Honam Mathematical Journal (호남수학학술지)

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DYNAMICS OF GUN VIOLENCE BY LEGAL AND ILLEGAL FIREARMS: A FRACTIONAL DERIVATIVE APPROACH

  • Chandrali, Baishya (Department of Studies and Research in Mathematics, Tumkur University) ;
  • P., Veeresha (Department of Mathematics, Center for Mathematical Needs, CHRIST (Deemed to be University))
  • Received : 2022.05.23
  • Accepted : 2022.08.16
  • Published : 2022.12.25
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Abstract

Crime committed by civilians and criminals using legal and illegal firearms and conversion of legal firearms into illegal ones has become a common practice around the world. As a result, policies to control civilian gun ownership have been debated in several countries. The issue arose because the linkages between firearm-related mortality, weapon accessibility, and violent crime data can imply diverse options for addressing criminality. In this paper, we have projected a mathematical model in terms of the Caputo fractional derivative to address the issues viz. input of legal guns, crime committed by legal and illegal guns, and strict government policies to monitor the license of legal guns, strict action against violent crime. The boundedness, existence and uniqueness of solutions and the stability of points of equilibrium are examined. It is observed that violent crime increases with the increase of crime committed by illegal guns, crime committed by legal guns and, decreases with the increase of legal guns, the deterrent effect of civilian gun ownership, and action of law against crime. Further, legal guns increase with the increase of the limitation of trade of illegal guns and decrease with the increase of conversion of legal guns into illegal guns and increase of the growth rate of illegal guns. Again, as crime is committed by legal guns also, the policy of illegal gun control does not assure a crime-free society. Weak gun control can lead to a society with less crime. Theoretical aspects are numerically verified in the present work.

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References

  1. S. J. Achar, C. Baishya, and M. K. A. Kaabar, Dynamics of the worm transmission in wireless sensor network in the framework of fractional derivatives, Math. Meth. Appl. Sci. (2021), 1-17.
  2. E. Ahmed and A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Physica A, 379 (2007), no. 2, 607-614. https://doi.org/10.1016/j.physa.2007.01.010
  3. A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Therm. Sci. 20 (2016), 763-769. https://doi.org/10.2298/tsci160111018a
  4. C. Baishya, S. J. Achar, P. Veeresha, and D. G. Prakasha, Dynamics of a fractional epidemiological model with disease infection in both the populations. Chaos 31 (2021), no. 4, 043130.
  5. C. Baishya, Dynamics of Fractional Holling Type-II Predator-Prey Model with Prey Refuge and Additional Food to Predator, J. Appl. Nonlinear Dyn. 10 (2021), no. 2, 315-328. https://doi.org/10.5890/JAND.2021.06.010
  6. C. Baishya and P. Veeresha, Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel, Proc. A 477 (2021), no. 2253, 20210438.
  7. C. Baishya, An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative, SeMA J. 79 (2022), 699-717. https://doi.org/10.1007/s40324-021-00268-9
  8. K. Bansal, S. Arora, K. S. Pritam, T. Mathur. and S. Agarwal. Dynamics of Crime Transmission Using Fractional-Order Differential Equations, Fractals 30 (2022), no. 1, 2250012.
  9. B. Beisner, D. Haydon, and K. Cuddington, Hysteresis, Encyclopedia of Ecology, 1930-1935, 2008.
  10. R. Cantrell, C. Cosner, and R. Manasevich, Global bifurcation of solutions for crime modeling equations. SIAM J. Math. Anal. 44 (2012), no. 3, 1340-1358. https://doi.org/10.1137/110843356
  11. M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International 13 (1967), no. 5, 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  12. M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015), 73-85.
  13. K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1997), no. 1, 1-6.
  14. K. Diethelm, N. J. Ford, and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002), no. 1, 3-22. https://doi.org/10.1023/A:1016592219341
  15. G. Gonzalez-Parra, B. Chen-Charpentier, and H.V. Kojouharov, Mathematical modeling of crime as a social epidemic, J. Interdiscip. Math. 21 (2018), 623-643. https://doi.org/10.1080/09720502.2015.1132574
  16. M. R. D'Orsogna and M. Perc, Statistical physics of crime: a review, Phys. Life Rev. 12 (2015), 1-21. https://doi.org/10.1016/j.plrev.2014.11.001
  17. A. Goyal, J. Shukla, A. Misra, and A. Shukla, Modeling the role of government efforts in controlling extremism in a society, Math. Methods Appl. Sci. (2014), 3368.
  18. M. Hellenbach, S. Elliott, F. J. Gerard, B. Crookes, T. Stamos, H. Poole, and E. Bowen, The detection and policing of gun crime: challenges to the effective policing of gun crime in europe, Eur. J. Criminol. 15 (2018), 172-196. https://doi.org/10.1177/1477370816686122
  19. D. Hemenway and M. Miller, Firearm availability and homicide rates across 26 highincome countries, J. Trauma 49 (2000), 985-988. https://doi.org/10.1097/00005373-200012000-00001
  20. https://worldpopulationreview.com/country-rankings/gun-deaths-by-country.
  21. A. A. Kilbas, H. M. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland Mathematics Studies, 2006.
  22. M. Konty and B. Schaefer, Small arms mortality: access to firearms and lethal violence, Sociol. Spectrum 32 (2012), 475-490. https://doi.org/10.1080/02732173.2012.700832
  23. A. Lankford, Public mass shooters and firearms: a cross-national study of 171 countries, Violence Vict. 31 (2016), 187-199. https://doi.org/10.1891/0886-6708.vv-d-15-00093
  24. L. K. Lee, E. W. Fleegler, C. Farrell, E. Avakame, S. Srinivasan, D. Hemenway, and M. C. Monuteaux, Firearm laws and firearm homicides: a systematic review, JAMA Intern. Med. 177 (2017), 106-119.
  25. H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, and Z. Teng, Dynamical analysis of a fractionalorder predator-prey model incorporating a prey refuge, J. Appl. Math. Comput. 54 (2017), no. 1-2, 435-449. https://doi.org/10.1007/s12190-016-1017-8
  26. R. Manasevich, Q. H. Phan, and P. Souplet, Global existence of solutions for a chemotaxis-type system arising in crime modelling, Eur J Appl Math. 24 (2013), 273-296. https://doi.org/10.1017/S095679251200040X
  27. D. McMillon, C. P. Simon, and J. Morenoff, Modeling the underlying dynamics of the spread of crime, PLOS ONE 9 (2014), e88923.
  28. A. Misra. Modeling the effect of police deterrence on the prevalence of crime in the society, Appl. Math. Comput. 237 (2014), 531-545. https://doi.org/10.1016/j.amc.2014.03.136
  29. L. H. A. Monteiro, More guns, less crime? A dynamical systems approach, Appl. Math. Comput. 369 (2019), 124804.
  30. M. Naghavi, L. B. Marczak, M. Kutz, K.A. Shackelford, and M. Arora, Global mortality from firearms, JAMA J. Am. Med. Assoc. 320 (2018), 1990-2016.
  31. F. Nyabadza, C. P. Ogbogbo, and J. Mushanyu, Modelling the role of correctional services on gangs: insights through a mathematical model, Roy. Soc. Open Sci. 4 (2017), 170511.
  32. C. Oliveiraand and G. B. Neto, The deterrence effects of gun laws in games with asymmetric skills and information, Rev. L. Econ. 11 (2015), 435-452.
  33. M. Perc, K. Donnay, and D. Helbing, Understanding recurrent crime as systemimmanent collective behavior, PLOS One 8 (2013), e76063.
  34. A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. OCallaghan, A. V. Pokrovskii, and D. Rachinskii, Memory effects in population dynamics, Spread of infectious disease as a case study 7 (2012), 204-226.
  35. A. B. Pitcher, Adding police to a mathematical model of burglary, Eur. J. Appl. Math. 21 (2010), no. 4-5, 401-419. https://doi.org/10.1017/S0956792510000112
  36. I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
  37. K. S. Pritam, Sugandha, T. Mathur, and S. Agarwal, Underlying dynamics of crime transmission with memory, Chaos, Solitons and Fractals 146 (2021), 110838.
  38. S. Resnick, R. N. Smith, J. H. Beard, D. Holena, P. M. Reilly, C. W. Schwab, and M. J. Seamon, Firearm deaths in america: can we learn from 462,000 lives lost?, Ann. Surg. 266 (2017), 432-440. https://doi.org/10.1097/SLA.0000000000002376
  39. A. P. Ribeiro, E. R. de Souza, and C. A. M. de Sousa, Injuries caused by firearms treated at brazilian urgent and emergency healthcare services, Cien. Saude Colet. 22 (2017), 2851-2860. https://doi.org/10.1590/1413-81232017229.16492017
  40. N. Rodriguez and A. Bertozzi, Local existence and uniqueness of solutions to a pde model for criminal behavior, Math Models Methods Appl Sci 20(supp01) (2010), 1425-1457,. https://doi.org/10.1142/S0218202510004696
  41. B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, in Fractional Calculus and Its Applications, Proceedings of the International Conference Held at the University of New Haven, 1-36, Springer, 1975.
  42. M. Short, A. Bertozzi, and P. Brantingham. Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst. 9 (2010), no. 2, 462-483. https://doi.org/10.1137/090759069
  43. M. B. Short, P. J. Brantingham, and M.R. D'Orsogna, Cooperation and punishment in an adversarial game: how defectors pave the way to a peaceful society, Phys. Rev. E 82 (2010), 066114.
  44. W. Stroebe, Firearm availability and violent death: the need for a culture change in attitudes toward guns, Anal. Soc. Issues Public Policy 16 (2016), 7-35. https://doi.org/10.1111/asap.12100
  45. B. Wang and L. Q. Chen, Asymptotic stability analysis with numerical confirmation of an axially accelerating beam constituted by the standard linear solid model, Journal of Sound and Vibration 328 (2009), no. 4-5, 456-466. https://doi.org/10.1016/j.jsv.2009.08.016
  46. D. Wodarz and N. L. Komarova, Dependence of the firearm-related homicide rate on gun availability: a mathematical analysis, PLOS One 8 (2013), e71606.
  47. T. K. Yamana, X. Qiu, and E. A. B. Eltahir, Hysteresis in simulations of malaria transmission, Advances in Water Resources 108 (2017), 416-422.  https://doi.org/10.1016/j.advwatres.2016.10.003