https://orcid.org/0000-0003-3468-5581
Figures
Abstract
Crime remains a persistent socio-economic challenge in South Africa, where the incarceration of petty offenders often exacerbates criminal behaviour. Empirical evidence suggests custodial sentencing for minor offences may accelerate progression to serious crime due to exposure to hardened offenders and criminogenic prison conditions. To investigate this, we develop a compartmental ODE model tracking eight population states: susceptibility, petty and serious crime, custodial and non-custodial sentencing, rehabilitation, and relapse pathways. In particular, the model has an explicit non-custodial sentencing compartment, positioned in a distinct community-supervision stratum that is neither the general community nor the prison system, enabling direct comparison of custodial and non-custodial policy scenarios. The model is parameterised using the South African crime and prison data. Sensitivity analysis shows that sentencing rates, rehabilitation efficacy, and relapse probabilities drive long-term dynamics. Numerical simulations reveal that over-reliance on custodial sentencing perpetuates recidivism and incarceration, while prioritising non-custodial disposals and rehabilitation reduces both and CCR. Findings indicate that indiscriminate incarceration of petty offenders entrenches criminal cycles. Evidence-based interventions focused on rehabilitation and relapse prevention offer durable crime reduction. This quantitative framework evaluates justice policies, highlighting reforms that integrate deterrence with effective rehabilitation.
Citation: Nyabadza F (2026) Modelling the dark side of prison dynamics: Quantifying the criminogenic effects of petty crime offenders’ incarceration. PLoS One 21(9): e0356366. https://doi.org/10.1371/journal.pone.0356366
Editor: Claudio Terranova, University of Padova: Universita degli Studi di Padova, ITALY
Received: October 24, 2025; Accepted: July 31, 2026; Published: September 16, 2026
Copyright: © 2026 Farai Nyabadza. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The data underlying the results presented in the study are available from the provided citations in the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Recent data from the South African Police Service (SAPS) for the 2022/2023 reporting period document a marked escalation in violent criminal activity, characterised by a substantial rise in cash-in-transit heists [1]. Concurrently, a surge in mass shooting incidents underscores the intensification of high-impact violent crime across the country [2]. While these phenomena attract the greatest public and policy attention, petty crimes remain pervasive and exert a substantial influence on the broader criminological landscape [3]. SAPS statistics for the same period indicate that petty offences, primarily theft and burglary, accounted for between 60% and 75% of all reported criminal cases, while serious crimes, including aggravated robbery and homicide, continued to impose a heavy societal burden, with more than 27,000 homicides recorded in 2022/2023 [1].
Crime, encompassing both minor (petty) and major (serious) offences, constitute a multifaceted and enduring challenge to global societal stability. Petty crimes, typically classified as low-severity infractions, include acts such as shoplifting, vandalism, and possession of small quantities of illicit substances [4]. Serious crimes, by contrast, involve high-severity offences such as armed robbery, assault, and homicide [5]. Within the South African context, both categories are highly prevalent and together contribute to elevated public safety concerns and deepening doubts regarding the efficacy of the criminal justice system.
Incarceration has historically been justified on the grounds of rehabilitation and deterrence [6], yet accumulating evidence challenges its effectiveness. Research suggests that imprisonment may not only fail to reduce crime but can, under certain conditions, intensify criminal behaviour [7]. In South Africa, chronically overcrowded prisons routinely house minor offenders alongside hardened criminals [8,9], creating criminogenic environments that substantially elevate recidivism risk [10]. This dynamic is consistent with the Prison Industrial Complex thesis, which holds that the carceral system sustains, rather than disrupts, cycles of criminality within broader socio-economic structures [11].
Mathematical modelling, particularly through compartmental systems of ordinary differential equations, provides a powerful and tractable framework for analysing the dynamics of criminal behaviour [12,13]. Such models partition the population into distinct subgroups and specify transition rates between compartments, thereby enabling simulation of crime progression and systematic evaluation of policy interventions [12]. Building on the foundational economic theories of crime and deterrence due to Becker [4] and Ehrlich [14], compartmental crime models incorporate deterrence mechanisms, rehabilitation processes, and socio-economic influences. More recent extensions have incorporated incarceration-induced behavioural change and recidivism dynamics [15].
Significant progress in mathematical modelling has shed light on the mechanisms through which incarceration shapes criminal behaviour. Chakra and Hilbe [16] developed a compartmental model demonstrating that prolonged incarceration can escalate serious criminality through prison socialisation. Accinelli et al. [17] applied predator-prey dynamics, framing incarceration as “predation” and establishing that excessive imprisonment yields diminishing deterrence returns. Kwofie et al. [18] integrated rehabilitation explicitly, showing that balanced investment in both incarceration and rehabilitation achieves optimal crime control. Collectively, these studies highlight the nonlinear and often counterproductive consequences of exclusively punitive policies [12].
The present study advances the existing literature in three key respects. First, unlike Abou Chakra and Hilbe [16], who model escalation without deriving a CCR threshold or incorporating a non-custodial pathway, we introduce the compartment as a distinct supervised probation excluded from crime initiation, together with the associated CSER metric
, to enable direct comparison of sentencing regimes. Second, in contrast to Accinelli et al. [17], whose predator-prey framework yields neither a crime generation number nor a carceral criminality threshold, we derive both
,
, and
analytically. Third, unlike Kwofie et al. [18], who include rehabilitation but do not distinguish custodial from non-custodial pathways, our model facilitates evaluation of differential criminogenic risk by sentencing type. Recent complementary contributions include compartmental models with optimal control [19], recidivism-focused frameworks [20] and fractional-order extensions [21].
The contagion analogy employed in this model rests on three complementary foundations. First, an extensive body of empirical criminological research documents strong peer-influence and social-learning dynamics in the onset and persistence of criminal behaviour, particularly among urban youth. These dynamics are structurally analogous to the transmission of infectious disease [15,22–25]. The criminal career literature [15] further demonstrates that offending is both initiated and sustained through social exposure, directly mirroring the epidemiological force-of-infection concept. Second, the chronically overcrowded South African prison environment, which places petty offenders in sustained contact with serious offenders [8,9], generates a “super-spreader” dynamic that is captured explicitly by the carceral escalation pathway in the model. Third, the contagion analogy necessarily abstracts away individual agency, economic incentives [4,14], and structural drivers such as poverty and unemployment, each of which is acknowledged explicitly in the Limitations section.
Building on this body of evidence, the present study addresses a gap in the literature by constructing a model that simultaneously tracks custodial and non-custodial sentencing pathways, derives explicit escalation thresholds, and calibrates findings to South African data. This study is motivated by the need to evaluate the systemic consequences of incarcerating petty offenders in South Africa. The primary objective is to quantify the effects of sentencing and rehabilitation policies on long-run crime dynamics through the derivation and analysis of three threshold parameters: the crime generation number (), the CCR (
), and the community-supervision escalation ratio (CSER,
). Through global sensitivity analysis and numerical simulation, we assess the impact of a range of policy interventions, with particular attention to the roles of non-custodial sentencing and structured rehabilitation in mitigating recidivism and reducing aggregate criminal activity.
Throughout this paper, we use the pairing petty and serious as the primary descriptive terms for the two offence-severity categories, so that the severity ordering is transparent to readers across jurisdictions; “petty” and “serious” are descriptive rather than statutory categories in South African law, which distinguishes offences by schedule [26] and by Magistrates’/High Court jurisdiction [27].
Materials and methods
Mathematical model
The compartmental model describes the population dynamics of individuals transitioning among states defined by criminal behaviour and rehabilitation status. The model comprises eight compartments: susceptibles (S); active petty offenders (); active serious (serious) offenders (
); petty offenders serving non-custodial sentences (
); petty offenders serving custodial sentences (
); serious offenders serving custodial sentences (
); individuals rehabilitated from a petty-crime sentence (
); and individuals rehabilitated from a serious-crime custodial sentence (
). The compartments S,
,
,
, and
describe community-level dynamics, while
and
represent the incarcerated (prison) population. The compartment
occupies a distinct community-supervision stratum where individuals in
are neither freely active in the general community nor detained in prison. They reside in the community under formal judicial supervision (e.g., community service, probation, electronic monitoring). It is important to clarify the meaning of the rehabilitated classes.
denotes individuals who have completed either a custodial sentence via
or a non-custodial sentence via
for a petty offence and have re-entered the community without active criminal engagement.
denotes individuals rehabilitated following a serious-crime custodial sentence (including petty offenders who escalated due to imprisonment in
). These classes are treated as distinct states because post-release reintegration outcomes, recidivism risk, and supervision requirements differ systematically between minor and serious offenders [10].
Rehabilitation classes are defined by offending severity at release rather than by the original offence. Individuals leaving at the petty-offending level enter , whereas those leaving at the serious-offending level enter
. Thus, a petty offender may enter
if escalation occurs during custody. This process is governed by the carceral escalation probability,
: of offenders released from
at rate
, a fraction
exits into
due to prison-induced socialisation into serious offending, while the remaining fraction
exits into
. No such split occurs along the non-custodial pathway
, since it bypasses the prison environment. Hence,
represents the probability that imprisonment itself generates a serious offender, irrespective of the initial offence. At the population level, this effect is captured by the carceral-induced criminality ratio (CCR,
), which incorporates
together with the probabilities of custodial sentencing and subsequent relapse into serious crime. The empirical justification for this escalation mechanism is provided by evidence of prison socialisation and overcrowded co-detention in South Africa, as discussed earlier in the Introduction and modelling assumptions. The total population of individuals in the community, those under custodial sentences and prison, is thus given by
The model is constructed under the following explicit assumptions. Criminal behaviour spreads through social contagion, analogous to the transmission of an infectious disease [22,23]. All individuals within a given compartment are assumed homogeneous; demographic and socio-economic heterogeneity within compartments is neglected. Rehabilitation is imperfect: individuals in and
may relapse into criminal activity. There is no pre-trial detention delay; convicted individuals transition directly from
or
to the appropriate sentencing compartment, a standard assumption in first-generation compartmental crime models [12]. Individuals in the community-supervision compartment
are under active judicial oversight and are therefore do not contribute to crime generation. This assumption reflects the empirical observation that supervised diversion programmes substantially curtail the unsupervised peer-influence contacts that drive criminal contagion [28,29]. Finally, the custodial pathways for petty and serious offenders remain distinct throughout sentencing; no transfer between
and
is permitted.
The escalation mechanism is grounded in documented conditions within the South African correctional system rather than being purely a modelling assumption. Although the Correctional Services Act 111 of 1998 provides for the classification and separation of offenders by risk level, persistent overcrowding, remand–sentenced mixing, and shared communal accommodation often undermine these provisions, resulting in sustained contact between petty and serious offenders [9,28–31]. This co-detention environment provides the empirical basis for the custody-induced escalation mechanism represented by . In contrast, the prison compartments
and
remain separate accounting classes without administrative transfer, a simplification adopted for modelling tractability. In settings where petty and serious offenders are effectively segregated, the escalation parameter
would be expected to decrease, with the limiting case
corresponding to a system in which imprisonment generates no criminogenic escalation.
The susceptible population is recruited at a constant rate with a proportion
of individuals committing petty offences, while the complementary proportion
commits serious offences upon recruitment into criminality driven by the force of crime transmission
Crime generation is assumed to be driven by actively offending individuals in the community and individuals in the community-supervision stratum
, the prison compartments
,
, and the rehabilitated classes
,
do not directly generate new criminal contacts. So,
Here, denotes the effective contact rate and
captures the greater criminogenic influence of serious offenders relative to petty offenders. Among convicted petty offenders, a proportion
receive a non-custodial sentence, joining
at rate
, while the remainder
receive a custodial sentence, joining
. All serious offenders receive custodial sentences and enter
at rate
. Individuals in
complete their supervised sentence and transition to
at rate
. Most importantly, this passage from
to
bypasses the prison environment entirely, so no carceral escalation can occur along this route. Individuals sentenced custodially and in compartment
transition to
upon release at rate
.
The transition from petty incarceration is modelled as a probabilistic escalation process. Upon release at rate
, a fraction
of individuals is placed into the serious-offender rehabilitated class
, reflecting prison-induced escalation in offending severity, while the complementary fraction
transitions to the petty rehabilitated class
. The limiting case
corresponds to a fully escalatory custodial environment, which is consistent with empirical evidence from the South African context [32,33].
Relapse from the rehabilitated classes is governed by four direct transition rates, each corresponding to a distinct pathway in the model diagram. To motivate the parameterisation, let and
denote the aggregate relapse rates from
and
, respectively, and let
denote the cross-escalation probabilities. Within each rehabilitated class, a fraction of relapses returns the individual to the same offending level (own-class relapse), while the complementary fraction transitions to the other offending level (cross-class escalation or de-escalation). Specifically, the four pathway-specific rates are defined by
so that the four rates admit the following interpretations; denotes the rate at which individuals in
relapse to petty crime,
denotes the rate at which individuals in
escalate to serious crime,
denotes the rate at which individuals in
relapse to serious crime, and
denotes the rate at which individuals in
de-escalate to petty crime. By construction, the decomposition (1) satisfies the aggregate consistency conditions
ensuring that the total outflow rate from each rehabilitated class is preserved. The baseline configuration (equivalently,
) corresponds to strictly class-specific relapse with no cross-pathway transitions. The escalation-dominant regime,
and
(equivalently,
and
), captures the empirically documented asymmetry in recidivism whereby upward transitions to more serious crimes substantially predominate over de-escalation [10,32].
A flow diagram of the model is shown in Fig 1.
The diagram illustrates the transitions between population compartments: Susceptibles (S), Criminals committing petty () or serious crimes (
), jailed individuals (
,
), those serving non-custodial sentences (
), and rehabilitated individuals (
,
). Arrows represent the movement between states, driven by the model’s parameters.
The dynamics of the model are governed by the following system of ordinary differential equations:
where the composite outflow rates are defined as
The system is subject to the initial conditions
All parameters are assumed strictly positive unless otherwise stated.
Model properties
Positivity of solutions
Theorem 1.
Let be the solution of system (3)–(10) with positive initial conditions as given in (11). Then all components remain strictly positive for all t > 0.
Proof.
We examine the vector field on the boundary of the non-negative orthant. For each compartment, when that compartment is zero and all other compartments are non-negative, the corresponding derivative is non-negative:
All right-hand sides are continuous and satisfy the quasi-positivity condition (i.e., whenever a variable is zero, its derivative is non-negative provided the other variables are non-negative). Since the vector field satisfies the quasi-positivity condition (each derivative is non-negative whenever the corresponding variable is zero and all other variables are non-negative), the non-negative orthant is positively invariant. Starting from strictly positive initial conditions, a standard continuity argument further implies that every component remains strictly positive for all t > 0. □
Boundedness and Invariant Region
Let the feasible region for model (3)–(10) be
Theorem 2. The feasible region is positively invariant and attracting under the flow of system (3)–(10).
Proof.
Summing Eqs (3)–(10), the dynamics of the total population satisfy
This is a linear first-order ODE whose solution is
It follows that whenever
, establishing positive invariance of
. Moreover,
monotonically as
, regardless of the initial value N(0), so
is attracting. This completes the proof.
Theorems 1 and 2 together establish that system (3)–(10) is mathematically well-posed: solutions exist, are unique, remain strictly positive, and are confined to the compact, epidemiologically meaningful domain .
Steady states
To determine the steady states, we set all crime-related state variables to zero and solve the resulting algebraic system. Setting the right-hand sides of Eqs (3)–(10) equal to zero gives
where
We first extract all downstream compartments in terms of the criminal stocks and
by solving Eqs (12)–(19) in sequence.
From (16) we have
From (17) we have
From (15) we observe that
From (18), substituting and
we have
From (19), substituting the state variables and
we have
From (12) we have
Crime-free equilibrium
The crime-free equilibrium (CFE) is obtained by setting all crime-related compartments to zero and solving the resulting algebraic system. Equating the right-hand sides of (3)–(10) to zero and setting yields the unique CFE:
This equilibrium is biologically feasible and lies on the boundary of .
Crime generation number
Geometric loop approach
To aid interpretation, we first describe the feedback loops that underlie the crime generation number . The crime generation number
quantifies the average number of secondary criminal cases produced by a single offender introduced into an otherwise crime-free, fully susceptible population. We derive
via a geometric series (transmission-loop) approach grounded in the next-generation matrix (NGM) framework of Diekmann et al. [34,35] and van den Driessche & Watmough [36]. Appendix B provides a formal proof that the spectral radius
of the standard NGM coincides with the
derived below. The geometric approach is preferred here because it renders the contribution of each transmission loop explicitly and directly interpretable for policy analysis, a transparency not afforded by the aggregate spectral radius alone.
The force of crime initiation that is driven solely by and
through
All downstream compartments (
) feed back into
or
via relapse or escalation pathways, but none of these compartments, including the community-supervision compartment
, appear in
directly. The loops below trace the indirect routes by which diversion through
can eventually return individuals to active offending. We identify and analyse these loops individually by considering the transitions of criminals as described in the model diagram:
Petty non-custodial cycle:
A petty offender departs at rate
to receive a non-custodial sentence and enters the community-supervision stratum
, where they are removed from the ability to initiate new infections. Upon completing the supervised sentence, they transition to
at rate
, and may subsequently relapse directly to petty crime
at rate
(or escalate to serious crime
at rate
). The probability of completing this relapse loop, accounting for competing outflows at each stage, is
Petty custodial, non-escalating branch:
A proportion of petty offenders enter custodial sentence
; of those released, a fraction
transition to
at rate
and subsequently relapse to petty crime
at rate
. The associated cycle factor is
Petty custodial, escalating then de-escalating:
The remaining fraction of released
individuals escalate to
at rate
but subsequently de-escalate back to petty crime
at rate
. The cycle factor is
The total petty-crime internal cycle factor, aggregating all three loops, is
For this to be well-defined as a probability, we require , which holds under the biological constraint that not all petty offenders cycle indefinitely. The mean total time a petty offender spends in
across all cycles is then
Serious-crime custodial cycle:
A serious offender enters at rate
, is released to
at rate
, and relapses to
at rate
. The serious crime internal cycle factor is
with a mean total residence time in given by
Cross-pathway escalation: (). Three distinct pathways allow a petty offender to escalate into the serious crime compartment
:
Here, represents the carceral escalation pathway
a petty offender receives a custodial sentence, escalates upon release to
, and subsequently relapses into serious crime.
represents escalation via the non-custodial route
, and
represents escalation via the non-escalating custodial route
.
The composite cross-pathway escalation factor is
and the mean time, a petty offender contributes to via these escalation pathways is
.
At the CFE , only the susceptible class
is non-zero. The next-generation matrix
has rank one (see Appendix B), so its spectral radius reduces to a scalar expression. Accounting for both the direct (primary) crime-generation pathways and the feedback loops, the crime generation number is
The first component accounts for new petty-crime cases generated by a typical petty offender cycling through the petty-crime pathway. The second component
accounts for new serious-crime cases arising from two sources: direct serious-crime recruitment (proportion
), and petty offenders who escalate to serious crime via the composite cross-pathway
(proportion q, weighted by the escalation factor).
Remark:
The carceral-induced criminality ratio (CCR), denoted and defined in (31), quantifies the probability that a custodially sentenced petty offender follows the escalation loop
, thereby generating serious crime. It is a product of three conditional probabilities: receiving a custodial sentence
, escalating to
upon release
, and relapsing from
into serious crime
. Unlike the composite
, which also responds to
and additional pathways,
is reducible by exactly two direct policy levers: increasing prison rehabilitation efficacy (
) or decreasing serious-offender relapse (
). Thus, the CCR provides a targeted measure of the justice system’s capacity to inadvertently escalate petty offenders to serious criminals.
The community-supervision escalation ratio quantifies the probability that a petty offender diverted from custody progresses through the loop
and escalates to serious crime.
is a product of three policy-addressable probabilities; non-custodial sentencing, successful supervised rehabilitation, and cross-class escalation from
to
.
The CCR () and CSER (
) represent the escalation risks of custodial versus non-custodial routes, respectively. They respond to different levers:
(post-diversion supervision) affects only
;
(prison rehabilitation) affects only
; increasing
(diversion rate) reduces
but raises
, creating a policy tension measured by
. At baseline,
, confirming that custody carries greater escalation risk and supporting diversion policies provided that relapse prevention (
decreasing) is implemented concurrently.
Crime-persistent equilibrium.
To determine the crime-persistent equilibrium (CPE) we substitute the expressions (20)-(25) into the criminal-Eqs (13)–(14). Substituting
and
into (13) we obtain
Defining the effective removal rate
this becomes
Substituting the same expressions into (14) gives
Defining
this becomes
Expanding and
explicitly in (36), we factor Q1 from
so that
where, substituting (22)–(24), yields
where
Each term in the expression for has a direct interpretation with regards to the movement of individuals.
is the
loop,
is the
loop (custodial, non-escalating fraction
) and
is the
loop (custodial, escalating fraction
, relapse back to petty crime at rate
).
Similarly, factoring Q2 from gives
corresponding to the single loop .
Dividing Eq (a) by q and (b) by gives the same left-hand side
. Given that
we can solve for so that
where
Now substituting and
into Eq (a) yields
We thus have corresponding to the crime-free equilibrium and the non-trivial case
gives the crime-persistent equilibrium. We note that
where
and
where
is the total cross-escalation probability from petty to serious crime.
A back substitution of the result in (39) into the remaining expressions gives the CPE.
Stability of equilibria
Stability of the crime-free equilibrium
Standard theory for compartmental models (van den Driessche & Watmough [36]) establishes the following result:
Theorem 3. The CFE is locally asymptotically stable when
and unstable whenever
.
Existence and stability of the crime-persistent equilibrium
The crime-persistent equilibrium components are thus obtained by back-substitution of (39), so that
We thus have the following result on the existence of the crime-persistent equilibrium.
Theorem 4.
If , there exists a unique crime-persistent equilibrium
If , the only non-negative equilibrium is E0.
Stability of the crime-persistent equilibrium
The stability of the crime-persistent equilibrium is analysed via the block Jacobian matrix at
. Partitioning the Jacobian into submatrices for susceptible, criminal, and rehabilitated classes simplifies the characteristic equation. Using Schur complement arguments to compute
isolates the eigenvalue spectrum and yields explicit stability conditions.
Consider the Jacobian matrix at the crime-persistent equilibrium
. The variables are ordered as
. The Jacobian can be written in the following
block matrix form:
where the blocks are defined by the partial derivatives of the system (3)–(10) evaluated at . The block matrices are thus given by
The characteristic polynomial is given by . Using the Schur complement formula for block matrices, given that
is invertible, we have:
Hence, the eigenvalues of are obtained from:
Given that the block matrix is a lower triangular matrix, the eigenvalues obtained from
are simply its diagonal entries:
These eigenvalues are clearly negative since
Considering the correction term , we note that
and
are sparse; only a few entries of the correction term are nonzero. First, we compute
. Since
is lower triangular, its inverse is also lower triangular and multiplying
by
, then left-multiplying by
, yields a sparse correction matrix
. After some tedious algebraic manipulations, we obtain the following nonzero entries:
We now define the matrix
. Substituting the expressions above results in the following
where and
To compute , we can perform a cofactor expansion along the fourth row, which has the form:
where is the minor obtained by deleting the fourth row and second column, and
is the minor obtained by deleting the fourth row and fourth column.
The minor is the
matrix obtained by deleting the fourth row and second column:
Therefore, the characteristic equation for the remaining eigenvalues is given by
Eq (42) is a transcendental equation in due to the rational functions
appearing in the minors.
Local asymptotic stability of is assessed by linearising system (3)–(10) about
. The Jacobian
yields a characteristic polynomial, and the Routh–Hurwitz conditions for local asymptotic stability require that all coefficients of the characteristic polynomial be positive and that all relevant Hurwitz determinants be positive. Denoting the characteristic polynomial of the reduced
Jacobian block
as
, the three Routh–Hurwitz conditions are:
- (i) a1 > 0, (ii) a3 > 0, and (iii)
.
These conditions have been verified numerically over a Latin Hypercube sample of 1,000 parameter vectors drawn from the ranges in Table 2; across all samples satisfying , all three conditions hold. The eigenvalues of
at the baseline parameterisation are reported in Table 1, and all have strictly negative real parts, confirming local asymptotic stability.
Proposition 1. If , the unique CPE
is locally asymptotically stable for the given baseline parameter set and across the parameter ranges in Table 1, as confirmed by the Routh-Hurwitz conditions and numerical eigenvalue analysis [37].
Numerical simulations
To investigate the impact of sentencing and rehabilitation policies on long-run crime dynamics in South Africa, we perform numerical simulations of system (3)–(10) using parameter values derived from the available literature and official statistical reports. The model is implemented in MATLAB using a standard fourth-order Runge–Kutta scheme and integrated over a 30-year horizon. The simulations are initialised using demographic and criminal justice data for Gauteng province, whose total population was approximately 16 million in 2023.
Parameter estimation
The recruitment rate yr-1 follows from a 2% annual growth rate for Gauteng’s 16 million population [38]. The contact rate
(criminal susceptible)
is calibrated to yield an equilibrium petty-crime prevalence
, consistent with SAPS data [1]. The non-custodial rehabilitation rate
yr-1 is based on DCS community corrections [29] and Muntingh [28]. Custodial rehabilitation rates
yr-1 and
yr-1 reflect lower prison efficacy, consistent with DCS statistics [29] and Singh [39]. Aggregate relapse rates
yr-1 and
yr-1 come directly from South African studies [32,33]. For parameters drawn from international literature (e.g., the model architecture and peer-influence factor
), the qualitative dynamics are structurally robust across contexts [12,16], and the sensitivity analysis confirms stability of conclusions over the full plausible parameter range.
A complete table of model parameters, their baseline values, and sources is given in Table 2, see also Appendix A.
The model is initialised using 2023 Gauteng demographic and criminal justice data from the Department of Correctional Services, with the assumption that Gauteng accounts for more than 25% of South Africa’s total prison population [42,43]. The initial conditions are set as follows:
Global sensitivity analysis
Global sensitivity analysis (GSA) employed Latin Hypercube Sampling (LHS) with Partial Rank Correlation Coefficients (PRCC) [44,45]. Five hundred parameter sets were drawn from the ranges in Table 2, and the model was simulated over a 100-year horizon per set. Time-varying PRCC values were computed for the crime compartments and
; PRCC ranks inputs and outputs, measures correlation while controlling for other parameters, and captures monotonic (including nonlinear) relationships [45]. Bootstrap confidence intervals (100 resamples per time point) assessed statistical robustness; shaded regions denote 95% confidence bands, and a grey band between
and 0.25 indicates negligible influence. Figs 2–4 present the resulting PRCC profiles. Tornado plots at the final simulation time rank parameters by the magnitude of their PRCC values; positive values indicate direct effects, negative values inverse relationships, and the magnitude reflects relative importance.
Shaded bands represent 95% confidence intervals.
(a) Tornado plot of final-time PRCCs for . (b) Tornado plot of final-time PRCCs for
.
Positive PRCC values (rightward bars) indicate a direct (increasing) effect, negative values (leftward bars) an inverse (reducing) effect. For petty crime , the strongest positive drivers are
(non-custodial sentencing proportion) and
(recruitment rate), while
(mortality) and
(conviction rate) exert the strongest reducing influence. For serious crime
, the dominant positive drivers are
,
(serious-crime relapse rate), and
(serious-crime conviction rate). Bar length ranks each parameter by absolute PRCC magnitude.
To enhance policy relevance, parameters are classified as either direct policy targets (modifiable by legislation or administration) or systemic outcomes (emergent from broader institutional processes). The non-custodial sentencing proportion and the non-custodial rehabilitation rate
are direct policy targets with strong PRCC effects and immediate feasibility. In contrast, the petty-crime conviction rate
is systemic, alterable only through longer-term institutional change. The newly introduced CSER (
) identifies
as an additional direct policy target: it governs escalation from the petty rehabilitated class
to serious crime
, a risk pathway activated by successful diversion through
. Reducing
via structured post-diversion supervision complements prison-rehabilitation strategies that target the CCR.
The PRCC analysis identifies and q as the strongest positive drivers of petty crime
, while
and
act as reducers, see Table 3. The parameter
exhibits the largest positive PRCC on
, and
,
,
,
, and
are moderate curbing factors. For serious crime
:
,
, and
are the dominant positive drivers, while
and
act to reduce serious crime. Fig 2 displays the time-varying PRCC profiles for
, and Fig 3 displays the corresponding profiles for
.
Model calibration and projection
Model fitting to data
The model was calibrated against annual South African prison population data spanning 1995–2024 (Table 4). Parameter estimation was carried out by minimising the sum of squared residuals between model predictions for the total incarcerated population and the observed data, using a nonlinear least-squares routine in MATLAB.
Fig 5 presents the calibration of the model to South African incarceration data. The model underestimates the sharp peak observed near 1999–2002, when the prison population reached approximately 187,640. This discrepancy is attributable to two concurrent policy shocks: (i) the Criminal Procedure Second Amendment Act (Act 85 of 1997), which introduced mandatory minimum sentences effective from 1998, precipitating an abrupt increase in custodial sentences [46]; and (ii) post-apartheid policing reforms that substantially increased arrest and conviction rates between 1998 and 2002 [2]. These discrete, non-smooth policy shocks cannot be fully reproduced by the continuous ODE framework without incorporating time-varying parameters, a recognised limitation of the present model.
Solid line: model prediction; circles: observed data. Horizontal axis shows calendar year, with t = 0 corresponding to 1995. Fitted parameters: ,
, q = 0.0035229,
,
,
,
,
,
,
,
,
,
.
Fig 6 presents the best-fit model alongside a five-year out-of-sample projection through 2029.
Solid line: model fit to observed data; dotted line: out-of-sample projection for 2024–2029, demarcated by a vertical dashed line. Under continuation of the 2024 fitted-parameter regime, the total incarcerated population is projected to decline gradually, consistent with the long-run attractor dynamics of the crime-persistent equilibrium at the estimated
.
Fig 7 illustrates the impact of improving the custodial rehabilitation rate on the total incarcerated population over the simulation time. Fig 8 shows projected incarceration trajectories under a policy-enhanced rehabilitation scenario, with the grey-shaded region quantifying the cumulative incarcerations averted.
Horizontal axis: calendar year. Baseline: yr−1 (solid black line; data up to 2024, thereafter model projection); dashed and dotted lines show scenarios with
increased by factors of 1.1 and 1.2, respectively. Even modest improvements in prison rehabilitation efficacy generate a sustained downward trajectory in the incarcerated population over the simulation period, illustrating the cumulative long-run benefit of investment in custodial rehabilitation programmes.
Blue solid line: baseline trajectory with yr-1 (calibrated value); red dashed line: improved scenario with
yr-1 (a 50% increase above baseline), intervention effective from
. The grey-shaded region quantifies cumulative incarcerations averted; its widening over time illustrates the compounding nature of rehabilitation gains.
Relationship between incarceration and crime rates
The model makes the link between incarceration and crime explicit. At equilibrium, the incarcerated populations are proportional to the petty- and serious-crime rates,
so total incarceration is a severity-weighted reflection of active offending, with the proportionality determined by the custodial-sentencing fraction , conviction rates
, and release/rehabilitation rates
. Away from equilibrium,
and
act as lagged accumulations of convictions and releases, producing characteristic delays of approximately
and
, respectively. This explains why incarceration trajectories follow underlying crime dynamics with a delay. Increasing the custodial rehabilitation rate
reduces incarceration both directly, by decreasing the petty-custody stock, and indirectly, by weakening the carceral escalation pathway
(captured by the CCR,
), thereby lowering serious crime and the associated serious-custody population. As a result, reductions in incarceration reflect simultaneous reductions in imprisonment and serious-crime prevalence rather than merely shifting offenders out of custody. The cumulative incarcerations averted therefore provide a conservative indicator of the broader crime-reduction benefit and can be linked to the corresponding petty- and serious-crime trajectories.
Asymmetric cross-pathway escalation
A central feature of the relapse dynamics is the asymmetry between upward escalation from the petty rehabilitated class to serious crime
(rate
) and downward de-escalation from the serious rehabilitated class
back to petty crime
(rate
). This asymmetry is structurally embedded in the composite cross-pathway escalation factor
which aggregates all routes by which petty offenders feed into serious-crime dynamics. Fig 9 explores this asymmetry over the full parameter domain of . The figure further shows that the community-supervision escalation ratio (CSER,
) is highly sensitive to
: even modest increases in
markedly expand the region where
, whereas de-escalation (
) provides only weak attenuation. This asymmetry implies that effective post-diversion supervision must target
to contain the residual escalation risk of non-custodial sentencing.
Colour intensity encodes ; warmer hues indicate higher escalation risk. The surface is strongly asymmetric:
rises sharply with
but is nearly insensitive to
, confirming the empirically documented escalation-dominant regime [32,33]. The horizontal gradient far exceeds the vertical gradient, implying that policies reducing
(strengthening relapse prevention for petty rehabilitated offenders) yield substantially greater reductions in
than equivalent efforts to lower de-escalation barriers from
.
Influence of sentencing and rehabilitation parameters on 
This section examines the dependence of the carceral-induced criminality ratio on key sentencing, rehabilitation, and relapse parameters. Recall that
, defined in Eq (31), measures the probability that a single custodially sentenced petty offender traverses the full escalation loop
and thereby generates serious criminal activity. Its three constituent probabilities, the probability of receiving a custodial sentence, the probability of escalating to
upon release, and the probability of relapsing into serious crime from
, each admit direct policy intervention.
Fig 10 reveals that an increase in can paradoxically elevate
, whereas an increase in
consistently reduces it. Increases in
and
both amplify
. The analytical expression
does not depend directly on . However, dynamically, a higher
increases throughput through
, amplifying the relapse flux
back into
and
. The enlarged
pool elevates the flow through the carceral escalation pathway
, thereby raising
dynamically when relapse prevention is inadequate. The policy implication is clear: expanding non-custodial diversion (
increasing) without simultaneously strengthening relapse prevention (
decreasing,
decreasing) may inadvertently sustain, or even amplify, the carceral escalation cycle.
The subplots correspond to three levels of the custodial rehabilitation rate
(rows, increasing downward) and three levels of the non-custodial rehabilitation rate
(columns, increasing rightward). Three patterns emerge: (i)
increases monotonically with both
and
; (ii) increasing
suppresses
; (iii) counter-intuitively, increasing
can elevate
by enlarging the
pool and amplifying relapse flux into
.
Impact of rehabilitation and relapse rates on 
We now examine the sensitivity of the crime generation number to the full set of rehabilitation and relapse parameters. Figs 11 and 12 present complementary heatmap analyses: Fig 11 varies the structural throughput parameters
and
across levels of
and
, while Fig 12 varies the four pathway-specific relapse rates
,
,
, and
.
The nine subplots vary (rows, increasing downward) and
(columns, increasing rightward). Four main findings: (1) increasing
reduces
; (2) increasing
raises
; (3) increasing
(custodial rehabilitation) strongly compresses
toward elimination; (4) increasing
can paradoxically raise
when
by enlarging the petty-rehabilitated pool and amplifying cross-class escalation.
Within each subplot, increases with both
and
, confirming that reducing own-class relapse suppresses crime generation. Across subplots, increasing
shifts the
boundary rightward, enlarging the crime-persistence region, whereas varying
has a modest effect. Thus, cross-pathway escalation (
) is a disproportionately influential driver of
, and policies that reduce it offer the highest leverage for reducing aggregate crime propagation.
Fig 11 reveals a delicate parameter balance: increasing reduces
; increasing
modestly raises it; increasing
consistently lowers
; and increasing
can raise
in the absence of adequate relapse prevention, consistent with the mechanism identified for
above.
At the baseline parameterisation (Table 2), , indicating a regime of sustained but moderate crime propagation in which each offender generates approximately 1.21 new offenders in a fully susceptible population. Doubling the non-custodial sentencing proportion
from 0.40 to 0.80, representing a major legislative shift, reduces
to approximately 0.87 < 1, implying that crime would decline toward the CFE. At Gauteng’s population scale, this corresponds to averting approximately 6,200 new petty-crime cases per year, providing a concrete quantitative illustration of the public safety benefit of sentencing reform.
Fig 13 displays the contour surface of as a joint function of the two custodial rehabilitation rates, identifying the rehabilitation frontier that separates crime-persistence from crime-elimination regimes.
The plane partitions the space into an upper crime-persistence region (
, shaded red) and a lower crime-elimination region (
, shaded blue).
decreases steeply with
, confirming that custodial petty rehabilitation is the primary lever for crime elimination, while the surface is comparatively flat with respect to
(serious offenders are a minority, q = 0.75).
Conclusion
This study developed and analysed an eight-compartment ordinary differential equation model to examine the criminogenic effects of incarcerating petty offenders in South Africa. A central structural innovation is the placement of the non-custodial sentencing class within a distinct community-supervision stratum that is excluded from the force of crime
, reflecting reduced peer-contagion under supervised diversion. The introduction of the crime generation number
, the carceral-induced criminality ratio (CCR,
), and the community-supervision escalation ratio (CSER,
) provides quantitative metrics for evaluating crime persistence and escalation across sentencing regimes. The model was parameterised using South African data from 1995–2024, calibrated to incarceration trends, and validated via global sensitivity analysis.
Results demonstrate that indiscriminate custodial sentencing of petty offenders can amplify criminal activity. The CCR quantifies the probability that custody induces escalation from petty to serious crime through the pathway . The CSER captures the analogous but markedly smaller risk along the diversion route
, with the empirical ordering
confirming the substantially lower escalation risk of community supervision. These findings support expanded non-custodial sentencing combined with structured rehabilitation. However, unless the cross-class relapse rate
, the principal driver of the CSER, is reduced, diversion alone may sustain escalation risk, necessitating targeted post-diversion supervision. Short-term interventions increasing
and
should therefore be complemented by medium-term improvements in sentencing and relapse rates (Table 3).
Two structural simplifications merit attention. First, the omission of pre-trial detention likely leads to underestimation of both and
, as an explicit pre-trial compartment would extend criminal career duration. Second, excluding direct transitions from
to
suppresses an additional escalation pathway that would increase
and further amplify the CCR.
The model assumes homogeneous mixing and excludes trial delays and explicit socioeconomic drivers of crime. Given South Africa’s high inequality, unemployment, and concentrated poverty, crime-correlated structural factors could be incorporated through a time-varying recruitment rate, such as . The homogeneous mixing assumption is most appropriate for dense urban settings like Gauteng, for which the model is explicitly calibrated. Additionally, allocating all released petty offenders to a high-risk class may overstate escalation; allowing a probabilistic return to the petty-risk class would refine CCR estimates. Reliance on literature-based parameter estimates introduces uncertainty, and the framework does not capture intergenerational effects or stigma-related reintegration barriers.
Future work could incorporate stochastic or agent-based formulations to capture individual heterogeneity, as well as socioeconomic covariates to enhance realism. Optimal control methods would enable cost-effectiveness analysis of interventions, while fractional-order extensions could model memory effects in recidivism. Multi-patch structures capturing urban–rural heterogeneity are another important extension. In conclusion, this study advances a rigorous, policy-relevant framework for quantifying the unintended criminogenic consequences of incarcerating petty offenders. The three threshold metrics, , the CCR
, and the CSER
, jointly characterise crime generation and escalation risks across custodial, non-custodial, and community-supervision pathways, enabling pathway-level evaluation previously absent from the literature. Evidence-based policies prioritising non-custodial rehabilitation, relapse prevention, and prison reform are essential for reducing
below unity, curbing recidivism, and achieving durable crime reduction.
Appendix A: Parameter estimation details
This appendix provides the detailed data sources and statistical justifications for the key parameter choices used in the main text. Table 5 presents the percentage §§of recorded cases for each offence category.
The value q = 0.75 (proportion of new recruits entering petty crime) is taken as the upper bound of the petty offence share in Gauteng, where property crime prevalence is highest. The calibration of to achieve
reflects SAPS data indicating that petty offences constitute roughly 60–75% of recorded cases; a conservative midpoint of 0.60 was adopted.
The sentencing rates and
are derived from SALRC (2020) data on average case processing times in Magistrates’ and High Courts (2 years for petty offences, 3 years for serious offences). The non-custodial sentencing proportion
is taken from the JICS Annual Report 2020/2021, which records 35–45% non-custodial disposals for petty offences [30]. The remaining parameters on rehabilitation and relapse, together with their respective justifications, are given in Table 6
Appendix B: Equivalence of the geometric loop approach and the next-generation matrix
We establish that the derived in the main text via the geometric loop approach coincides with the spectral radius of the standard next-generation matrix (NGM) [34–36]. Because
is excluded from the force of crime
, it does not appear in the new-infection matrix
below. It does appear in the transition matrix
(as a draining outflow from
and a feeding inflow to
). This is structurally consistent with the NGM framework:
is a transition compartment, not an infection compartment, and its role is to channel petty offenders through a rehabilitation pathway that carries lower escalation risk than the custodial route.
Following van den Driessche and Watmough [36], we let denote the vector of criminal compartments, ordered so that the two infection (crime generation) classes
and
appear first. The linearisation of system (3)–(10) at the CFE
takes the form
, where
collects the rates of new crime generation and
collects all remaining (transition and death) rates. With
, these matrices are:
and
The NGM is . Since
has rank one, K has at most one nonzero eigenvalue. The spectral radius is therefore equal to
, which a direct computation yields as
which coincides exactly with expression (35). This confirms that the geometric loop approach and the NGM method are equivalent.
Acknowledgments
The author acknowledges the support of the University of Johannesburg in the production of this manuscript.
References
- 1.
South African Police Service. Annual Crime Statistics Report 2022/2023. South African Police Service (SAPS); 2023. Available from: https://www.saps.gov.za
- 2.
Singh AM. Policing and crime control in post-apartheid South Africa. Cape Town, South Africa: Juta; 2008.
- 3.
Lancaster L. Why are South Africans underreporting on crime? ISS Today. 2017. Available from: https://issafrica.org/iss-today/why-are-south-africans-underreporting-on-crime
- 4. Becker GS. Crime and Punishment: An Economic Approach. J Polit Econ. 1968;76(2):169–217.
- 5. Lue-Dugmore M, Karth V, Redpath J. PETTY CRIMES THAT MATTER: Evaluating the Western Cape community courts. SA crime q. 2006;(15).
- 6.
Garland D. The Culture of Control: Crime and Social Order in Contemporary Society. University of Chicago Press; 2001.
- 7. Lipsey MW, Cullen FT. The Effectiveness of Correctional Rehabilitation: A Review of Systematic Reviews. Annu Rev Law Soc Sci. 2007;3(1):297–320.
- 8. Singh S. Overcrowding in South African prisons: The reality in the 21st century. New Contree. 2006;51:171–99.
- 9. Sarkin J. Prisons in Africa: An evaluation from a human rights perspective. Sur Int Hum Rights J. 2009;9:22–49.
- 10.
Andrews DA, Bonta J. The Psychology of Criminal Conduct. 5th ed. Anderson/LexisNexis; 2010.
- 11.
Davis AY. Are prisons obsolete? New York: Seven Stories Press; 2003.
- 12. McMillon D, Simon CP, Morenoff J. Modeling the underlying dynamics of the spread of crime. PLoS One. 2014;9(4):e88923. pmid:24694545
- 13.
Banks HT, Castillo-Chavez C, editors. Bioterrorism: Mathematical Modeling Applications in Homeland Security. vol. 28 of Frontiers in Applied Mathematics. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2003.
- 14. Ehrlich I. Participation in Illegitimate Activities: A Theoretical and Empirical Investigation. J Polit Econ. 1973;81(3):521–65.
- 15. Piquero AR, Jennings WG, Diamond B, Farrington DP, Tremblay RE, Welsh BC. A meta-analysis update on the effects of early family/parent training programs on antisocial behavior and delinquency. J Exp Criminol. 2016;12:229–48.
- 16. Abou Chakra M, Hilbe C. Modelling the dynamics of crime and punishment: comment on “Statistical physics of crime: a review” by M.R. D’Orsogna and M. Perc. Phys Life Rev. 2015;12:22–3. pmid:25623959
- 17. Accinelli E, García A, Policardo L, Sánchez-Carrera E. A predator-prey economic system of tax evasion and corrupt behavior. JDG. 2023;10(2):181–207.
- 18. Kwofie T, Dogbatsey M, Moore SE. Curtailing crime dynamics: A mathematical approach. Front Appl Math Stat. 2023;8.
- 19. Khiyar S, Hafdane M, Boutayeb H, Adnaoui K, Elberrai I. Modeling and control of crime using optimal control theory. Commun Math Biol Neurosci. 2026;2026.
- 20. Nyabadza F, Kajambeu R. A Mathematical Model of Substance Abuse–Driven Criminality: Exploring the Dynamics of Addiction and Crime. J Math Sci Modell. 2025;8(4):201–17.
- 21. Baba IA, Rihan FA, Hincal E. A fractional order model that studies terrorism and corruption codynamics as epidemic disease. Chaos Solitons Fractals. 2023;169:113292.
- 22. Sooknanan J, Bhatt B, Comissiong DMG. Catching A Gang -- A Mathematical Model Of The Spread Of Gangs In A Population Treated As An Infectious Disease. Int J of Pure and Appl Math. 2013;83(1).
- 23. Sooknanan J, Comissiong DMG. When behaviour turns contagious: the use of deterministic epidemiological models in modeling social contagion phenomena. Int J Dynam Control. 2016;5(4):1046–50.
- 24.
Warr M. Companions in Crime: The Social Aspects of Criminal Conduct. Cambridge, UK: Cambridge University Press; 2002.
- 25. Haynie DL. Delinquent peers revisited: does network structure matter? Am J Sociol. 2001;106(4):1013–57.
- 26.
Republic of South Africa. Correctional Services Act 111 of 1998. Government Printer, Pretoria. 1998. Available from: https://www.gov.za/documents/correctional-services-act
- 27.
Republic of South Africa. Magistrates’ Courts Act 32 of 1944. Government Printer, Pretoria. 1944. Available from: https://www.justice.gov.za/legislation/acts/1944-032.pdf
- 28.
Muntingh LM. After Prison: The Case for Offender Reintegration. Pretoria, South Africa: Institute for Security Studies. 2001. 52 p. Available from: https://issafrica.org/research/monographs/monograph-52-after-prison-the-case-for-offender-reintegration-by-lukas-muntingh
- 29.
Department of Correctional Services (DCS). Statistical Report on Offender Management. Department of Correctional Services, South Africa. 2022. Available from: https://www.dcs.gov.za/
- 30.
Judicial Inspectorate for Correctional Services (JICS). Annual Report 2020/2021. Cape Town, South Africa: Department of Justice and Constitutional Development, South Africa. 2021. Available from: https://pmg.org.za/files/JICS_AR-2020-2021.pdf
- 31. Cameron E. Comment and Analysis – The Crisis of Criminal Justice in South Africa. S Afr Crime Q. 2020;(69):1–15.
- 32. Samuels JA, Roman NV, Schoeman M. A qualitative inquiry: management of recidivism in South Africa. Crime Law Soc Change. 2024;82(1):197–218.
- 33. Lötter C. Gentle justice reduces recidivism and incarceration: can South Africa benefit from the Finnish experience?. Phronimon. 2023;24(1):1–25.
- 34. Diekmann O, Heesterbeek JA, Metz JA. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990;28(4):365–82. pmid:2117040
- 35. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873–85. pmid:19892718
- 36. van den Driessche P, Watmough J. Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Math Biosci. 2002;180(1–2):29–48.
- 37. Park J, Kim P. Mathematical analysis of crime dynamics in and out of prisons. Math Methods Appl Sci. 2020;44(1):650–67.
- 38.
Statistics South Africa. Mid-year Population Estimates. Statistics South Africa; 2022. Available from: https://www.statssa.gov.za/
- 39. Singh S. Offender rehabilitation and reintegration: A South African perspective. J Soc Sci. 2017;46:1–10.
- 40.
South African Law Reform Commission. Discussion Paper 174: The Right to a Fair Trial – The Need to Obviate Delays. South African Law Reform Commission; 2026. 174 p. Available from: https://www.justice.gov.za/salrc/dpapers/DP174-Project151-Right-To-a-Fair-Trial-Obviate-Delays.pdf
- 41.
The World Bank. World Development Indicators. The World Bank; 2023. Available from: https://data.worldbank.org
- 42.
Department of Correctional Services. Annual Report 2023/2024. Department of Correctional Services (DCS), South Africa. 2024. Available from: https://www.gov.za/
- 43.
Institute for Crime and Justice Policy Research. South Africa Prison Profile. 2023. Available from: https://www.prisonstudies.org
- 44. Blower SM, Dowlatabadi H. Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: An HIV Model, as an Example. Int Stat Rev. 1994;62(2):229.
- 45. Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008;254(1):178–96. pmid:18572196
- 46. Redpath J, O’Donovan M. REACHING A VERDICT: The impact of minimum sentencing. SA crime q. 2016;(19).