Skip to main content
SpringerLink
Log in

Disease control through provision of alternative food to predator: a model based study

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This paper focuses on the study of a mathematical predator-prey epidemic model supplying alternative food to predator. The main objective is to study the consequences of providing alternative food on the epidemic system. Different basic reproduction numbers are determined and the impacts of alternative food in reducing infected individuals are discussed. The dynamical behaviour of the system is investigated from the point of view of stability and persistence both analytically and numerically. To achieve the control of disease, an optimal control problem is formulated and solved. Numerical results illustrate that there exists a critical infection rate above which disease free system can not be reached in absence of alternative food, but suitable alternative food leads the system disease free with higher infection rate. The disease free system will be also obtained in presence of seasonally varying contact rate. This study is aimed to introduce a new non-chemical method for controlling disease in a predator-prey system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Kermack WO, McKendrick AG (1927) A contributions to the mathematical theory of epidemics. Proc R Soc Edinb Sect A Math 115:700–721

    Article  MATH  Google Scholar 

  2. Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics, ii—the problem of endemicity. Proc R Soc Edinb Sect A Math 138:55–83

    Article  MATH  Google Scholar 

  3. Kermack WO, McKendrick AG (1933) Contributions to the mathematical theory of epidemics, iii—further studies of the problem of endemicity. Proc R Soc Edinb Sect A Math 141:94–122

    Article  MATH  Google Scholar 

  4. Anderson RM, May RM (1986) The invasion, persistence and spread of infectious diseases within animal and plant communities. Philos Trans R Soc Lond B 314:533–570

    Article  Google Scholar 

  5. Hadeler KP, Freedman HI (1989) Predator-prey populations with parasite infection. J Math Biol 27:609–631

    Article  MathSciNet  MATH  Google Scholar 

  6. Haque M, Chattopadhyay J (2007) Role of transmissible disease in an infected prey-dependent predator-prey system. Math Comput Model Dyn Syst 13:163–178

    Article  MathSciNet  MATH  Google Scholar 

  7. Haque M, Venturino E (2006) The role of transmissible diseases in the Holling–Tanner predator-prey model. Theor Popul Biol 70:273–288

    Article  MATH  Google Scholar 

  8. Haque M, Venturino E (2007) An ecoepidemiological model with disease in predator: the ratio-dependent case. Math Methods Appl Sci 30:1791–1809

    Article  MathSciNet  MATH  Google Scholar 

  9. Haque M, Greenhalgh D (2010) When a predator avoids infected prey: a model-based theoretical study. Math Med Biol 27:75–94

    Article  MathSciNet  MATH  Google Scholar 

  10. Sahoo B, Poria S (2013) Disease control in a food chain model supplying alternative food. Appl Math Model 37:5653–5663

    Article  MathSciNet  MATH  Google Scholar 

  11. London WP, Yorke JA (1973) Recurrent outbreaks of measles, chickenpox and mumps. 1. Seasonal variation in contact rates. Am J Epidemiol 98:453–468

    Google Scholar 

  12. Yorke JA, Nathanson N, Pianigiani G, Martin J (1979) Seasonality and the requirements for perpetuation and eradication of viruses in populations. Am J Epidemiol 109:103–123

  13. Buonomo B (2011) On the optimal vaccination strategies for horizontally and vertically transmitted infectious diseases. J Biol Syst 19:263–279

    Article  MathSciNet  MATH  Google Scholar 

  14. Das KP, Chatterjee S, Chattopadhyay J (2009) Disease in prey population and body size of intermediate predator reduce the prevalence of chaos-conclusion drawn from Hastings–Powell model. Ecol Complex 6:363–374

    Article  Google Scholar 

  15. van Rijn PCJ, van Houten YM, Sabelis MW (2002) How plants benefit from providing food to predators even when it is also edible to herbivores. Ecology 83:2664–2679

    Article  Google Scholar 

  16. Srinivasua PDN, Prasada BSRV, Venkatesulub M (2007) Biological control through provision of additional food to predators: a theoretical study. Theor Popul Biol 72:111–120

    Article  Google Scholar 

  17. Srinivasu PDN, Prasad BSRV (2010) Role of quantity of additional food to predators as a control in predator-prey systems with relevance to pest management and biological conservation. Bull Math Biol. doi:10.1007/s11538-010-9601-9

  18. Haque M (2011) A detailed study of the Beddington–DeAngelis predator-prey model. Math Biosci 234:1–16

    Article  MathSciNet  MATH  Google Scholar 

  19. Haque M (2009) Ratio-dependent predator-prey models of interacting populations. Bull Math Biol 71:430–452

    Article  MathSciNet  MATH  Google Scholar 

  20. Hsu SB, Huang TW (1995) Global stability for a class of predator-prey systems. SIAM J Appl Math 55:763–783

    Article  MathSciNet  MATH  Google Scholar 

  21. Freedman HI, Waltman P (1984) Persistence in models of three interacting predator-prey populations. Math Biosci 68:213–231

    Article  MathSciNet  MATH  Google Scholar 

  22. Kumar R, Freedman H (1989) A mathematical model of facultative mutualism with populations interacting in a food chain. Math Biosci 97:235–261

    Article  MathSciNet  MATH  Google Scholar 

  23. Butler GJ, Freedman H, Walteman P (1986) Uniformly persistent systems. Proc Am Math Soc 96:425–429

    Article  MathSciNet  MATH  Google Scholar 

  24. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New York

    Google Scholar 

  25. Clark CW (1990) Mathematical bioeconomics: the optimal management of renewable resources. Wiley, New York

    MATH  Google Scholar 

  26. Lenhart S, Workman JT (2007) Optimal control applied to biological models. Mathematical and computational biology series. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  27. Joshi HR (2002) Optimal control of an HIV immunology model. Optim Control Appl Methods 23:199–213

    Article  MathSciNet  MATH  Google Scholar 

  28. Birkoff G, Rota GC (1982) Ordinary differential equations. Ginn, Boston

    Google Scholar 

  29. Zaman G, Kang YH, Jung IH (2008) Stability analysis and optimal vaccination of an SIR epidemic model. Biosystems 93:240–249

    Article  Google Scholar 

  30. Taylor N, Walters C (2010) Estimation of bioenergetics parameters for a stunted northern Pikeminnow population of south central British Columbia. Open Fish Sci J 3:110–121

    Article  Google Scholar 

  31. Steele JH, Henderson EW (1992) The role of predation in plankton models. J Plankton Res 14:157–172

    Article  Google Scholar 

  32. Hastings A, Powell T (1991) Chaos in a three-species food chain. Ecology 72:896–903

    Article  Google Scholar 

  33. Butzel HM, Bolten AB (1968) The relationship of the nutritive state of the prey organism Paramecium aurelia to the growth and encystment of Didinium nasutum. J Protozool 15:256–258

    Article  Google Scholar 

  34. Cheney DL et al (2004) Factors affecting reproduction and mortality among baboons in the Okavango Delta, Botswana. Int J Primatol 25:401–428

  35. Stone L, Olinky R, Huppert A (2007) Seasonal dynamics of recurrent epidemics. Nature 446:533–536

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Calcutta, Kolkata, West Bengal, India

    Banshidhar Sahoo

Authors
  1. Banshidhar Sahoo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Banshidhar Sahoo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sahoo, B. Disease control through provision of alternative food to predator: a model based study. Int. J. Dynam. Control 4, 239–253 (2016). https://doi.org/10.1007/s40435-014-0099-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-014-0099-0

Keywords

Search

Navigation