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 point to the fact that; sole militarization of 
counter insurgency interventions is 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 counterinsurgency 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-population. 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 describes the interaction between; active, 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.