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Mathematical Modelling and Analysis of Infiltration of Firearms and Its
Insecurity Implications in the Northern Region of Nigeria
*Mutah Wadai
1
; Ibekwe Jacob John
2
and Idongesit Nnammonso Akpan
3
1&2
Department of Mathematics, Federal University of Health Sciences, Otukpo, Benue State, Nigeria
3
Department of Chemistry, Federal University of Health Sciences, Otukpo, Benue State, Nigeria
*Corresponding Author
1
https://orcid.org/0009-0006-6075-4340;
3
https://orcid.org/0009-0009-2168-3596
DOI: https://doi.org/10.51583/IJLTEMAS.2026.150100049
Received: 13 December 2025; Accepted: 19 December 2025; Published: 31 January 2026
ABSTRACT
This research produces and analyzes an innovative nonlinear compartmental model to explore the infiltration of
firearms into Nigeria and its resulting insecurity dynamics in Northern Nigeria. The model incorporates four
interacting components: susceptible individuals, violent actors, arms and ammunition, and recovered individuals,
capturing the bidirectional feedback between arms infiltration and violence recruitment. An effective
reproduction number, R
e
, was derived to characterize the threshold conditions governing the persistence or
elimination of violence. Stability analyses, including Lyapunov-based proofs (
AwVwAVL
21
),( +=
), create the
conditions for global stability of both the violence-free and endemic equilibria. Normalized sensitivity analysis
using Python revealed that arms-related parameters, particularly arms inflow (σ =+0.312), arms to violence
activation (α= -0.020), and arms-induced susceptibility (γ=+0.254), are the most influential drivers of violence
propagation. Numerical simulations were used to validate the investigative results, which showed how reducing
arms availability can substantially lower the long-term levels of violent actors. These findings showed that illicit
arms proliferation is the primary catalyst sustaining endemic violence. The necessary interventions, such as
addressing arms inflow, distribution networks, and de-escalation processes, are most effective for violence
reduction. The simulation results establish that the long-term dynamics of violence in Northern Nigeria are
strongly driven by the circulation and availability of illicit firearms. The trajectories for S(t), V(t), and A(t)
showed that the reductions in arms-related parameters produce meaningful declines in violence, confirming the
sensitivity analysis results. This study provides a quantitative framework and actionable insights to support
evidence-based security policy, arms-control strategies, and conflict-mitigation efforts in Northern Nigeria.
Keywords: Violence, Simulation, Dynamics, Arms-Control
INTRODUCTION
The infiltration of firearms into Nigeria, particularly through the Sahel region and Central African countries, has
emerged as one of the most critical drivers of insecurity in northern region of Nigeria (Bassey et al., 2025). The
proliferation of small arms and light weapons is adjudged as the most immediate security challenge to
individuals, societies, and states worldwide, fuelling civil wars, organized criminal violence, insurgency, and
terrorist activities, posing great obstacles to sustainable security and development of nations (Bashir, 2014). The
infiltration of small arms and light weapons (SALW) in Nigeria has become a critical concern for national
security and socio-economic stability, particularly in the northern part of Nigeria (Bassey et al., 2025).
Unfortunately, too, the Northern region of Nigeria faces an array of violent threats, including banditry, killings
by Boko Harm, farmer-herder clashes, terrorism, and communal conflicts that have grown in scale, frequency,
and lethality over the past decade (Agaba and Upkabio, 2023; Isah et al., 2024). Nigeria’s porous borders,
spanning more than 4,000 km, serve as major entry points for illegal smuggling of small arms and light weapons
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(SALWs), that have been estimated to exceed 6 million in circulation within the country (Agaba and Upkabio,
2023; Isah et al., 2024).
Worrisomely, empirical evidence has continued to link the availability of small arms to increased insecurity,
banditry, kidnapping, and terrorism in across the West Africa countries (Abiodun et al., 2018; Small Arms
Survey, 2019). In Nigeria, illicit firearms flow mainly from Libya, Chad, Niger, and Cameroon as a result of
regional instability (Akinola, 2020) has continued to remain the simplest means through which all kinds of
violence groups access their weapons including armed robbers, kidnappers, Boko Haram, killer herdsmen to
foment, promoted and sustained the general insecurity in the northern region of Nigeria and worst still, the
ECOWAS Convention on SALWs has achieved limited success due to weak enforcement mechanisms (Bah,
2017). Fundamentally, it is important to note that the infiltration of firearms creates a reinforcing cycle of
violence: increased access to weapons encourages the formation and recruitment of armed groups, which in turn
escalates conflict intensity and insecurity, thereby increasing demand for more weapons (Karp, 2018; Afuzie et
al., 2021). Interestingly, mathematical modeling provides a rigorous framework for understanding how these
interconnected processes evolve and how targeted interventions can disrupt the violence-arms cycle (Brauer &
Dunne, 2011). This study develops a dynamic compartmental model that captures the interactions among
susceptible individuals S(t), Violent groups V(t), the inflow of illegal firearms/ammunition A(t), and recovery
individuals R(t). The model incorporates sociopolitical drivers unique to northern region of Nigeria and provides
quantitative insights into how firearm inflow amplifies insecurity in the region.
Currently, research applying formal mathematical models to arms proliferation and the violence it enables or
enhances in Nigeria is an emergent but interesting growing field of research. Nevertheless, although there are
different modelling traditions apparently in the literature, such as agent-based and network modelling, game-
theoretic and economic models of demand and diversion (Morgan, 2020; Alusala, 2023), compartmental or
deterministic population modelling methodology was employed in the present investigation. Meanwhile, many
researchers and authors have so far reported on the use of mathematical models and simulations to study the
infiltration and spreading of firearms and weapons with respect to kidnapping, banditry, terrorists and Islamic
extremists’ activities and the implications for West African regional security. In that regard, Kambai (2023)
reported on a mathematical model and analysis of the proliferation of arms and weapons in Nigeria and control,
and a seven-compartmental model with well-defined classes was utilized as his designed model for analysis of
the influence of proliferation of arms and weapons used and kidnapping activities on population dynamics. In
addition, Kambai (2023) employed the Runge-Kutta method of order four to derive the numerical solutions of
the model’s hypotheses.
In another report, Akpienbi and Ibrahim (2024) have reported the mathematical modelling of security forces –
insurgent dynamics in Nigeria. In the work, the authors developed a compartmental model where the populace
was compartmentalized into protection forces (security forces), insurgents, and rehabilitated compartments. The
model's equilibrium points for local approach stability of equilibrium were determined using the Routh-Hurwitz
condition, and the model's local stability and numerical simulations were performed by employing parameter
values with codes implemented using MATLAB 2012b software to generate a diagram of population trends. In
conclusion, the study recommended a modified version of the prey-predator model of security forces and
criminal dynamics, which was proposed as reported by Oduro et al. (2015). In other related reports, Bassey et
al. (2025) reported small arms and light weapons proliferation in Nigeria: problems and prospects, and Bashir
(2014) reported small arms and light weapons proliferation and its implications for West African regional
security, respectively. In their submission, both authors maintained that the infiltration of small arms and light
weapons (SALW) into Nigeria has become a critical concern for nationwide security and socio-economic
stability by fuelling all kinds of violence activities including civil wars, organized criminal violence, insurgency,
and terrorist activities, thereby constituting great hindrances to sustainable security, growth and developmental
progress in Nigeria. Furthermore, the authors have adjudged the infiltration and spreading of small arms and
light weapons (SALW) in Nigeria as the greatest direct security challenge not only to individuals and societies,
but also to states and nations worldwide (Bashir, 2014; Bassey et al., 2025).
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Meanwhile, despite the numerous reports and mathematical modellings and simulations of the effect of
proliferation of arms and weapons on kidnapping activities and its general consequences on the West African
regional security elsewhere in the literature, there appears to be very little or none of such information about the
mathematical modelling and analysis of infiltration of firearms and its insecurity implications in the Northern
part of Nigeria. Based on the foregoing, this work therefore is aimed at utilizing the mathematical tools of
modelling and simulation to analyze the possible means of infiltration of firearms and their insecurity
implications in Northern Nigeria, and possible means of reducing the inflow of firearms into Nigeria using
normalized sensitivity analysis and simulation of a designed model to curtail the rate of insecurity in the Northern
part of Nigeria. This, we believe, will go a long way to form a quantitative framework and actionable insights to
support evidence-based security policy, arms-control strategies, and conflict-mitigation efforts by governments
at various levels toward the Northern part of Nigeria.
This paper, in addition also argues that using a mathematical model and simulation, the possession of illegal
small and light weapons through infiltration of firearms across the breadth and length of the northern part of
Nigeria in particular and Nigeria at large, which fuels insecurity, insurgency, banditry, farmers-herders clashes,
Boko haram, killers herd men and terrorism in the northern Nigeria must be addressed along the line of
controlling the infiltration into Nigeria and the spreading of illegal firearms and light weapons cross the country
and other west African countries must cooperate and collaborate to address this huge and complex insecurity
challenge bedevilling the northern part of Nigeria in particular and Nigeria in general.
MATERIALS AND METHODS
Mathematical Models of Conflict Dynamics
This study utilizes a model by including northern Nigeria–specific parameters such as border permeability and
armed group recruitment. Mathematical modeling and analysis of infiltration of firearms into Nigeria and its
insecurity implications in the Northern region is developed and analyzed. The invariant region was determined,
and the positivity of the solution of the model was analyzed, in order to ascertain that the model is well- posed
mathematically and bounded.
The Designed Model
To undertake the study, the proposed designed mathematical model is subdivided
into four compartments at
time
t
: Susceptible population
)(tS
(populations not directly involved in violence),
)(tV
(violence actors such
as bandit, Boko Haram, kidnapping, Asaru etc.),
)(tA
(arms/ammunitions that have infiltrated the region,
treated as a resource that fuels
)(tV
),
)(tR
(population who have escaped the violence zone or have been
rehabilitated(recovered).
The birth and death rate for susceptible populations are π and µ respectively. The susceptible population, S(t)
becomes violence actors at the rate of
SV
and is been reintegrated back to S(t) at the rate of
and dies out
at the rate of µ.
and
AV
are the permanent removal from V and the removal from V via A’s action
respectively. The
and
SV
are the de-mobilization/outflow from A and mobilization from S due to V’s
presence respectively. The arm/ammunitions are recruited at the rate σ and fuels the violence actors at the rate
α and decay or seized at the rate µ.
Model Hypotheses
i. Do firearms increase the recruitment rate of new violence actors?
ii. Does violent conflict decrease as arms are removed or actors are demobilized?
iii. Do arms enter the system through smuggling, theft, and black-market trade?
iv. Can Government interventions reduce firearm stock and violent groups?
The systematic diagram as well as the detailed descriptions of the model’s variables and the parameters are given
in the Figure 1, and Tables 1 and 2.
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Figure 1: Schematic Diagram of the Conflict Flow Model
Variables
Interpretations
S
Susceptible individuals.
V
Violence actors.
A
Arms/ammunitions.
R
Recovered individuals.
Table 1: State variables and their interpretations
Parameters
Interpretations
Recruitment or birth rate into S
Displacement rate (effect of violent class V reducing susceptible).
Rate at which arms get to susceptible individuals.
baseline rate by which susceptible move into violence class
Rate at which the violent actor’s ‘V’ produces additional armed individual’s
‘A’ via contact with susceptible (term εSV).
Natural exit rate (death, migration, ageing out, decay, seizures).
Rate at which arms/ammunitions fuels violence groups ‘V’.
Transition rate from violence groups ‘V’ to recovered class R(capture,
death, reintegration).
Coupling coefficient representing additional removal/transition intensity
that depends on A interacting with V.
External inflow to the arms/ammunitions class ‘A’ (e.g., firearms
importation or supply).
Table 2: Model Parameters and their Interpretations
The Model Equations
Considering the systematic diagram, Figure 1, the possible differential equations derived are:
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SSVSAV
dt
dS
−−−+−=
, (1)
VAAS
dt
dV
)(
++−+=
, (2)
ASV
dt
dA
)(
++−+=
, (3)
RAVV
dt
dR
−+=
, (4)
Basic Mathematical Properties of the Designed Model
The fundamental properties of the equations (1) to (4) of our designed model such as the invariant region,
positivity of solutions of the model, the equilibrium state (violence-free equilibrium states and its local and global
stabilities) were considered, the effective reproduction number were obtained, and the sensitivity analysis of the
model equations was also computed.
The Invariant Region/Boundedness
The invariant region that the model solutions are bounded was obtained, and the total population,
)(tN
was
given by
;
RVStN ++=)(
, ………………………………………(5a)
Since A(arms/ammunitions) is a stock or inhuman, with initial condition;
,)0(
0
NN =
Hence, the differentiation of
)(tN
with regards to ‘𝑡’ leads to;
dt
dR
dt
dV
dt
dS
dt
dN
++=
. …………………………………………………(5b)
Substituting equations (8) to (15) into (16) and simplifying further, we have;
( )
)()( SVARVS
dt
dN
−+++++−=
; ………………………………(6)
Since
,)( RVStN ++=
Then equation (6) reduces to,
)()( SVAN
dt
dN
−+++−=
…………………………………….(7)
In the absence of violence and arms (since its inhuman), V = A = 0, and
Introducing inequality in (7), we have;
,N
dt
dN
−
……………………………………………………(8)
Applying method of integrating factor and further simplification of (8), leads to;
( )
,1)(
0
tt
eNetN
−−
+−
…………………………………………(9)
This result confirms the boundness of the total population as; 𝑡 → ∞, the term
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0→
− t
e
(assuming µ > 0) , in equation (9);
,)(
→tN
………………………………………………………..(10)
This means the feasible solution of the model of Equations (1) to (4) is in the region,
.)(:0,,,;),,,({
4
=
+
tNRAVSRRAVS
; ………………………..(10b)
Hence, this implies that the proposed conflict model systems (1) to (4) are well modelled mathematically and
with boundedness. Hence, they are sufficient enough to analyze and ascertain the dynamics of the model in this
region, Ω
Positivity of Solution
The solution of the model is shown to be positive as it represents a human population that cannot be negative.
To validate this, it has been assumed that the given initial condition of the model is non-negative, and the
illustration has shown that the solutions of the model are positive. In that mathematical sense of consideration,
the following theorems were proposed:
Let
4
},,,{
+
= RRAVS
be solution set such that
0
)0( SS =
,
0
)0( VV =
,
0
)0( AA =
,
0
)0( RR =
, are positive, then the
solution of the set
are all positive for
.0t
From equation (1), we have:
SSVSAV
dt
dS
−−−+−=
; ………… (11)
Which means that;
,][ SV
dt
dS
++−
…………………………………..(11b)
Using the exponential growth criterion and integrating (11) with initial conditions
0
)0( SS =
gives;
,0)(
)(
0
++− dtV
eStS
,0)(
)(
0
++− tV
eStS
Also, from equation (2), we have:
VAAS
dt
dV
)(
++−+=
; which means that;
VAAS
dt
dV
)(
++−+
; …………………………… (12a)
Using the exponential growth criterion and integrating (12) with initial conditions
0
)0( VV =
gives;
0)(
)(
0
++− dtA
eVtV
0)(
)(
0
++− tA
eVtV
From equation (3), we have;
ASV
dt
dA
)(
++−+=
; ……………………………………(12b)
which means that;
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A
dt
dA
)(
++−
……………………………………………... (13)
Using the exponential growth criterion and integrating (13) with initial conditions
eAA
0
)0( =
gives:
0)(
)(
0
++− t
eAtA
From equation (4), we have;
RAVV
dt
dR
−+=
which means that;
R
dt
dR
−
……………………………………(14)
Using the exponential growth criterion and integrating (14) with initial conditions
0
)0( RR =
gives:
t
eRtR
−
0
)(
Therefore, the set solution identified as
},,,{ RAVS
for the systems of the Equations (1) to (4)) are positive only
for all
0t
; since the exponential functions and their initial conditions have been proved to be positive.
The Violence-Free Equilibrium State of the Model
The state where there is no conflicts (violence) is referred to as the violence-free equilibrium. Therefore, the
point was obtained by equating the right-hand side of all the equations (1) – (4) of the system to zero and then
imposing V
0
= 0. Here, we let E
0
be the violence -free point.
Then, substituting, V
0
= 0 into (3), we have;
,0)( =++−
A
………………………………………(15a)
Therefore,
)(
0
++
=A
…………………….(15b)
Substituting V
0
and A
0
into (1), we have;
0)(
00
=+−+ SA
,
)(
0
0
+
+
=
A
S
…………………………………………….(16)
Therefore,
.0,
)(
,0,
)(
),,,(
0
00000
+++
+
==
A
RAVSE
, ………(17)
Effective Reproduction Number
We treat the violent class “V” and the arms/ammunition class “A” as the infected compartment. Thus, the
infected vector is X = (V, A)
T
We follow the standard decomposition (Driessche & Watmough, 2002). For each infected compartment i, write
x
i
= F
i
(x) – V
i
(x) where F
i
= is the rate of appearance of new voilence in compartment i and V
i
is the net
transmission (other gains minus losses) in that compartment. Evaluate Jacobians at the CFE E
0
and compute the
next-generation K = FV
-1
. The Effective Reproduction Number R
e
is the spectral radius ρ(K).
From the model: New infections that are directly produced by infected classes:
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New V from arms: F
v
= αA;
Where; ‘A’ receives new contributions from ‘V’ through εSV. So F
A
= εSV.
,
=
SV
A
F
……………………………………………….(18)
The remaining terms define V(x) (losses and transfers excluding F).
AVAxV
−++= )()(
, and
AxV )()(
++=
,
Therefore,
++
−++
=
A
AVA
xV
)(
)(
)(
………………………………..(19 )
Taking the partial derivative of F and V with regards to the violence vector (V, A) at VFE, we have the Jacobian
matrices
,
0
0
)(
0
0
=
S
EF
………………………….(20)
and
++
++
=
0
0
)(
0
0
A
EV
.
……..(21a)
Taking the inverse of V, we have;
++
++
=
−
1
0
0
1
0
1
A
V
The next generation matrix is K = FV
-1
.
Thus, by computing the eigen-values to evaluate the effective reproduction number
c
R
by taking the spectral
radius (dominant eigen-value) of the matrix
K
.
This is computed by
0=− IK
, and after further simplification, we have:
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))((
0
0
A
S
++++
=
………………(21b)
Therefore, the effective reproduction number for violence is the spectral radius (largest absolute eigen-
value) of the matrix. Substituting the expression for S
0
and A
0
we have;
)))(()((
))((
++++++
+++
=
e
R
………………..(22a)
The Stability Threshold.
The effective reproduction number,
c
R
, was determined by the decomposition technique in van den
Driessche and Watmough, (2002). Thus, an effective reproduction number attained by this method
defines the local stability of the Conflict (violence)-free equilibrium point which is locally
asymptotically stable for
1
c
R
and unstable for
1
c
R
.
Thus, the condition for long-term control of firearm-driven violence is to ensure that
1
)))(()((
))((
++++++
+++
……….(22b)
The Existence of Endemic Equilibrium States of the Model
The endemic equilibrium state of the model is a steady-state solution where violence exists in the susceptible
population. For the determination of the endemic equilibrium point E
*
, the steady state of the system from
Equations (1) to (4) of all state variables was considered. Then;
0
******
=−−−+− SVSSAV
, ……………………………….(23)
0)(
****
=++−+ VAAS
,……………………………………….(24)
0)(
***
=++−+ AVS
,…………………………………………(25)
0)(
***
=−+ RVA
,………………………………………………….(26)
Since we are at endemic equilibrium,
0
*
V
From (26), solving for R
*
*
*
*
)(
V
A
R
+
=
………………………..(27)
From (24), solve for S
*
****
)( AVAS
−++=
***
*
)( AVA
S
−++
=
………………………(28)
From (25), solving for A
*
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***
)( VSA
+=++
)(
**
*
++
+
=
VS
A
, ……………………………………(29)
Substitute (28) into (23), we have;
−++
+=++
***
**
)(
)(
AVA
VA
……….(30)
( )
( ) ( )
**
2
**
2
**
)( VAVAVA
−+++=++
, ………………(31)
Substituting (29) into (30) and simplifying further will yield a high complex equation and can be solved
numerically for specific parameter values. Therefore, the actual value of V
*
is the positive root of a high-degree
polynomial function F(V
*
) = 0
The equation (31) is a polynomial of high degree in V
*
, which cannot be solved analytically. The state of stability
of endemic equilibrium is often related to the trans-critical bifurcation that occurs at R
0
= 1. If R
0
> 1; the endemic
equilibrium E
*
generally occurs and is locally asymptotically stable and if R
0
< 1; the endemic equilibrium E
*
is
either non-existent or unstable and the system moves towards the violence free equilibrium E
0
.
Global Stability of Violence Free Equilibrium
At this point, our target is to investigate the global asymptomatic stability of the violence free equilibrium
state. In that regard, we proposed the theorem 1: if
1
c
R
, then E
0
is globally asymptomatically stable in the
feasible region
0
))(),(),(),(( EtRtAtVtS →
as
→t
an unstable if
1
c
R
.
Proof: Since the violence variables are V(t) and A(t) and these form the next-generation matrix from which R
e
is derived, a natural Lyapunov function is the weighted linear combination of V and A using the left eigenvector
of the next-generation matrix.
Let; K = FV
-1
; where F and V are the new violence and transition matrices evaluated at E
0
.
However, let w = (w
1
+ w
2
) > 0 be the left perron eigenvector satisfying; wF = R
e
w.
Then, define the Lyapunov function;
AwVwAVL
21
),( +=
. Where; w
1
> 0, w
2
> 0.
Clearly,
).0,0(),(0,0 == AVLL
This isolates the violence subsystem and is the standard structure for global stability proofs in epidemic systems.
Let
dt
dA
w
dt
dV
w
dt
dL
21
+=
…………………………………………………(32)
Substitute the derivative of V and A in (32), we have;
)())((
21
DASVwVAASw
dt
dL
−++++−+=
…………………..(33)
Where D = γ + α +µ. And from (33), we rearranging the terms in the form (34);
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)())(()()(
211202201
DwwVwSwAADwwSSw
dt
dL
−++−+−−+−=
(34)
At the VFE:
,0)(
000
=+=+ VAS
.0=− DA
Therefore, the first bracket vanishes when (S, A) = (S
0
, A
0
).
(for stability analysis, we use the fact that (S(t), A(t) remain bounded and positive)
Using the next generation matrix relation, that is eigenvalue relation, we have;
.wVRwF
e
=
This yields the inequality (standard in the van den Driessche-Watmough framework).
).)(()()(
2100201
DwwRASwSw
e
++=+
………………………….…(35)
Because
0
)( StS
and
0
)( AtA
for all t (due to boundness of solution), we obtain;
).)((
2121
DwwRSAwSw
e
+++
…………………………………...(36)
This inequality is the central device allowing us to control
•
L
. Therefore, using the inequality (36), the
derivative satisfies,
.))()(1(
121
AVwDAwVwRL
e
−++−
•
………………………………….(37)
From (25) the last term
.0
1
− AVw
; and the factor multiplying the first bracket is
)1( −
e
R
. Thus, if;
1
e
R
,
then;
.0))()(1(
121
−++−
•
AVwDAwVwRL
e
Equality
0=
•
L
occurs only when V = 0, A = 0.
Because L is radially unbounded in (V, A), all solutions approach the largest invariant set in
,0:).(
=
•
LAV
which is the only the singleton
.)0,0(
Thus:
).0,0())(),.(( →tAtV
Once
)0,0())(),.(( →tAtV
; the remaining subsystem reduces to the linear equations
,)( SS
+−=
•
,RR
−=
•
whose global solution is
,)(
0
StS →
.0)( →tR
Thus;
.)0,,0,())(),(),(),((
000
EAStRtAtVtS =→
Conclusively, if
1
e
R
, then the violence-free equilibrium E
0
is globally asymptotically stable.
Global Stability of Endemic Equilibrium Point
For this section, our target is to examine the global asymptomatic stability of the endemic equilibrium. Here, we
treat the important and instructive special case where the ‘A’ dependent removal term ϕAV in V and R equation
is absent (ϕ = 0). This removes of nonlinear coupling and allows construction of a Volterra-type Lyapunov
function and a full algebraic verification of negative definiteness of its derivative. Hence, the next theorem has
been proposed;
Theorem 2: if ϕ = 0 and all the parameters > 0. There occurs a unique interior equilibrium; E
*
= (S
*
, V
*
, A
*
, R
*
)
with V
*
> 0.
Proof: Here, we have proven that, E
*
is globally asymptotically stable in the interior of Ω. At E
*
the steady
equations give (ϕ = 0):
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******
0 SVSSAV
−−−+−=
, (38)
***
)(0 VAS
+−+=
, (39)
***
0 DAVS −+=
, (40)
**
0 RV
−=
, (41)
Where D = γ + α +µ,
From (40) and (41),
,
)(
**
*
D
VS
A
+
=
and
*
*
V
R =
Lyapunov function (Volterra-type) is defined by;
( )
)42.....(..........ln
lnlnln,,,
*
**
4
*
**
3
*
**
2
*
**
1
−−
+
−−+
−−+
−−=
R
R
RRRc
A
A
AAAc
V
V
VVVc
S
S
SSScRAVSL
Where c
1
, c
2
, c
3
, c
4
> 0 are all constants to be chosen. This function is nonnegative in the interior of Ω and
varnishes if and only if (S, V, A, R) = (S
*
, V
*
, A
*
, R
*
). We will choose c
i
so that
0
•
L
and
0=
•
L
only at
equilibrium. On differentiating (42), we have;
)43.......(1111
*
4
*
3
*
2
*
1
dt
dR
R
R
c
dt
dA
A
A
c
dt
dV
V
V
c
dt
dS
S
S
c
dt
dL
−+
−+
−+
−=
substituting (1) to (4) into (43), we have;
)(1)(1
))((1)(1
*
4
*
3
*
2
*
1
RAVV
R
R
cDASV
A
A
c
VAAS
V
V
cSSVSAV
S
S
c
dt
dL
−+
−+−+
−
+++−+
−+−−−+−
−=
…(43b)
After expanding and cancellation of common opposite terms in Equation 43b, we will be left with;
),,,( RAVSQ
dt
dL
−=
, ……………………………………………….. (44)
Where Q is a sum of nonnegative terms (quadratic and product terms) and Q = 0, only at the equilibrium.
Because
L
is positive definite and
0
•
L
with
0=L
only at E
*
, by LaSalle’s invariance principle the interior solution
trajectories converge to E
*
. Hence E
*
is globally asymptotically stable in the interior of Ω.
RESULTS AND DISCUSSION
Sensitivity Analysis, Numerical Simulation and Effective Reproduction Number
In this section, results of sensitivity analysis and numerical simulation are presented using parameter estimates
in Table 3. In order to establish the impact of the model parameters on the mathematical modelling and analysis
of infiltration of firearms and Its implications in Northern Nigeria, it is important that we consider and conduct
sensitivity analysis using the normalized forward sensitivity index (22). Following (22), the relative importance
of the input parameters associated with the output variable
c
R
is evaluated using the formula:
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e
e
R
p
R
p
p
R
X
e
=
,
Using the values in Table 3, from Equations (1) to (4), we obtained the sensitivity indices by employing python
code and the results are as shown in Table 4. To this end, Table 4 shows the sensitivity index of each parameter
in the effective reproduction number
e
R
. The sensitivity index in the Table 4 sequence shows the parameter
with the highest sensitivity to the lowest sensitivity. The parameters
,,,
have the most positive sensitivity
contributors while
,,
have the most sensitive negative contributors. Notably, Table 3 and Table 4 show the
parameter value and indices for the model system (1) to (4). The utmost positive index is ε (arms/contact
influence). This implies that ‘ε’ have the strongest promoter of violence reproduction. The most negative index
is β (violence removal rate). This shows that the more the population practices these procedures β,
keeping other
parameters constant, the more the decrease in violence. Therefore, it is the most powerful suppressor.
Table 3: Values of Parameters of the Model
Model
Parameters
Values
Source of Each Value
0.033yr
-
1
World Bank birth rate ≈ 33/1000 (Nigeria, 2023)
0.012yr
-
1
World Bank death rate ≈ 12/1000 (Nigeria, 2023)
0.30
Literature on radicalization & arms inflow (Okoli &
Iortyer, 2014)
0.20
Demobilization rate estimates (UNDP, 2020 DDR report)
0.08
Small Arms Survey (2019): ~12-year average circulation
lifespan
0.10
UNODC (2022) reports on voluntary surrender programs
0.50
Small Arms Survey (2021) estimates for West Africa's
illicit arms inflow
0.25
Nigerian Armed Forces Annual Report (2021)
0.60
Studies on weaponization effect (Kwaja, 2020; Eze, 2020)
Table 4: Values of Sensitivity Indices of Some Parameters.
Parameter
Sensitivity indices
+0.0085
-0.205
-0.020
+0.254
-0.403
-0.320
+0.312
-0.103
+0.5
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Figure 2: Shows the Normalized Forward Sensitivity Index of R
e
with Respect to it Parameters
Bar Chat (Bakare & Nwozo, 2017).
Figure 3: The Graph of Effect of R
e
on π; (Mahboobtosi et al., 2025),
Figure 4: The Graph of Effect of R
e
on µ.
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Figure 5: The Graph of Effect of R
e
on α.
Figure 6: The Graph of Effect of R
e
on γ.
Figure 7: The Graph of Effect of R
e
on β.
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Figure 8: The Graph of Effect of R
e
on δ.
Figure 9: The Graph of Effect of R
e
on σ.
Figure 10: The Graph of Effect of R
e
on ϕ.
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Figure 11: The Graph of Effect of R
e
on ε.
The Stability Simulation of Endemic Equilibrium State (EES)
The analysis of stability of the numerical simulations was performed to validate analytical results. Numerical
integration of the full model from five widely dispersed interior initial conditions shows convergence of all state
variables to a common interior equilibrium (S
*
, V
*
, A
*
and R
*
) ≈ (0.4944, 1.0113, 1.5625, 41.0380) (Figures 12
– 16). The susceptible, violent and arms/ammunitions compartments equilibrate rapidly (within t ≈ 20, time
units) while the recovered class accumulates more slowly. The phase plot (Figure 16) confirms joint attraction
of the (A, V) subsystem. These findings numerically corroborate the analytic stability results and support the
conclusion that, for table 4 parameterization, the endemic equilibrium is globally attractive. Policy simulations
(sensitivity analysis) indicate that reducing arms inflow (σ) and contact-driven arming (ε) are the most effective
levers to lower equilibrium violence. In this particular situation, the trajectories are represented by colours such
as; blue, red, green, purple and solid plots that congregate towards the equilibrium state as the time approaches
infinity as shown in Figure 12 to Figure 16 below.
Figure 12: A Graph of Time Series of S(t) of the Model from Simulation
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Figure 13: A Graph of Time Series of V(t) of the Model from Simulation
Figure 14: A Graph of Time Series of A(t) of the Model Obtained Through Simulation.
Figure 15: A Graph of Time Series of R(t) of the Model Obtained by Simulation
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Figure 16: A Graph of V versus A-Time Series of the Model from Simulation.
In this section, we discussed some of the effects of the parameters of the model.
Figure 3: Effect of R
e
on π.
The curve is monotone increasing and essentially linear over the 0.5 – 1.5× range. As π increases from 0.5×
baseline to 1.5× baseline, R
e,
rises steadily.
π appears in the numerator of the term inside the square root through π(γ + α + μ) + γσ. Increasing π increases
available susceptibles at VFE (S₀), which increases the two-step arms to violence cycle that determines R
e
.
Because π enters linearly in the numerator under the square root, the effect on R
e
is sub-linear (square-root
dampening), producing a near-linear slope over this narrow range.
Magnitude (elasticity): normalized index ≈ +0.085, a 1% increase in π increases R
e
by 0.085%. Thus, π is a
weak positive driver relative to arms parameters.
Interpretation/policy: Higher population recruitment modestly increases violence potential but is not the most
effective lever for short-term arms/violence control. Long-term planning (youth employment, education) matters
but will have smaller immediate effects on R
e
than arms specific interventions.
Figure 4: Effect of R
e
on µ.
R
e
falls as μ increases (negative slope). μ appears in the denominator both directly and inside the (γ +α + μ)
terms, raising μ increases exit rates that shorten the effective lifetimes of arms or violent actors, thereby reducing
the reproduction potential. Because μ appears in denominator inside the square root, the effect is stronger than
for π.
Magnitude (elasticity): normalized index ≈ −0.205, a 1% increase in μ reduces, R
e
by 0.205%. (μ is non-policy
in the short term but mathematically a damping factor.)
Interpretation/policy: μ is not a realistic direct policy lever, however, it shows the system’s sensitivity to rates
that reduce the duration of exposure (e.g. accelerated removal/incapacitation has a similar dampening effect).
Thus, interventions that decrease the active duration of violent actors (capturing, de-radicalization and
reintegration) have tangible effects similar to increasing μ.
Figure 5: Effect of R
e
on α;
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The curve increases with α but with diminishing slope (concave up then flattening), the increase is real but sub-
linear. α multiplies ε and appears both in numerator and as part of (γ + α + μ) term in the denominator. Increasing
α raises direct conversion capability (favors growth) but also increases (γ + α + μ) (shortens arm lifetime via the
pathway A to V leaving A), producing partially offsetting effects. Net effect here is positive but moderated.
Magnitude (elasticity): normalized index ≈ −0.020 in one prior local calculation (small negative) but the sweep
here shows a small positive effect, this difference stems from baseline choices and algebraic cancellations (α
appears in numerator and denominator). For the baseline used for the plots we observe a modest positive effect.
Interpretation/policy: α is mechanistically important, if arms are more likely to convert holders into violent
actors, the system becomes more permissive. Interventions that reduce the conversion probability (education,
normative change, targeted policing) reduce R
e
, but because α is entangled with other rates its marginal effect
may be smaller than direct supply interventions.
Figure 6: Effect of R
e
on;
Increasing γ raises R
e
in the plotted range (curve is increasing), at first glance raising γ (disarmament or arms
decay) might be expected to reduce risk. In this model γ plays two roles: (i) it increases the denominator (γ + α
+ μ) (which would reduce R
e
), but (ii) it contributes positively to the S₀ term via π(γ + α + μ) + γσ (because A₀
= σ/(γ + α + μ) and S₀ = (π + γA₀)/(β + μ). In our baseline, the second effect dominates slightly, raising γ increases
the susceptible pool in the VFE (through the γσ term), which increases opportunities for the V to A to V cycle.
The net result is a mild positive slope.
Magnitude (elasticity): normalized index ≈ +0.154 (moderate positive sensitivity).
Interpretation/policy: This is a model structure effect “disarmament” in the model includes an element that
replenishes a susceptible pool; real disarmament programs that remove weapons without increasing
susceptibility will reduce risk. In practical policy terms, disarmament must be paired with reintegration and
measures that decrease recruitment (reduce ε) to avoid unintended increases in S vulnerability.
Figure 7: Effect of R
e
on β
The curve R
e
falls sharply as β increases, β appears in the denominator as (β + μ). In this particular algebraic
arrangement β effectively reduces S₀ (since S₀ = (π + γA₀)/(β + μ)), so raising β reduces the susceptible pool in
the VFE and therefore reduces the arms to violence throughput. The negative sign is large because β sits directly
in that denominator factor.
Magnitude (elasticity): normalized index ≈ −0.403, one of the largest magnitudes (suppressive) elasticities.
Interpretation/policy: raising β reduces S₀ because of how VFE was defined (conversion out of susceptible into
V), policies that change β will have a strong impact on R
e
.
Figure 8: Effect of R
e
on δ;
Increasing δ reduces R
e
steadily. δ increases the removal term (δ + μ) that appears in the denominator factor (δ
+ μ)(γ + α + μ) + ϕσ. Larger δ shortens violent actor duration and reduces secondary production.
Magnitude (elasticity): normalized index ≈ −0.320, a strong negative sensitivity.
Interpretation/policy: Increasing δ (faster neutralization, arrests, rehabilitation) is an effective lever. Unlike μ,
δ is a realistic policy lever through security operations, targeted arrests with due process, and robust reintegration
programs.
Figure 9: Effect of R
e
on σ.
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It is monotone increasing and gently concave (sub-linear growth), σ appears in numerator via γσ and in the
denominator via φσ inside the additive term. Increasing σ raises A₀ (arms stock), raising potential conversions
and the numerator; the denominator also increases through φσ but the numerator effect dominates for the
baseline, so R
e
increases.
Magnitude (elasticity): normalized index ≈ +0.312, it has a large positive sensitivity.
Interpretation/policy: This confirms intuition, arms inflow is a powerful driver. Policies that reduce cross-
border smuggling and illicit inflows (border control, regional cooperation, arms tracing) will lower R
e
substantially.
Figure 10: Effect of R
e
on ϕ;
It is monotone decreasing, raising ϕ reduces R
e
. ϕ multiplies σ in the denominator term
(δ + μ)(γ + α + μ) + φσ) Higher ϕ increases the rate at which arms amplify removal of violent activity (or
increase conversion into R), effectively reducing reproduction. The effect here is moderate.
Magnitude (elasticity): normalized index ≈ −0.103 (it has small to moderate negative sensitivity).
Interpretation/policy: ϕ captures how effective interactions between arms stock and removal mechanisms are;
policies that increase the effectiveness of weapon seizure and removal (e.g., intelligence-led seizures, improved
interdiction) will reduce R
e
but ϕ marginal effect is smaller than σ or e.
Figure 11: Effect of R
e
on ε.
It has strongly monotone increasing, almost linear over the plotted range, increasing ε substantially increases R
e
.
Mathematical reason: ε appears multiplicatively in the numerator, so its effect on R
e
is direct and scales as the
square root of ε for the 50–150% variation this appears near linear in the plot.
Magnitude (elasticity): normalized index ≈ +0.50, the single most influential positive parameter in the baseline
sensitivity analysis.
Interpretation/policy: ε represents the rate at which violent actors interacting with susceptible lead to arms
acquisition (the contact transmission of armament). Reducing ε (through community protection, reducing contact
opportunities, targeted policing, community engagement) is one of the highest-impact short-term levers to reduce
R
e
.
Figure 12: Time series of S(t) (Susceptible population)
The plot shows five trajectories of S(t), each starting from a different initial condition (colors: blue, orange,
green, red, purple). Rapid transient behavior in the early period (first few time units) then drop or rise depending
on initial conditions. All trajectories settle to the same steady level S
*
≈ 0.4944 by about t ≈ 20 and remain there
for the remainder of the simulation.
Figure 13: Time series of V(t) (Violence actors)
The plot shows the V trajectories show an initial fast transient (either a spike or rapid increase/decrease
depending on starting V and A), followed by a slower relaxation toward V
*
≈ 1.0113. After t ≈ 50 – 100, all
curves are effectively indistinguishable and sit at the same steady value.
Figure 14: Time series of A(t)(Arms /Ammunitions)
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The plot shows A(t) exhibits early rise for some initial conditions (especially when initial A or V is large), then
decays to A
*
≈ 1.5625. Convergence is similar to S and V; transients die out and trajectories single group.
Figure 15: Time series of R(t) (Recovered Individuals)
The plot shows R(t) increases monotonically from near zero to a large steady level R
*
≈ 41.038; curves
through various initial conditions join to essentially the identical final value, but the approach is slower
(timescale from100 – 300). The large steady R is explained by the small natural exit rate μ and the fact that R
accumulates removals from V (the formula
**
*
)( VA
R
+
=
gives a large R when μ is small).
Figure 16: Phase plot ‘V’ versus ‘A’
The plot shows trajectories in the A to V plane starting from different initial points all spiral/curve into the same
small neighborhood around (A
*
, V
*
). One trajectory (the green one) shows an extended excursion from a large
initial A, moving down toward the common attractor.
CONCLUSION AND RECOMMENDATION
Conclusion
In this work, mathematical modelling and analysis of infiltration of firearms and its implications in northern
region of Nigeria was is formulated. The effective reproduction number was computed by utilizing the next
generation operation methodology. Additionally, the model’s violence free equilibrium has shown to be
asymptotically stable globally wherever the associated reproduction number appears to be less than unitary value
of one. Again, the sensitivity analysis was done on the parameters connected to the control reproduction number.
The values of α and σ obtained demonstrate the highest magnitude of sensitivity, meaning that controlling the
rate at which arms empower violence actors and the inflow of firearms yields the strongest and fastest reduction
in violence reproduction rate. Conversely, parameters with negative sensitivity indices particularly the violence-
free/removal parameter (μ), the violence de-escalation rate (δ), and the arms deactivation/recovery rate (ϕ)
reduce R
e
when increased. This means that strengthening law-enforcement removal, demobilization programs,
and arms-recovery interventions substantially contributes to the long-term destabilization of illicit networks.
In addition, the numerical simulation of the model has shown that reducing arms inflow (σ), the arms to violence
activation rate (α), and the arms-induced susceptibility rate (γ) results in a noticeable decline in both arms stock
and violent actors over time. The violence curve becomes flatter and stabilizes at a significantly lower
equilibrium, illustrating the strong impact of arms-control measures. Conversely, increasing de-escalation and
recovery rates (δ, φ) accelerates the decline of violent actors, confirming the effectiveness of reintegration
programs, community policing, and de-radicalization initiatives.
Recommendations
To effectively reduce violence and instability in Northern Nigeria, interventions must focus on controlling illicit
arms inflow, reducing arms availability in communities, disrupting arms to violence conversion mechanisms,
and strengthening removal and rehabilitation initiatives. Mathematical and numerical evidence demonstrate that
strategies targeting arms’ infiltration into Nigeria to yield the greatest reduction in long-term violence must make
arms control the most critical and impactful policy direction going forward. Additionally, Nigeria’s borders’
control should be tightened by various security agencies, including the military, police, customs, immigration,
civil defense, and others, with the application of the newest technologies.
Contributions to Knowledge
This work provides the first comprehensive mathematical framework that jointly models arms infiltration and
violence spread as an integrated dynamical system, establishes threshold and stability conditions, quantifies
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parameter sensitivities, and demonstrates through simulation the most effective strategies for reducing insecurity
in Northern Nigeria. It thereby contributes novel theoretical foundations, analytical tools, and practical policy
insights to the literature on conflict dynamics, arms control, and mathematical modeling.
Authors’ Contributions
Wadai, Mutah conceptualized the ideas, composed the topic, and also supervised the compilation of the paper’s
manuscript. Ibekwe John Jacob performed the tasks of producing the model framework, simulation, data
analysis, and interpreting the variables of the designed model, while Idongesit Nnammonso Akpan performed
the tasks of preparation, type setting, editing, and proper referencing, as well as the production of the final draft
of the paper’s manuscript.
Funding: This research was not sponsored by any internal or external institutional-based sponsors.
Data Availability
All sources of secondary data explored, collected, reviewed, analysed, and utilized in this work are duly
acknowledged. The data backing up the findings of this investigation will be made readily accessible by the
corresponding author upon realistic request.
Consent and Ethical Approval
Since all the sources of secondary data used in this investigation, which are in the public domain, have been duly
acknowledged, additional ethical approval and consent were not obtained, and therefore, there was no ethical
violation in this work.
Declarations of Conflict of Interests
The authors of this paper declare that they have no known existed competing interest during and after the
production of the paper.
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