Modelling financial crime population dynamics: optimal control and cost-effectiveness analysis

  • J. O. Akanni
  • F. O. Akinpelu
  • S. OlaniyiEmail author
  • A. T. Oladipo
  • A. W. Ogunsola


This work is designed to formulate and analyse a mathematical model for population dynamics of financial crime with optimal control measures. Necessary conditions for the existence and stability of financial crime steady states are derived. The financial crime reproduction number is determined. Based on construction of suitable Lyapunov functionals, crime-free equilibrium point of the formulated model is shown to be globally asymptotically stable when the crime reproduction number is below unity, while a unique crime-present equilibrium is proved to be globally asymptotically stable whenever the crime reproduction number exceeds unity. Sensitivity analysis is carried out to determine the relative importance of model parameters in financial crime spread. Furthermore, optimal control theory is employed to assess the impact of two time-dependent optimal control strategies, including public enlightenment campaign (preventive) and corrective measure, on the financial crime dynamics in a population. The cost-effectiveness analysis is carried out to determine the least costly and most effective strategy of the singular and combined implementations of the intervention strategies, when the available resources to combat the spread of financial crime are limited.


Financial crime model Crime reproduction number Lyapunov functionals Sensitivity analysis Optimal control measures Cost-effective intervention 



The authors express thanks to the editor and anonymous reviewers whose insightful suggestions enhanced the original manuscript.


  1. 1.
    Jung J, Lee J (2017) Contemporary financial crime. J Public Adm Gov 7(2):88–97. Google Scholar
  2. 2.
    Gottschalk P (2010) Categories of financial crime. J Financ Crime 17(4):441–4587. CrossRefGoogle Scholar
  3. 3.
    Pickett KS, Pickett JM (2002) Financial crime investigation and control. Wiley, New YorkzbMATHGoogle Scholar
  4. 4.
    Thieme HR (2003) Mathematics in population biology. Princeton University Press, PrincetonCrossRefzbMATHGoogle Scholar
  5. 5.
    Zhao H, Feng Z, Castillo-Chávez C (2014) The dynamics of poverty and crime. J Shangai Normal Univ 43(5):486–495Google Scholar
  6. 6.
    Nuño JC, Herrero MA, Primicero M (2008) A triangle model of criminality. Physica A 387:2926–2936CrossRefGoogle Scholar
  7. 7.
    Nuño JC, Herrero MA, Primicero M (2010) Fighting cheaters: how and how much to invest. Eur J Appl Math 21:459–478. MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Shukla JB, Goyal A, Agrawal K et al (2013) Role of technology in combating social crimes: a modeling study. Eur J Appl Math 24:501–514. CrossRefzbMATHGoogle Scholar
  9. 9.
    Livni J, Stone L (2015) The stabilizing role of the Sabbath in pre-monarchic Israel: a mathematical model. J Biol Phys 41:203–221. CrossRefGoogle Scholar
  10. 10.
    González-Parra G, Chen-Charpentier B, Kojouharov HV (2018) Mathematical modeling of crime as a social epidemic. J Interdiscip Math 21(3):623–643. CrossRefGoogle Scholar
  11. 11.
    Athithan S, Ghosh M, Li X-Z (2018) Mathematical modeling and optimal control of corruption dynamics. Asian Eur J Math 11(6):1–12. MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chaharborj FS, Pourghahramani B, Chaharborj SS (2017) A dynamic economic model of criminal activity in the criminal law. Int J Basic Appl Sci 6(4):73–76CrossRefGoogle Scholar
  13. 13.
    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. MathSciNetCrossRefGoogle Scholar
  14. 14.
    Roslan UAM, Zakaria S, Alias A, Malik SM (2018) A mathematical model on the dynamics of poverty, poor and crime in west malaysia. Far East J Math Sci 107(2):309–319. Google Scholar
  15. 15.
    Sooknanan J, Bhatt B, Comissiong DMG (2016) A modified predator–prey model for the interaction of police and gangs. R Soc Open Sci 3:160083. MathSciNetCrossRefGoogle Scholar
  16. 16.
    Korobeinikov A (2004) Lyapunov functions and global properties for SEIS epidemic models. Math Med Biol 21:75–83CrossRefzbMATHGoogle Scholar
  17. 17.
    van den Driessche P, Wang L, Zou X (2007) Modeling diseases with latency and relapse. Math Biosci Eng 4(2):205–219. MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599–653MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    LaSalle JP (1976) The stability of dynamical systems. In: Regional conference series in applied mathematics. SIAM, Philadelphia, PaGoogle Scholar
  21. 21.
    Okuonghae D, Gumel AB, Ikhimwin BO, Iboi E (2018) Mathematical assessment of the role of early latent infections and targeted control strategies on syphilis transmission dynamics. Acta Biotheor. Google Scholar
  22. 22.
    Obabiyi OS, Olaniyi S (2019) Global stability analysis of malaria transmission dynamics with vigilant compartment. Electron J Differ Equ 2019(09):1–10MathSciNetzbMATHGoogle Scholar
  23. 23.
    Karamzadeh OAS (2011) One-line proof of the AM-GM inequality. Math Intell 33(2):3. CrossRefzbMATHGoogle Scholar
  24. 24.
    Chitnis N, Hyman JM, Cushing JM (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull Math Biol 70(5):1272–1296. MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Olaniyi S, Obabiyi OS (2014) Qualitative analysis of malaria dynamics with nonlinear incidence function. Appl Math Sci 8(74):3889–3904. Google Scholar
  26. 26.
    Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkzbMATHGoogle Scholar
  27. 27.
    Gaoue OG, Jiang J, Ding W, Agusto FB, Lenhart S (2016) Optimal harvesting strategies for timber and non-timber forest products in tropical ecosystems. Theor Ecol. Google Scholar
  28. 28.
    Hugo A, Makinde OD, Kumar S, Chibwana FF (2017) Optimal control and cost effectiveness analysis for Newcastle disease eco-epidemiological model in Tanzania. J Biol Dyn 11(1):190–209. MathSciNetCrossRefGoogle Scholar
  29. 29.
    Olaniyi S (2018) Dynamics of Zika virus model with nonlinear incidence and optimal control strategies. Appl Math Inf Sci 12(5):969–982. CrossRefGoogle Scholar
  30. 30.
    Olaniyi S, Okosun KO, Adesanyan SO, Areo EA (2018) Global stability and optimal control analysis of malaria dynamics in the presence of human travelers. Open Infect Dis 10:166–186. CrossRefGoogle Scholar
  31. 31.
    Bonyah E, Khan MA, Okosun KO, Gómez-Aguilar JF (2019) On the co-infection of dengue fever and Zika virus. Optim Control Appl Meth. MathSciNetGoogle Scholar
  32. 32.
    Fleming WH, Richel RW (1975) Deterministic and stochastic optimal control. Springer, New YorkCrossRefGoogle Scholar
  33. 33.
    Magombedze G, Chiyaka C, Mukandavire Z (2011) Optimal control of malaria chemotherapy. Nonlinear Anal Model Control 16(4):415–434MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman & Hall, LondonzbMATHGoogle Scholar
  35. 35.
    Alhassan A, Momoh AA, Abdullahi AS, Kadzai MTY (2017) Optimal control strategies and cost effectiveness analysis of a malaria transmission model. Math Theory Model 7(6):123–138Google Scholar
  36. 36.
    Berhe HW, Makinde OD, Theuri DM (2018) Optimal control and cost-effectiveness analysis for dysentery epidemic model. Appl Math Inf Sci 12(6):1183–1195. MathSciNetCrossRefGoogle Scholar
  37. 37.
    Oke SI, Matadi MB, Xulu SS (2018) Cost-effectiveness analysis of optimal control strategies for breast cancer treatment with ketogenic diet. Far East J Math Sci 109(2):303–342. Google Scholar
  38. 38.
    Okosun KO, Rachid O, Marcus N (2013) Optimal control strategies and cost-effectiveness analysis of a malaria model. BioSystems 111:83–101. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physical Sciences (Mathematics)Precious Cornerstone UniversityIbadanNigeria
  2. 2.Department of Pure and Applied MathematicsLadoke Akintola University of TechnologyOgbomosoNigeria
  3. 3.Department of MathematicsUniversity of LagosLagosNigeria

Personalised recommendations