Mathematical Model for the Dynamics of Insurgents-Security Forces Interaction under the Combined Effects of Counseling and Rehabilitation

Abstract

Due to the complex underlying roots of Insurgency, it has become a global issue in 
the 21st Century, efforts to curtail insurgency (and, to a wider scale, terrorism), in 
the form of militarized counter-insurgency and counter-terrorism come with huge 
capital budgetary expenditure on the states and concerned parties. In some cases, 
the states and concerned parties have to source for loans and conditional financial 
support from foreign powers with its attendant consequences on their economic 
and socio-cultural ethics and values. Despite these efforts, insurgency and 
terrorism have not been tamed appropriately; in fact, the menace is none 
decreasing, particularly in Africa. This points to the fact that the sole militarization of 
Counter-insurgency interventions are not enough to tame insurgency. In this study, 
We look into the effect(s) of rehabilitation through counseling on insurgents. The 
population of insurgents was decoupled into inactive and active insurgents because 
of the collaborative interactions between the two sub-populations to develop a 
mathematical model for the dynamics of interactions between security-forces and 
insurgents under the influence of rehabilitation through counseling as counter
insurgency approaches. In developing the model, the interaction between the 
security-forces and the insurgents was assumed to be prey-predator like, in which 
the former is the predator and the latter is the prey, or vice-versa, and assumed 
logistic growth capacity for each of the decoupled insurgent sub-populations. The 
security-forces deployment was assumed to be a function of the insurgents’ 
activities.  On the basis of this, we proposed a system of three non-liner ordinary 
differential equations that describe the interaction between active and non-active 
insurgents and security forces. We carried out a qualitative analysis of the model, 
where we established the equilibrium point solution, positivity and invariant region 
for the solution for the model. We computed five equilibria for the model, namely: 
trivial, endemic insurgents, inactive-insurgents-free, active-insurgents-free, 
security-forces-free equilibrium points. The trivial and security-forces-free 
equilibrium were found to be locally asymptotically unstable. The inactive
insurgents-free and the active-insurgents-free equilibria were proved to be 
conditionally locally asymptotically stable.