Abstract
Crime and corruptions are two major problems in developing nations. They spread like infectious disease in the population. Here, a nonlinear mathematical model is formulated and analyzed to study the interaction between criminal population and non-criminal population by considering non-monotone incidence rate. Non-monotone incidence rate shows a behavioral change in non-criminal populations. First, we consider constant recruitment and death type demography and then we take logistic growth type demography. For both the models the basic reproduction number \(R_0\) is computed. All possible equilibria of the models are obtained and stabilities of these equilibria are discussed in details. Further these models are extended to optimal control problem to study the impact of time-dependent law enforcement rate on the size of criminal population. Numerical simulations are presented to illustrate our analytical findings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbas S, Tripathi JP, Neha AA (2017) Dynamical analysis of a model of social behavior: criminal vs non-criminal population. Chaos Solitons Fract 98:121–129
Akanni JO, Akinpelu FO, Olaniyi S, Oladipo AT, Ogunsola AW (2019) Modelling financial crime population dynamics: optimal control and cost-effectiveness analysis. Int J Dyn Control. https://doi.org/10.1007/s40435-019-00572-3
Athithan S, Ghosh M, Li X-Z (2018) Mathematical modeling and optimal control of corruption dynamics. Asian-Eur J Math. https://doi.org/10.1142/S1793557118500900
Becker G (1968) Crime and punishment: an economic approach. J Polit Econ 76:169–217
Castillo-Chavez C, Song B (2004) Dynamical model of tuberculosis and their applications. Math Biosci Eng 1:361–404
Castillo-Chavez C, Feng Z, Huang W (2002) On the computation of R0 and its role on global stability. Springer, New York, pp 229–250
Chelmis C, Srivastava A, Prasanna VK (2014) Computational models of technology adoption at the workplace. Soc Netw Anal Min 4:199. https://doi.org/10.1007/s13278-014-0199-z
Clinard MB, Meier RF (2010) Sociology of deviant behaviour. Wadsworth Publishing, Belmont
Comissiong DMG, Sooknanan J (2018) A review of the use of optimal control in social models. Int J Dyn Control 6:1841–1846. https://doi.org/10.1007/s40435-018-0405-3
Dhar J, Jain A, Gupta VK (2016) A mathematical model of news propagation on online social network and a control strategy for rumor spreading. Soc Netw Anal Min 6:57. https://doi.org/10.1007/s13278-016-0366
Driessche PV, Watmough J (2002) Reproduction numbers and sub threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48
Escalante R, Odehnal MA (2020) A deterministic mathematical model for the spread of two rumors. Afr Mat 31:315–331. https://doi.org/10.1007/s13370-019-00726-8
Fernandez A (2005) Crime prevention. Wilan Publishing, New York
Gao LQ, Hethcote HW (1992) Disease transmission models with density-dependent demographics. J Math Biol 30:717–731
González-Parra G, Chen-Charpentier B, Kojouharov HV (2018) Mathematical modeling of crime as a social epidemic. J Interdiscip Math 21(3):623–643
Gordon MB (2010) A random walk in the literature on criminality: a partial and critical view on some statistical analyses and modelling approaches. Eur J Appl Math 21:283–306
Gordon MB, Iglesias JR, Semeshenko V, Nadal J-P (2009) Crime and punishment: the economic burden of impunity. Eur Phys J B 68(1):133–44
Harries K (2006) Property crimes and violence in united states: an analysis of the influence of population density. Int J Crim Justice Sci 1(2):24–34
Iglesias JR, Semeshenko V, Schneider EM, Gordon MB (2012) Crime and punishment: does it pay to punish? Phys A 391(15):3942–50
Lakshmikantham V, Leela S, Martynyuk AA (1989) Stability analysis of nonlinear systems. Marcel Dekker Inc., New York
Lenhart S, Workman JT (2007) Optimal control applied to biological models. CRC Press, Boca Raton
Levitt S (1996) The effect of prison population size on crime rates: evidence from overcrowding litigation. Q J Econ 111(2):319–51
MATLAB (2018) R2018a, version MATLAB 9.4. The MathWorks Inc., Natick
McMillon D, Simon CP, Morenoff J (2014) Modeling the underlying dynamics of the spread of crime. PLoS ONE 9(4):e88923. https://doi.org/10.1371/journal.pone.0088923
Mushayabasa S (2017) Modeling optimal intervention strategies for property crime. Int J Dyn Control 2017(5):832–841. https://doi.org/10.1007/s40435-015-0201-2
Mussumeci E, Coelho FC (2018) Reconstructing news spread networks and studying its dynamics. Soc Netw Anal Min 8:6
Nadal J-P, Gordon MB, Iglesias JR, Semeshenko V (2010) Modelling the individual and collective dynamics of the propensity to offend. Eur J Appl Math 21(4–5):421–40
Nyabadza F, Ogbogbo CP, Mushanyu J (2017) Modelling the role of correctional services on gangs: insights through a mathematical model. R Soc Open Sci 4:170511
Phan D, Gordon MB, Nadal JP (2004) Social interactions in economic theory: an insight from statistical mechanics. In: Bourgine P, Nadal JP (eds) Cognitive economics. Springer, Berlin, Heidelberg, pp 335–358
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Interscience Publishers, Geneva
Smith RJ, Blower SM (2004) Could disease modifying HIV vaccine cause population-level pervasity? Lancet Infect Dis 4:636–639
Acknowledgements
The authors thank the handling editor and anonymous referees for their valuable comments and suggestions that led to an improvement of our original manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Srivastav, A.K., Athithan, S. & Ghosh, M. Modeling and analysis of crime prediction and prevention. Soc. Netw. Anal. Min. 10, 26 (2020). https://doi.org/10.1007/s13278-020-00637-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13278-020-00637-8