Modeling and Analysis of a Single Species Population with Viral Infection in Polluted Environment

Affiliation(s)

IMS Engineering College, Adyatmik Nagar, Ghaziabad, India.

School of Mathematics and Allied Sciences, Jiwaji University, Gwalior, India.

IMS Engineering College, Adyatmik Nagar, Ghaziabad, India.

School of Mathematics and Allied Sciences, Jiwaji University, Gwalior, India.

ABSTRACT

In this paper, a mathematical model is proposed to study the effect of pollutant and virus induced disease on single species animal population and its essential mathematical features are analyzed. It is observed that the susceptible population does not vanish when it is only under the effect of infection but in the polluted environment, it can go to extinction. Also, it has been observed that the replication threshold obtained, increases on account of pollutant concentration consequently decreasing the susceptible population. Further persistence results for the proposed model are obtained and the condition for the existence of the Hopf-bifurcation is derived. Finally, numerical simulation in support of analytical results is carried out.

In this paper, a mathematical model is proposed to study the effect of pollutant and virus induced disease on single species animal population and its essential mathematical features are analyzed. It is observed that the susceptible population does not vanish when it is only under the effect of infection but in the polluted environment, it can go to extinction. Also, it has been observed that the replication threshold obtained, increases on account of pollutant concentration consequently decreasing the susceptible population. Further persistence results for the proposed model are obtained and the condition for the existence of the Hopf-bifurcation is derived. Finally, numerical simulation in support of analytical results is carried out.

Cite this paper

S. Chauhan and O. Misra, "Modeling and Analysis of a Single Species Population with Viral Infection in Polluted Environment,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 662-672. doi: 10.4236/am.2012.36100.

S. Chauhan and O. Misra, "Modeling and Analysis of a Single Species Population with Viral Infection in Polluted Environment,"

References

[1] R. M. Anderson and R. M May, “Regulation and Stability of Host-Parasite Population Interactions I: Regultory Processes,” Journal of Animal Ecology, Vol. 47, 1978, pp. 219-247. doi:10.2307/3933

[2] R. M. Anderson and R. M. May, “The Invasion, Persistence, and Spread of Infectious Disease within Animal and Plant Communities,” Transactions of the Royal Society of London, Vol. B 314, No. 1167, 1986, pp. 533-570.

[3] R. M. May and R. M. Anderson, “Regulations and Stability of Host-Parasite Population Interactions II, Destabilizing Processes,” Journal of Animal Ecology, Vol. 47, 1978, pp. 249-267. doi:10.2307/3934

[4] H. W. Hethcote and S. A. Levin, “Periodicity in Epidemiological Models,” In: L. Gross, T. G. Hallam and S. A. Levin, Eds., Applied Mathematical Ecology, SpringerVerlag, Berlin, 1989, pp. 193-211. doi:10.1007/978-3-642-61317-3_8

[5] M. Begon and R. G. Bowers, “Beyond Host-Parasite Dynamics,” In: B. T. Grenfell and A. P. Dobson, Eds., Ecology of Disease in Natural Populations, Cambridge University Press, Cambridge, 1995, pp. 479-509.

[6] B. T. Grenfell and A. P. Dobson, Eds., “Ecology of Disease in Natural Populations,” Cambridge University press, Cambridge, 1995.

[7] H. W. Hethcote, W. Wang and L. Yi, “Species Coexistence and Periodicity in Host-Host Pathogen Model,” Journal of Mathematical Biology, Vol. 51, No. 6, 2005, pp. 629-660.

[8] T.-W. Hwang and Y. Kuang, “Deterministic Extinction Effect of Parasite on Host Populations,” Journal of Mathematical Biology, Vol. 46, 2003, pp. 17-30. doi:10.1007/s00285-002-0165-7

[9] H. Mccallum and A. P. Dobson, “Detecting Diseases and Parasite Threats to Endangered Species Ecosystems,” Trends in Ecology and Evolution, Vol. 19, 1995, pp. 190-194. doi:10.1016/S0169-5347(00)89050-3

[10] M. Zhien, B. J. Song and T. G. Hallam, “The Threshold of Survival for the System in Fluctuating Environment,” Bulletin of Mathematical Biology, Vol. 57, No. 3, 1989, pp. 311-323.

[11] H. P. Liu and M. Zhien, “The Threshold of Survival for the System of Two Species in a Polluted Environment,” Journal of Mathematical Biology, Vol. 30, No. 1, 1991, pp. 49-61.

[12] L. Zhan, Z. S. Shun and W. Ke, “Persistence and Extinction of Single Population in a Polluted Environment, Electronic,” Journal of Differential Equations, Vol. 108, 2004, pp. 1-5.

[13] N. Nuraini, E. Soewono and K. A. Sidarto, “A Mathematical Model of Dengue Internal Transmission Process,” Journal of Indonesia Mathematical Society, Vol. 13, No. 1, 2007, pp. 123-132.

[14] N. M. May, “Los Alamos Mathematical Model Gauges Epidemic Potential of Emerging Diseases,” LOS ALAMOS, New Mexico, 27 May 2008.

[15] W. M. Liu, “Criterion of Hopf-Bifurcation without Using Eigenvalues,” Journal of Mathematical Analysis and Application, Vol. 250, 1994.

[16] D. Greenhalgh and M. Haque, “A Predator-Prey Model with Disease in Prey Species Only,” Mathematical Methods of Applied Sciences, Vol. 30, 2007, pp. 911-929. doi:10.1002/mma.815

[17] K. P. Hadeler and H. I. Freedman, “Predator-Prey Populations with Parasitic Infection,” Journal of Mathematical Biology, Vol. 27, No. 6, 1989, pp. 609-631.

[18] M. Haque and E. Venturino, “The Role of Transmissible Disease in Holling-Tanner Predator-Prey Model,” Theoritical Population Biology, Vol. 70, No. 3, 2006, pp. 273-863.

[19] E. Beltrami and T. O. Carroll, “Modelling the Role of Viral Disease in Recurrent Phytoplankton Blooms,” Journal of Mathematical Biology, Vol. 32, 1994, pp. 857-863. doi:10.1007/BF00168802

[20] S. Sinha, O. P. Misra and J. Dhar, “Study of a Prey-Predator Dynamics under the Simultaneous Effect of Toxicant and Disease,” The Journal of Nonlinear Analysis and its Applications, Vol. 1, No. 2, 2008, pp. 102-117.

[21] S. Sinha, O. P. Misra and J. Dhar, “A Two Species Competition Model under the Simultaneous Effect of Toxicant and Disease,” Non-Linear Analysis-Real World Application, Elsevier Publication, Vol. 11, 2010, pp. 1131-1142.

[22] S. Sinha, O. P. Misra and J. Dhar, “Modeling a Predator Prey System with Infected Prey in Polluted Environment,” Applied Mathematical Modeling, Elsevier Publication, Vol. 34, 2010, pp. 1861-1872.

[23] J. K. Hale, “Ordinary Differential Equations,” 2nd Edition, Kriegor, Basel, 1980.

[1] R. M. Anderson and R. M May, “Regulation and Stability of Host-Parasite Population Interactions I: Regultory Processes,” Journal of Animal Ecology, Vol. 47, 1978, pp. 219-247. doi:10.2307/3933

[2] R. M. Anderson and R. M. May, “The Invasion, Persistence, and Spread of Infectious Disease within Animal and Plant Communities,” Transactions of the Royal Society of London, Vol. B 314, No. 1167, 1986, pp. 533-570.

[3] R. M. May and R. M. Anderson, “Regulations and Stability of Host-Parasite Population Interactions II, Destabilizing Processes,” Journal of Animal Ecology, Vol. 47, 1978, pp. 249-267. doi:10.2307/3934

[4] H. W. Hethcote and S. A. Levin, “Periodicity in Epidemiological Models,” In: L. Gross, T. G. Hallam and S. A. Levin, Eds., Applied Mathematical Ecology, SpringerVerlag, Berlin, 1989, pp. 193-211. doi:10.1007/978-3-642-61317-3_8

[5] M. Begon and R. G. Bowers, “Beyond Host-Parasite Dynamics,” In: B. T. Grenfell and A. P. Dobson, Eds., Ecology of Disease in Natural Populations, Cambridge University Press, Cambridge, 1995, pp. 479-509.

[6] B. T. Grenfell and A. P. Dobson, Eds., “Ecology of Disease in Natural Populations,” Cambridge University press, Cambridge, 1995.

[7] H. W. Hethcote, W. Wang and L. Yi, “Species Coexistence and Periodicity in Host-Host Pathogen Model,” Journal of Mathematical Biology, Vol. 51, No. 6, 2005, pp. 629-660.

[8] T.-W. Hwang and Y. Kuang, “Deterministic Extinction Effect of Parasite on Host Populations,” Journal of Mathematical Biology, Vol. 46, 2003, pp. 17-30. doi:10.1007/s00285-002-0165-7

[9] H. Mccallum and A. P. Dobson, “Detecting Diseases and Parasite Threats to Endangered Species Ecosystems,” Trends in Ecology and Evolution, Vol. 19, 1995, pp. 190-194. doi:10.1016/S0169-5347(00)89050-3

[10] M. Zhien, B. J. Song and T. G. Hallam, “The Threshold of Survival for the System in Fluctuating Environment,” Bulletin of Mathematical Biology, Vol. 57, No. 3, 1989, pp. 311-323.

[11] H. P. Liu and M. Zhien, “The Threshold of Survival for the System of Two Species in a Polluted Environment,” Journal of Mathematical Biology, Vol. 30, No. 1, 1991, pp. 49-61.

[12] L. Zhan, Z. S. Shun and W. Ke, “Persistence and Extinction of Single Population in a Polluted Environment, Electronic,” Journal of Differential Equations, Vol. 108, 2004, pp. 1-5.

[13] N. Nuraini, E. Soewono and K. A. Sidarto, “A Mathematical Model of Dengue Internal Transmission Process,” Journal of Indonesia Mathematical Society, Vol. 13, No. 1, 2007, pp. 123-132.

[14] N. M. May, “Los Alamos Mathematical Model Gauges Epidemic Potential of Emerging Diseases,” LOS ALAMOS, New Mexico, 27 May 2008.

[15] W. M. Liu, “Criterion of Hopf-Bifurcation without Using Eigenvalues,” Journal of Mathematical Analysis and Application, Vol. 250, 1994.

[16] D. Greenhalgh and M. Haque, “A Predator-Prey Model with Disease in Prey Species Only,” Mathematical Methods of Applied Sciences, Vol. 30, 2007, pp. 911-929. doi:10.1002/mma.815

[17] K. P. Hadeler and H. I. Freedman, “Predator-Prey Populations with Parasitic Infection,” Journal of Mathematical Biology, Vol. 27, No. 6, 1989, pp. 609-631.

[18] M. Haque and E. Venturino, “The Role of Transmissible Disease in Holling-Tanner Predator-Prey Model,” Theoritical Population Biology, Vol. 70, No. 3, 2006, pp. 273-863.

[19] E. Beltrami and T. O. Carroll, “Modelling the Role of Viral Disease in Recurrent Phytoplankton Blooms,” Journal of Mathematical Biology, Vol. 32, 1994, pp. 857-863. doi:10.1007/BF00168802

[20] S. Sinha, O. P. Misra and J. Dhar, “Study of a Prey-Predator Dynamics under the Simultaneous Effect of Toxicant and Disease,” The Journal of Nonlinear Analysis and its Applications, Vol. 1, No. 2, 2008, pp. 102-117.

[21] S. Sinha, O. P. Misra and J. Dhar, “A Two Species Competition Model under the Simultaneous Effect of Toxicant and Disease,” Non-Linear Analysis-Real World Application, Elsevier Publication, Vol. 11, 2010, pp. 1131-1142.

[22] S. Sinha, O. P. Misra and J. Dhar, “Modeling a Predator Prey System with Infected Prey in Polluted Environment,” Applied Mathematical Modeling, Elsevier Publication, Vol. 34, 2010, pp. 1861-1872.

[23] J. K. Hale, “Ordinary Differential Equations,” 2nd Edition, Kriegor, Basel, 1980.