISSN (Online): 2682 - 5708
International Journal of Mathematical Analysis and Modelling
Volume 5, Issue 2, September 2022, Pages 121 – 136
121 IJMAM, Vol. 5, Issue 2 (2022) ©NSMB; www.tnsmb.org
(Formerly Journal of the Nigerian Society for Mathematical Biology)
Mathematical modeling of rape under the influence
of human disturbance and noise
C. Nkuturum
*
and M.N. Onwubuya
†
Abstract
To leverage rape decline, a guide from epidemiological modeling was used to develop a
new deterministic model for the dynamics of rape during covid-19 and the world at large.
This was used to qualitatively assess the role of convictions and loss of immunity to rape on
the transmission process of rape in a society. This study showed that loss of immunity to
rape and its rate of previous convicted and jailed rape culprit individuals regress to being
susceptible to rape cases induced a backward bifurcation when associated to rape
reproduction number (Ref <1) is less than one. This means that imprisonment of the
convicted rape individuals will help to eradicate rape out from the entire world even at covid-
19 pandemic. The study showcased that if the cause of backward bifurcation is removed,
the rape-free equilibrium (Re) of the model is globally asymptotically stable when the
reproduction number is less than one. This study also showed that the rape endemic
equilibrium (Ree) is locally-asymptotically stable when the reproduction number is greater
than one (Ref >1). The analytical solutions suggest that Policymakers should provide policies
that would check one time convicted and jailed individuals from being susceptible to rape
after their released from prison. Self-recovered individuals should be assisted and engaged
with employment as to avoid the stigmatization of rape.
Keywords
: rape; Covid-19; pandemic; SCPV model; noise and human disturbance.
1 Introduction
Rape is the abuse of humanity and personality trait to the victims and society at large. It is a sexual
assault that involves sexual intercourse or other forms of sexual penetration being carried out by
physical force or coercion against the person’s consent. Rape is an element of the crime of genocide
when committed with the intention to destroy a targeted ethnic group partly or completely. It can be
notable in various form as child rape and abuse, incest rape, sexual slavery, gang rape, prison rape,
marital rape, acquaintance rape, war rape and statutory rape, [16,18]. Rape is predominantly committed
by male factors against females. Globally, rape occurs in all continents of the world and the female
partners are always helpless when involved according to [20, 21, 22] stated that about 35% of women
worldwide have been raped in their life time.
*
Corresponding Author. Department of Mathematics, School of Foundation Studies, Rivers State
College of Health Science and Management Technology, Port Harcourt, Nigeria; E-mail:
nkuturumchristiana@rschst.edu.ng, christienkuturum245@gmail.com
†
Department of Statistics, Delta State Polytechnic, Oghara, Nigeria
C. Nkuturum and M.N. Onwubuya
122 IJMAM, Vol. 5, Issue 2 (2022) ©NSMB; www.tnsmb.org
(Formerly Journal of the Nigerian Society for Mathematical Biology)
Several reasons have been identified for the causes of rape in a society. Thus, it can be caused by
some of these factors drug abuse and alcohol addicts, lack of sex education, rape myths by men
against women, the desire to control, parental negligence, Peer and family factors, Societal factors
and poverty. Drug facilitated sexual assault is also known as predator-prey rape [12]. it is a sexual
assault carried out after the victim has become incapacitated due to having consumed alcoholic
beverages or other drugs, [13]. Alcohol has used to play an activating role during sexual assault,
when the rapists must have taken some drugs like Cocaine, Marijuana and Tramadol which
contributed to psychological havoc or trauma on the victims, [10,11, 14]. The major cost of rape
is risk of being caught, punished, convictions and jailed. Nevertheless, the probability of being
caught depends on the effectiveness of security and legal system of such countries, [15]. One may
wish to read the detail article on rape and its possible mode of control in [1].
The various body of literature existing on the social, political and management aspects of rape and
attempt to provide policies to curb rape. Scientifically, several attempts have been made to use
mathematical models to study the dynamics of rape in a population. Amongst others were [9,17]
used elementary model of population growth first order differential equation to study the growth
rate of domestic violence and predictions using differential equations in a population;
consideration was given in a community full of individuals prone to rape as case study. The work
done by [7] studied rape as a disease epidemic process and used [5] classical SIR epidemiological
mathematical models for modelling the spread of rape. He considered the population as made up
of individuals susceptible to rape (S), rapists (I) and individuals who recover and are immune to
rape (R) where the notations for the state variables were adjusted differently in [6]. [13] examined
the impact of chaos theory in social systems afflicted with rape.
Currently, this study presents a new deterministic model that studies the population dynamics of
rape in a population at covid-19 pandemic under the influence of human disturbance and noise.
The model treats rape as a disease as was done by [19], thus, the model could fit well for an
epidemiological setting even at covid-19 pandemic. In this present study, instead of having just
three classes used by [7, 21], a modification of the SIR models to SCPVR models was made and I
proposed dividing the population into five classes which includes the rape culprits (C), rape victims
or prey (P) and individuals who are convicted and jailed for rape they committed (V). Also, the
model considered human disturbance and noise factors. Hence, this paper is organized as follows:
the model formulation is section 2 and section 3 provide discussion and conclusions.
2 Model Formulation
The total population at time t, denoted by N(t) is subdivided into five mutually exclusive
compartments of susceptible individuals to rape (S(t)); individual for rape culprit (C(t)); individuals
who are victims of rape or prey to rape (P(t)), rape individuals who are convicted and jailed (V(t))
and individuals who recover from rape (R(t)) respectively. So that the total population becomes
N(t) = S(t) + C(t) + P(t) + V(t) + R(t).
Rape interaction in the population is modelled using a standard incidence function. The population
of individuals who are susceptible to rape S(t) is increased by the recruitment of a proportion ,
of immigrants (who are susceptible) into the population at a rate , ‘recovered’
individuals who become susceptible to rape again at a rate and convicted individuals who after
their release become susceptible to rape again at a rate . The susceptible population is decreased
by rape at a rate that is the rape force of infection, where
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Volume 5, Issue 2, September 2022, Pages 121 – 136
123 IJMAM, Vol. 5, Issue 2 (2022) ©NSMB; www.tnsmb.org
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,
and β is the rape transmission rate. The susceptible population is further decreased by natural
death at a rate ; natural removal occurs in all the compartments at this rate. Thus,
The population of rape culprit individuals C(t) is increased by the rape interaction that effects the
subsequently infects susceptible individuals at the rate . Because the human disturbance or
predation is negative on the culprit and positive on the victim or prey, so that it becomes
(q is the predating coefficient and E is total effort applied on predating on the population of the
culprit). The noise term is assumed to be sinusoidal with amplitude A and frequency is of the
noise, and phase shift . The population is decreased by trial, conviction and subsequently jailing
which leads to removal from the society at a rate , recovery at a rate and natural death . In
this study, the noise term from culprit is called black noise. This is colored noise because it does
not last long in the culprit and it shifts per time from when it was conceived and ends at the time
action has been executed by the culprit; gives
The population of rape victims or prey individuals P(t) is increased by the rape interaction, that
effects subsequently infect susceptible individuals at the rate . Because the human disturbance or
predation is positive on the victim or prey, so that it becomes (q is the predating
coefficient and E is total effort applied on predating on the population of the culprit as to stop the
action). The noise term is assumed to be sinusoidal with amplitude A and frequency is of the
noise The population is decreased by trial, conviction and subsequently trauma and sickness
which leads to removal from the society at a rate , recovery at a rate and natural death . In
this study, the noise coming from the victims or prey is called green noise. This green noise is a
limiting case of the black noise because it last long in the victims-prey; gives
The population of convicted and jailed individuals V(t) is increased by culprit individuals who are
caught and convicted at the rate (this population is in a form of isolation and without interaction
with the outside world) and decreased by such individuals who are later become susceptible to
rape after their release. Considering the period spent in jail at the rate Ω, convicted individuals who
after serving their become resistant to rape at rate and natural death . This gives
The population of rape individuals who were not caught and convicted but recover due to
counselling and public enlightenment programmes on the need to shun rape considering that the
society is an imperfect one where not all persons who commit rape face convictions. R(t) is
increased by rape individuals who abandon a rape lifestyle at a rate . This population is decreased
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by developing resistance to rape environment at a rate , becoming susceptible to rape again after
some time at a rate and natural death , yields
Based on the above formulations and assumptions, the model for the dynamics and spread of rape
at covid-19 in a population consist of the following system of nonlinear (deterministic) differential
equations. Table 1 describes the associated state variables and parameters in the model.
, (1)
, (2)
, (3)
, (4)
, (5)
with initial conditions
S(0)= S0, C(0) = C0, P(0) = P0, V(0) = V0, R(0) = R0.
Some of the assumptions and features of this model are below.
1. Assume that we are dealing an imperfect society whereby not all culprit individuals are
caught and convicted.
2. Assume that the culprit individuals who likely recovered due to counseling and
enlightenment campaigns abandon their rape lifestyle and recover from rape tendencies.
3. Assume that the culprit individuals are not rape victims or prey
4. Incorporating a class of convicted individuals who are jailed and are in complete isolation
from the outside.
5. Assume that the human disturbance is negative on the culprit and positive on the victim
or prey; hence it was ignored on victim class.
6. Assume the noise term (white and colored noise) to be sinusoidal with amplitude A,
frequency is and phase shift .
Table 1 shows the variables and parameters description.
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Dependent
Variables
Interpretation
S
Population of individuals susceptible to rape
C
Population of culprit individuals
P
Population of rape victims or prey individuals
V
Population convicted individuals who are jailed (isolated)
R
Population of individuals who recover from rape
Parameters
Interpretation
Recruitment rate
Proportion of immigrants who are susceptible to rape
Susceptible population is decreased by rape
Natural mortality rate in all the compartments
Rate of the ‘recovered’ individuals become susceptible to rape
Convicted individuals become susceptible to rape after release
Rape recovery rate
Rape transmission rate
Predating coefficient by human beings
Total effort applied on predating the population of the culprit
Amplitude of the noise from culprit and victim individuals
Frequency of the noise from culprit and victim individuals
Phase shift of the noise from culprit
Rate at which the victim individuals are removal from the society
Ω
Conviction rate
Rate at which convicted individuals become resistant to rape
Rate at which culprit individuals caught and convicted
Rate of developing resistance to rape environment
Table 1: Variables and parameters description
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2.1 Basic properties
For model (1-5) to be meaningful, it is important to prove that all the state variables are non-
negative for all time t. thus, the solutions of model (1-5) with positive initial data will remain
positive for all .
2.1.1 Well-posed model and positivity of the solutions
Model (1) controls and monitors human population therefore all the state variables and parameters
of the model are non-negative. Considering the domain
. (6)
It can be shown that the set of D is a positive invariant set and a global attractor of this system.
That is any phase trajectory initiated anywhere in the non-negative region
of the phase space
eventually enters the region D and remains in D thereafter.
Let . Then, there exists solutions S(t), C(t),
P(t), V(t), R(t) for the dynamical system (1) with initial data at t = 0, that defined for all . In
fact, S(t), C(t), P(t), V(t), R(t) are non-negative and
S(t) + C(t) + P(t) + V(t) + R(t) =N(t) for all t. if C0 =0, R0 =0, V0 =0, then and
.
Since N = S + C + P + V + R, it follows that the rate of change of the total population is given
by
, (7)
therefore, that
as . That is the total population is asymptotically constant. The
well-posedness of the model follows a straight forward application of the classical theory in [11].
Lemma 2.1 The region D is positively invariant for the model equations (1-5) and if , then ,
, and for all and V is bounded by
.
Proof. The right hand side of system (1-5) is continuously differentiable and hence it is locally
Lipschitz and therefore there exists a unique solution S(t), C(t), P(t), V(t), R(t) to system (1-5) with
the initial data that is defined on a maximal forward interval of existence, [11].
Consider the set
, since from system of equation (1-5), yields
(8)
(9)
(10)
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. (11)
So that , , , , , , where ,
, , .
To show that V is bounded by
, is to show whether , and
that
if . Suppose that the above inequalities do not hold, then there exists a time,
, with and . Using the equation for V in (1), gives
. (12)
Since and. This is the contradiction which implies that ,
. Suppose that
. In order to show that , we assume that
the last inequality does not hold. Hence, there exists a time such that and
.
Since
. Then
, but . (13)
Hence, we reach a contradiction and V(t) is bounded from above by , where
.
End of proof.
Lemma 2.1 guarantees the positivity of the different groups of individuals making up the total
population, S, C, P, V and R for all time, .
3 Asymptotic stability of rape-free equilibrium (RFE)
The rape-free equilibrium of the model (1-5) is given by
(14)
Setting , depicting no noise and no rape action.
The linear stability of E0 can be established using the next generation operator method on the
model equ (1-5), [5,2,10]. Using the notations in [4], it follows that F and V, which stands for the
new rape infection terms and remaining transition terms respectively are given by and
.
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It follows that the effective rape reproduction number of the model equation (1-5) denoted by
.
The result above follows from theorem 2 in [4].
Lemma 3.1 The CFE (E0) of the model equ (1-5) is locally-asymptotically stable (LAS) if and
unstable if .
The threshold quantity, is the rape reproduction number. It represents the average number of
secondary cases of rape generated by a typical rape individual in a population that is completely
susceptible to rape (using a typical language in epidemiology, [14]. The implication of Lemma 3.1
is that when Rrn is less than one rape can be eliminated from the population if the
initial sizes of the sub-population of the model are in basin of attraction of the rape-free
equilibrium Hence, a small influx of rape-infected individuals into the country will not
generate large outbreaks of rape and it dies out in time completely.
3.1 Backward Bifurcation Analysis
To describe the bifurcation type in the model equations (1-5), it will go a long way to help in
determining the factors that could hinder efforts in tackling rape in the population. Hence, the
following results were assumed.
Theorem 3.1 That the model equation (1-5) exhibits a backward bifurcation at whenever
a bifurcation coefficient denoted by a, is positive.
Proof.
Let represents any arbitrary endemic equilibrium of the model; that
is an equilibrium in which at least one of the infected components is non-zero. The existence of
backward bifurcation will be explored using the Manifold theory, [2,3]. To apply this theory, it is
appropriate to carry out the following change of variables. Let: , , , ,
. By using the vector notation, , the model can be written in the form
, with , yields:
(15)
(16)
(17)
(18)
(19)
where , , , ,
and the force of infection is given by
and .
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Put as a bifurcation parameter. Solving for from gives
. (20)
The Jacobian of the transformed model equation (15–19) evaluated at the rape-free equilibrium
(E0) with is given by
(21)
The matrix J* has a right eigenvector as , where
w1
(22)
Thus, J* has a left eigenvector satisfies that v.w = 1, with
, and .
This follows from Theorem 4.1 in [2]. To compute the associated non-zero partial derivatives of
F(x) evaluated at rape-free equilibrium, by defining the associated bifurcation coefficients, a
and b, giving by:
and
when computed gives
(23)
and
. (24)
Hence, the bifurcation coefficient b is positive, it follows from Theorem 4.1 in [2] that the model
equation (1-5) and the transformed model (15-19) will undergo a backward bifurcation if the
backward bifurcation coefficient a, given by equation (23) is positive.
The occurrence of backward bifurcation has been observed in numerous disease transmission
dynamics as in [2,3] showed a typical case of co-existence of a stable rape-free equilibrium and a
stable rape endemic equilibrium when the associated rape reproduction number of the model is
less than one. This implies that the model equation (1-5) is the typical tool of having a rape
reproduction number less than one, while the jailed equation is to deteriorated rape in the
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population where it is common. Putting = = 0 in (23) and setting all parameters of the model
(1-5) positive and * > 0, the bifurcation parameter a reduces to;
(25)
with v2> 0 and
From Theorem 4.1 in the model equation (1-5) does not undergo backward bifurcation, i.e. if =
= 0. Then it shows that the rate at which individuals ‘recover’ from rape practices due to certain
interventions like public enlightenment, counseling, jail e.t.c and previously convicted individuals
become susceptible to rape for 0 and 0 are the causes of backward bifurcation in the rape
transmission model. Therefore, the mathematical analysis in shows that previously convicted and
recovered individuals becoming susceptible to rape will make it difficult to have rape eradicated
from the population.
3.2 Global Asymptotic Stability: Special Case = = 0.
Considering the model equation (1-5) is with = = 0.
Theorem 3.2 The RFE of the model (1-5) with = = 0 is globally-asymptotically stable (GAS)
in D whenever Rrn <1.
Proof. Consider the model (1-5) with = = 0 and linear Lyapunow function
By Lyapunow derivative with respect to t.
(26)
From (26), S (t) < N(t) – V(t) in D for all t > 0, then
(27)
Thus, F < 0 if Ro < 1 with , if and only if C=0. Therefore, F is a Laypnow function in D
and it follows from LaSalle’s Invariance Principle [8], that every solution to the equations in (1-5)
with = = 0 with initial conditions in D converges to Eo as t → ∞ i.e.
(C(t), R (t), V(t) →(0.0,0) as . Substituting C = R = V = 0 into equation (1-
5) gives S (t) →
as . Hence
as for
so that the RFE, E0, is GAS in D if Rrn < 1 for the case = = 0.
The above result signifies that in the absence of recovered individuals and persons previously
convicted and jailed for rape becoming susceptible to rape again, rape will be eliminated from the
population of the society if the rape reproduction number or threshold Rrn <1.
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3.3 Existence and stability of rape endemic equilibrium
To create the existence of the rape endemic equilibrium of the model equation (1 –5), let
E1 = {S**, C**, P**, V**, R**} represents any arbitrary rape endemic equilibrium of the model (1 –5).
The equation (1-5) are solved in terms of the rape force of infection at a steady state to give
S**
(29)
C**
(30)
R**
(31)
Recall, that the rape force of infection at steady state,
is expressed as
where H= and further
reduces to
(32)
(33)
Hence, a unique rape endemic equilibrium exists (i.e.
when as far as H > 0.
Clearly, if then H > 0 and there exists a unique rape endemic equilibrium for this case.
Thus, the following result is established.
Theorem 3.3 The model (1-5) has a unique endemic (positive) equilibrium when = = 0
whenever
The possibility of the existence of a rape endemic equilibrium exists even This occurs
when which is possible when and since . This further proves the
possibility of a backward bifurcation when whereby the rape free equilibrium will co-exist
with a rape endemic equilibrium. Therefore, the general implication of this as stated before that
individual who ‘recover’ from rape after an intervention programme and previously convicted
individuals should strive not to become susceptible to rape again. The relevant government
agencies may determine and formulate polices that could help with preventive strategies cum
corrective measures.
3.4 Local Asymptotic Stability of the Rape Endemic Equilibrium
Since the total population is asymptotically constant, the result in [2] guarantee that the following
system of equations from (34-38).
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(34)
(35)
(36)
(37)
(38)
where
with
, has the same asymptotic qualitative dynamics as those of system.
Putting ,
The Jacobian matrix for system (34-38) at the endemic equilibrium, 1 is
where
and
Hence, the characteristic equation associated with the local stability of are:
(39)
where
= 1,
(A-B++
),
and represents the eigenvalues of the Jacobian with is the model’s parameter vector.
The local stability of the rape endemic equilibrium is geared towards the roots of equation (39).
Therefore, the rape endemic equilibrium point is locally asymptotically stable if the roots of
equations (39) are negative real roots.
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4 Numerical Simulations
Figure1: Dynamical Behavior of Individuals Susceptible to Rape in a Population
Thesecondscenarioshall
assumethefollowinThenumericalsimulationgraphdepictingthebehaviorofthemodelwithrespecttoth
e aboveassumedvaluesisdisplayedintheFigure 4.2fromwhichwecansee the behaviour of the
perpetrator (predator) and the victim (Prey) to
Figure 1: Graph of Susceptible Individuals to Rape cases in a Population. Parameter Values are:
π=3.142, P=0.5, δ=0.90, ρ=0.45, β=0.72, k1=0.75, k2=0.75, µ=0.005, φ=0.67, ϕ=0.81, γ=0.80,
=0.45, with initial condition (S(0), P(0), V(0), C(0), R(0)) = (100, 50, 50, 20, 10).
Figure 1 above showed that the susceptible population or individuals reduces slowly with respect
time and move to either the culprit-victim individuals asymptotically.
Figure 2: Graph of Rape Perpetrator-Victim cases in a Population. The parameter values are:
π=3.142, P=0.5, δ=0.90, ρ=0.45, β=0.72, k1=0.75, k2=0.75, µ= 0.005, φ=0.67, ϕ=0.81, γ=0.80,
=0.45, with initial condition (S(0), P(0), V(0), J(0), R(0)) = (100, 50, 50, 20, 10).
In figure 2, the graph depicts that rape victims do not go to extinction because in the natural the
stigma lives with them mostly females, while the perpetrators or culprits
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sometimes vanishes as a result rape control measures and if the culprit are males.
Figure 3: Graph of Recovered from Rape cases in a Population
Figure 3: Graph of Recovered Individuals from Rape cases in a Population. The parameter
values are: π=3.142, P=0.5, δ=0.90, ρ=0, β=0.72, k1=0.75, k2=0.75,
µ= 0.005, φ=0, ϕ=0.81, γ=0.80, τ=0.45, with initial condition
(S(0), P(0), V(0), C(0), R(0)) = (100, 50, 50, 20, 10).
Figure 3 above showed that the recovered individuals reduce to annihilation asymptotically with
respect time through natural death or migration to another environment.
4.1 Discussions of Findings
The study designed and discussed the asymptotic behavior of a new deterministic model for the
dynamics of rape in a population with noise and human disturbance. The model equation (1-5)
has a locally-asymptotically stable rape free equilibrium whenever the associated rape reproduction
number (Rrn) is less than one. This model undergoes the phenomenon of backward bifurcation,
where the stable rape-free equilibrium co-exists with a stable rape endemic equilibrium. The
analysis showed that this phenomenon is caused by the rates at which individuals who previously
recovered from rape, following some intervention programme and individuals who previously
were convicted and jailed for rape revert to being susceptible to rape.
It is also shown that the model has a globally-asymptotically stable rape-free equilibrium (DFE)
when individuals who recover from rape and those previously convicted and jailed for rape never
become susceptible to rape but have developed total resistance to rape. The Jailed model was
identified as the general measure to curb and control rape, both at Covid-19 and other crisis
respectively. To restrain rape globally, an SCPVR models was developed which implies Susceptible
to rape S, Culprit C, Prey P, Convicted and Jailed because of rape committed V and recover from
rape R, models. The study identifies two equilibrium points such as Pandemic free equilibrium
state and the rape free equilibrium state, Re. The MATLAB ODE45 software was used for the
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Volume 5, Issue 2, September 2022, Pages 121 – 136
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simulations. However, the graphs show that the susceptible class increases with time in a ratio of
1:1 and the recovered class goes into extinction with the presence of human disturbance and jail
class which is the control measure of rape cases in any given population. The results and
simulations mean that the rape culprit goes to life imprisonment because of the action. The
simulation of this study with noise showed a sinusoidal and an exponential increase; which implies
that the action of the rape culprit was high on the victim.
We can see from the line graph that the recovered class tends to zero with respect to time. We can
also affirm that the model is a good for the control of rape cases in a population within a stipulated
time. The numerical simulation graph of this is depicted in rape cases with respect to time. The
perpetrator-victim classes decline with respect to time also. This shows that the proposed model
can be used to control the high rise in number of rape cases during the Covid-19 which is realistic
to real life situation.
5 Conclusion
In conclusion, the findings from the model analysis, shows that the convicted and jail for life
imprisonment is the only possible measure to mitigate and control the impact of rape cases in any
rape society. This study could be extended to multiple rape scenarios that led to pregnancy.
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