Skip to main content
SpringerLink
Log in

Study of anthrax disease dynamics in multi-compartment with Grass and herbivores population

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

The seasonal temperature variations strongly affect anthrax disease in the herbivores population in wildlife. This article presents a grass-herbivores model incorporating anthrax disease in herbivores taking grass growth rate, grass death rate, and anthrax transmission rate among herbivores temperature-dependent parameters. The positivity, boundedness, and existence of solutions are derived. For the autonomous model, the stability of disease-free equilibrium is established for \(R_C<1\). Further, the stability of the coexistence state has been proven under conditions. The analysis of the non-autonomous model reveals the local stability of the periodic disease-free state for periodic basic reproduction number \(R_C(t)<1\) with the help of the monodromy matrix. Again, the global stability of periodic infection-free state has been shown \(R_C(t)<1\) with the help of comparison theory and the theory of periodic semi-flow. The existence and global attractivity of the periodic coexistence state have been shown at \(R_C(t)>1\) via the Poincare map and comparison theory. The significance of the parameters related to disease transmission and prevalence is described using sensitivity analysis. Numerical simulation proved that we could control anthrax disease by decreasing the anthrax transmission rate among predators, healthy herbivores’ death rate, and increasing the healthy grass predation rate. Also, an increase in temperature increases the periodic basic reproduction number, revealing that a temperature rise increases the chances of anthrax infection in herbivores.

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
Fig. 9

Similar content being viewed by others

References

  1. Goel, A.K.: Anthrax: a disease of biowarfare and public health importance. World J. Clin. Cases WJCC 3(1), 20 (2015)

    Google Scholar 

  2. Clegg, S., Kornberger, M., Rhodes, C.: Business ethics as practice. Br. J. Manag. 18(2), 107–122 (2007)

    Google Scholar 

  3. Mongoh, M.N., Dyer, N.W., Stoltenow, C.L., Khaitsa, M.L.: Risk factors associated with anthrax outbreak in animals in north dakota, 2005: a retrospective case-control study. Public Health Rep. 123(3), 352–359 (2008)

    Google Scholar 

  4. Miao, C., Chakraborty, M., Chen, S.: Impact of reaction conditions on the simultaneous production of polysaccharides and bio-oil from heterotrophically grown chlorella sorokiniana by a unique sequential hydrothermal liquefaction process. Bioresource Technol. 110, 617–627 (2012)

    Google Scholar 

  5. Charbon, M., Milzbrand, S.F.: Species affected

  6. Rezapour, S., Etemad, S., Mohammadi, H.: A mathematical analysis of a system of caputo-fabrizio fractional differential equations for the anthrax disease model in animals. Adv. Diff. Equ. 2020(1), 1–30 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Turnbull, P.C.B.: Anthrax in humans and animals. World Health Organization, Geneva, Switzerland (2008)

    Google Scholar 

  8. Walsh, M.G., de Smalen, A.W., Mor, S.M.: Climatic influence on anthrax suitability in warming northern latitudes. Sci. Rep. 8(1), 1–9 (2018)

    Google Scholar 

  9. Saad-Roy, C., Van den Driessche, P., Yakubu, A.-A.: A mathematical model of anthrax transmission in animal populations. Bulletin Math. Biol. 79(2), 303–324 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Turner, W.C., Imologhome, P., Havarua, Z., Kaaya, G.P., Mfune, J.K., Mpofu, I.D., Getz, W.M.: Soil ingestion, nutrition and the seasonality of anthrax in herbivores of Etosha national park. Ecosphere 4(1), 1–19 (2013)

    Google Scholar 

  11. Lembo, T., Hampson, K., Auty, H., Beesley, C.A., Bessell, P., Packer, C., Halliday, J., Fyumagwa, R., Hoare, R., Ernest, E., et al.: Serologic surveillance of anthrax in the serengeti ecosystem, Tanzania, 1996–2009. Emerg. Infect. Dis. 17(3), 387 (2011)

    Google Scholar 

  12. De Vos, V., Bryden, H.: Anthrax in the Kruger national park: temporal and spatial patterns of disease occurrence. Salisbury Med. Bull. 87(special suppl), 26–30 (1996)

    Google Scholar 

  13. Raza, A., Baleanu, D., Yousaf, M., Akhter, N., Mahmood, S.K., Rafiq, M.: Modeling of anthrax disease via efficient computing techniques

  14. Logan, N.A., Berge, O., Bishop, A., Busse, H.-J., De Vos, P., Fritze, D., Heyndrickx, M., Kämpfer, P., Rabinovitch, L., Salkinoja-Salonen, M., et al.: Proposed minimal standards for describing new taxa of Aerobic, endospore-forming bacteria. Int. J. Syst. Evol. Microbiol. 59(8), 2114–2121 (2009)

    Google Scholar 

  15. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-verlag, New york (1983)

    MATH  Google Scholar 

  16. Li, X., Ren, J., Campbell, S.A., Wolkowicz, G.S., Zhu, H.: How seasonal forcing influences the complexity of a predator-prey system. Discret Contin. Dyn. Syst. B 23(2), 785 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Niu, X., Zhang, T., Teng, Z.: The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey. Appl. Math. Modell. 35(1), 457–470 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Lu, Y., Wang, X., Liu, S.: A non-autonomous predator-prey model with infected prey. Discrete Contin. Dyn. Syst. B 23(9), 3817 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Misra, O., Dhar, J., Sisodiya, O.: Dynamical study of Svirb epidemic model for water-borne disease with seasonal variability. Dyn. Contin. Discrete Impulsive Syst. Ser. A Math. Anal. 27, 351–374 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Sahoo, B., Poria, S.: Dynamics of a predator-prey system with seasonal effects on additional food. Int. J. Ecosyst. 1(1), 10–13 (2011)

    Google Scholar 

  21. Roy, P., Upadhyay, R.K.: Assessment of rabbit hemorrhagic disease in controlling the population of red fox: a measure to preserve endangered species in Australia. Ecol. Complexi. 26, 6–20 (2016)

    Google Scholar 

  22. Hethcote, H.W., Wang, W., Han, L., Ma, Z.: A predator-prey model with infected prey. Theoretical Population Biol. 66(3), 259–268 (2004)

    Google Scholar 

  23. Venturino, E.: The influence of diseases on lotka-volterra systems. The Rocky Mountain Journal of Mathematics 381–402 (1994)

  24. Beltrami, E., Carroll, T.: Modeling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol. 32(8), 857–863 (1994)

    MATH  Google Scholar 

  25. Venturino, E.: The effects of diseases on competing species. Math. Biosci. 174(2), 111–131 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Chattopadhyay, J., Pal, S., Abdllaoui, A.E.: Classical predator-prey system with infection of prey population-a mathematical model. Math. Methods Appl. Sci. 26(14), 1211–1222 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Silva, C.J., Cruz, C., Torres, D.F., Munuzuri, A.P., Carballosa, A., Area, I., Nieto, J.J., Fonseca-Pinto, R., Passadouro, R., Santos, ESd., et al.: Optimal control of the covid-19 pandemic: controlled sanitary deconfinement in Portugal. Sci. Rep. 11(1), 1–5 (2021)

    Google Scholar 

  28. Srivastava, H.M., Area Carracedo, I.C., Nieto, J. et al.: Power-series solution of compartmental epidemiological models. Math. Biosci. Eng

  29. Sharma, S., Samanta, G.: A leslie-gower predator-prey model with disease in prey incorporating a prey refuge. Chaos Solitons Fractals 70, 69–84 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Murthy, M.R., Bahlool, D.K.: Modeling and analysis of a prey-predator system with disease in predator. IOSR J. Math. 12(1), 21–40 (2016)

    Google Scholar 

  31. pada Das, K., Kundu, K., Chattopadhyay, J.: A predator–prey mathematical model with both the populations affected by diseases. Ecol. Complex. 8(1), 68–80 (2011)

    Google Scholar 

  32. Hadeler, K., Freedman, H.: Predator-prey populations with parasitic infection. J. Math. Biol. 27(6), 609–631 (1989)

    MathSciNet  MATH  Google Scholar 

  33. Hsieh, Y.-H., Hsiao, C.-K.: Predator-prey model with disease infection in both populations. Math. Med. Biol. A J. IMA 25(3), 247–266 (2008)

    MATH  Google Scholar 

  34. Bera, S., Maiti, A., Samanta, G.: A prey-predator model with infection in both prey and predator. Filomat 29(8), 1753–1767 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Gao, X., Pan, Q., He, M., Kang, Y.: A predator-prey model with diseases in both prey and predator. Phys. A Stat. Mech. Appl. 392(23), 5898–5906 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Gupta, J., Dhar, J., Sinha, P.: Mathematical study of the influence of canine distemper virus on tigers: an eco-epidemic dynamics with incubation delay. Rend. del Circ. Mat. di Palermo Ser. 2, 1–23 (2021)

    MATH  Google Scholar 

  37. Tannoia, C., Torre, E., Venturino, E.: An incubating diseased-predator ecoepidemic model. J. Biol. Phys. 38(4), 705–720 (2012)

    Google Scholar 

  38. Das, D.K., Khajanchi, S., Kar, T.K.: The impact of the media awareness and optimal strategy on the prevalence of tuberculosis. Appl. Math. Comput. 366, 124732 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Das, D.K., Khajanchi, S., Kar, T.: Transmission dynamics of tuberculosis with multiple re-infections. Chaos Solitons Fractals 130, 109450 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Khajanchi, S., Das, D.K., Kar, T.K.: Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation. Phys. A Stat. Mech. Appl

  41. Das, D.K., Khajanchi, S., Kar, T.K.: Influence of Multiple Re-infections in Tuberculosis Transmission Dynamics: a Mathematical Approach. In: 8th International Conference on Modeling Simulation and Applied Optimization (ICMSAO). IEEE 2019, pp 1–5 (2019)

  42. Hahn, B.D., Furniss, P.R.: A deterministic model of an anthrax epizootic: threshold results. Ecol. Modell. 20(2–3), 233–241 (1983)

    Google Scholar 

  43. Furniss, P., Hahn, B.: A mathematical model of an anthrax epizoötic in the Kruger national park. Appl. Math. Modell. 5(3), 130–136 (1981)

    Google Scholar 

  44. Friedman, A., Yakubu, A.-A.: Anthrax epizootic and migration: persistence or extinction. Math. Biosci. 241(1), 137–144 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Heffernan, J., Smith, R., Wahi, L.: Perspective on basic reproduction ratio. J.R. Soc. Interface 2(4), 281–293 (2005)

    Google Scholar 

  46. Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Diff. Equ. 20(3), 699–717 (2008)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, F., Zhao, X.-Q.: A periodic epidemic model in a patchy environment. J. Math. Anal. Appl. 325(1), 496–516 (2007)

    MathSciNet  MATH  Google Scholar 

  48. Chitnis, N., Hyman, J.M., Cushing, J.M.: Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70(5), 1272 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. S.M.S. Govt. Model Science College, Gwalior, Madhya Pradesh, 474002, India

    Jyoti Gupta & Poonam Sinha

  2. ABV-Indian Institute of Information Technology and Management, Gwalior, Madhya Pradesh, 474015, India

    Joydip Dhar

Authors
  1. Jyoti Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joydip Dhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Poonam Sinha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jyoti Gupta.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gupta, J., Dhar, J. & Sinha, P. Study of anthrax disease dynamics in multi-compartment with Grass and herbivores population. Rend. Circ. Mat. Palermo, II. Ser 72, 3841–3867 (2023). https://doi.org/10.1007/s12215-022-00859-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-022-00859-z

Keywords

Search

Navigation