Optimal Campaign in Leptospirosis Epidemic by Multiple Control Variables

Affiliation(s)

Department of Mathematics, University of Malakand, Chakdara, Lower Dir District of Khyber Pakhtunkhwa, Pakistan.

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan.

Department of Business Administration and Accounting, Buraimi University College, Al-Buraimi, Oman.

Department of Mathematics, University of Malakand, Chakdara, Lower Dir District of Khyber Pakhtunkhwa, Pakistan.

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan.

Department of Business Administration and Accounting, Buraimi University College, Al-Buraimi, Oman.

ABSTRACT

In this paper, we consider a leptospirosis epidemic model to implement optimal campaign by using multiple control variables. First, we show the existence of the control problem. Then we derive the conditions under which it is optimal to eradicate the leptospirosis infection and examine the impact of a possible educatioal/vaccinaction campaign using Pontryagin’s Maximum Principle. We completely characterize the optimal control problem and compute the numerical solution of the optimality system using an iterative method. The results obtained from the numerical simulations of the model show that a possible educational/vaccinaction combined with effective treatment regime would reduce the spread of the leptospirosis infection appreciably.

In this paper, we consider a leptospirosis epidemic model to implement optimal campaign by using multiple control variables. First, we show the existence of the control problem. Then we derive the conditions under which it is optimal to eradicate the leptospirosis infection and examine the impact of a possible educatioal/vaccinaction campaign using Pontryagin’s Maximum Principle. We completely characterize the optimal control problem and compute the numerical solution of the optimality system using an iterative method. The results obtained from the numerical simulations of the model show that a possible educational/vaccinaction combined with effective treatment regime would reduce the spread of the leptospirosis infection appreciably.

Cite this paper

M. Khan, G. Zaman, S. Islam and M. Chohan, "Optimal Campaign in Leptospirosis Epidemic by Multiple Control Variables,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1655-1663. doi: 10.4236/am.2012.311229.

M. Khan, G. Zaman, S. Islam and M. Chohan, "Optimal Campaign in Leptospirosis Epidemic by Multiple Control Variables,"

References

[1] R. U. Palanippan, S. Ramanujam and Y.-F. Chang, “Leptospirosis: Pathogenesis, Immunity, and Diagnosis,” Current Opinion in Infectious Diseases, Vol. 20, No. 3, 2007, pp. 284-292. doi:10.1097/QCO.0b013e32814a5729

[2] R. Inada and Y. Ido, “The Etiology Mode of Infection and Specific Therapy of Weil’s Disease,” Journal of Experimental Medicine, Vol. 23, No. 3, 1916, pp. 377-402. doi:10.1084/jem.23.3.377

[3] R. C. Abdulkader, A. C. Seguro, P. S. Malheiro, et al., “Peculiar Electrolytic and Hormonal Abnormalities in Acute Renal Failure Due to Leptospirosis,” The American Journal of Tropical Medicine and Hygiene, Vol. 54, No. 1, 1996, pp. 1-6.

[4] V. M. Arean, G. Sarasin and J. H. Green, “The Pathogenesis of Leptospirosis: Toxin Production by Leptospira Icterohaemorrhagiae,” American Journal of Veterinary Research, Vol. 28, 1964, pp. 836-843.

[5] V. M. Arean, “Studies on the Pathogenesis of Leptospirosis. II. A Clinicopathologic Evaluation of Hepatic and Renal Function in Experimental Leptospiral Infections,” Laboratory Investigation, Vol. 11, 1962, pp. 273-288.

[6] S. Barkay and H. Garzozi, “Leptospirosis and Uveitis,” Anmnals of Ophthalmology, Vol. 16, No. 2, 1984, pp. 164-168.

[7] S. Faine, “Guideline for Control of Leptospirosis,” World Health Organization, Geneva, 1982, p. 129.

[8] N. Chitnis, T. Smith and R. Steketee, “A Mathematical Model for the Dynamics of Malaria in Mosquitoes Feeding on a Heterogeneous Host Population,” Journal of Biological Dynamics, Vol. 2, No. 3, 2008, pp. 259-285. doi:10.1080/17513750701769857

[9] M. Derouich and A. Boutayeb, “Dengue Fever: Mathematical Modelling and Computer Simulations,” Applied Mathematics and Computation, Vol. 177, No. 2, 2006, pp. 528-544. doi:10.1016/j.amc.2005.11.031

[10] L. Esteva and C. Vergas, “A Model for Dengue Disease with Variable Human Populations,” Journal of Mathematical Biology, Vol. 38, No. 3, 1999, pp. 220-240. doi:10.1007/s002850050147

[11] P. Pongsuumpun, T. Miami and R. Kongnuy, “Age Structural Transmission Model for Leptospirosis,” The 3rd International Symposium on Biomedical Engineering, Bangkok, 10-11 November 2008, pp. 411-416.

[12] G. Zaman, M. A. Khan, S. Islam, M. I. Chohan and I. H. Jung, “Modeling Dynamical Interactions between Leptospirosis Infected Vector and Human Population,” Applied Mathematical Sciences, Vol. 6, No. 26, 2012, pp. 1287-1302.

[13] W. Triampo, D. Baowan, I. M. Tang, N. Nuttavut, J. Wong-Ekkabut and G. Doungchawee, “A Simple Deterministic Model for the Spread of Leptospirosis in Thailand,” International Journal of Biological and Life Sciences, Vol. 2, No. 1, 2006, pp. 22-26.

[14] G. Zaman, “Dynamical Behavior of Leptospirosis Disease and Role of Optimal Control Theory,” International Journal of Mathematics and Computation, Vol. 7, No. J10, 2010, pp. 80-92.

[15] M. N. Ashra? and A. B. Gumel, “Mathematical Analysis of the Role of Repeated Exposure on Malaria Transmission Dynamics,” Differential Equations and Dynamical Systems, Vol. 16, No. 3, 2008, pp. 251-287. doi:10.1007/s12591-008-0015-1

[16] A. A. Lashari and G. Zaman, “Optimal Control of a Vector Borne Disease with Horizontal Transmission,” Nonlinear Analysis: Real World Applications, Vol. 13, No. 1, 2012, pp. 203-212. doi:10.1016/j.nonrwa.2011.07.026

[17] S. Lenhart and J. T. Workman, “Optimal Control Applied to Biological Models,” Champion and Hall/CRC, London, 2007.

[18] G. Zaman, Y. H. Kang and I. H. Jung, “Optimal Treatment of an SIR Epedemic Model with Time Delay,” Biosystems, Vol. 98, No. 1, 2009, pp. 43-50. doi:10.1016/j.biosystems.2009.05.006

[19] G. Zaman, M. A. Khan, et al., “Mathematical Formulation and Dynamical Interaction of Leptospirosis Disease,” Submitted 2012.

[20] O. Sharomi, C. N. Podder, A. B. Gumel, E. H. Elbasha and J. Watmough, “Role of Incidence Function in VaccineInduced Backward Bifurcation in Some HIV Models,” Mathematical Biosciences, Vol. 210, No. 2, 2007, pp. 436-463. doi:10.1016/j.mbs.2007.05.012

[21] K. W. Blayneh, Y. Cao and H. D. Kwon, “Optimal Control of Vector-Born Disease: Treatment and Prevention,” Discrete and Continuous Dynamical Systems Series B, Vol. 11, No. 3, 2009, pp. 587-611.

[22] G. Birkhoff and G.-C. Rota, “Ordinary Differential Equations,” 4th Edition, John Wiley and Sons, New York, 1989.

[23] D. L. Lukes, “Differential Equations: Classical to Controlled,” Academic Press, New York, 1982.

[24] M. I. Kamien and N. L. Schwartz, “Dynamics Optimization: The Calculus of Variations and Optimal Control in Economics and Management,” 2nd Edition, Elsevier Science, Amsterdam, 1991.

[1] R. U. Palanippan, S. Ramanujam and Y.-F. Chang, “Leptospirosis: Pathogenesis, Immunity, and Diagnosis,” Current Opinion in Infectious Diseases, Vol. 20, No. 3, 2007, pp. 284-292. doi:10.1097/QCO.0b013e32814a5729

[2] R. Inada and Y. Ido, “The Etiology Mode of Infection and Specific Therapy of Weil’s Disease,” Journal of Experimental Medicine, Vol. 23, No. 3, 1916, pp. 377-402. doi:10.1084/jem.23.3.377

[3] R. C. Abdulkader, A. C. Seguro, P. S. Malheiro, et al., “Peculiar Electrolytic and Hormonal Abnormalities in Acute Renal Failure Due to Leptospirosis,” The American Journal of Tropical Medicine and Hygiene, Vol. 54, No. 1, 1996, pp. 1-6.

[4] V. M. Arean, G. Sarasin and J. H. Green, “The Pathogenesis of Leptospirosis: Toxin Production by Leptospira Icterohaemorrhagiae,” American Journal of Veterinary Research, Vol. 28, 1964, pp. 836-843.

[5] V. M. Arean, “Studies on the Pathogenesis of Leptospirosis. II. A Clinicopathologic Evaluation of Hepatic and Renal Function in Experimental Leptospiral Infections,” Laboratory Investigation, Vol. 11, 1962, pp. 273-288.

[6] S. Barkay and H. Garzozi, “Leptospirosis and Uveitis,” Anmnals of Ophthalmology, Vol. 16, No. 2, 1984, pp. 164-168.

[7] S. Faine, “Guideline for Control of Leptospirosis,” World Health Organization, Geneva, 1982, p. 129.

[8] N. Chitnis, T. Smith and R. Steketee, “A Mathematical Model for the Dynamics of Malaria in Mosquitoes Feeding on a Heterogeneous Host Population,” Journal of Biological Dynamics, Vol. 2, No. 3, 2008, pp. 259-285. doi:10.1080/17513750701769857

[9] M. Derouich and A. Boutayeb, “Dengue Fever: Mathematical Modelling and Computer Simulations,” Applied Mathematics and Computation, Vol. 177, No. 2, 2006, pp. 528-544. doi:10.1016/j.amc.2005.11.031

[10] L. Esteva and C. Vergas, “A Model for Dengue Disease with Variable Human Populations,” Journal of Mathematical Biology, Vol. 38, No. 3, 1999, pp. 220-240. doi:10.1007/s002850050147

[11] P. Pongsuumpun, T. Miami and R. Kongnuy, “Age Structural Transmission Model for Leptospirosis,” The 3rd International Symposium on Biomedical Engineering, Bangkok, 10-11 November 2008, pp. 411-416.

[12] G. Zaman, M. A. Khan, S. Islam, M. I. Chohan and I. H. Jung, “Modeling Dynamical Interactions between Leptospirosis Infected Vector and Human Population,” Applied Mathematical Sciences, Vol. 6, No. 26, 2012, pp. 1287-1302.

[13] W. Triampo, D. Baowan, I. M. Tang, N. Nuttavut, J. Wong-Ekkabut and G. Doungchawee, “A Simple Deterministic Model for the Spread of Leptospirosis in Thailand,” International Journal of Biological and Life Sciences, Vol. 2, No. 1, 2006, pp. 22-26.

[14] G. Zaman, “Dynamical Behavior of Leptospirosis Disease and Role of Optimal Control Theory,” International Journal of Mathematics and Computation, Vol. 7, No. J10, 2010, pp. 80-92.

[15] M. N. Ashra? and A. B. Gumel, “Mathematical Analysis of the Role of Repeated Exposure on Malaria Transmission Dynamics,” Differential Equations and Dynamical Systems, Vol. 16, No. 3, 2008, pp. 251-287. doi:10.1007/s12591-008-0015-1

[16] A. A. Lashari and G. Zaman, “Optimal Control of a Vector Borne Disease with Horizontal Transmission,” Nonlinear Analysis: Real World Applications, Vol. 13, No. 1, 2012, pp. 203-212. doi:10.1016/j.nonrwa.2011.07.026

[17] S. Lenhart and J. T. Workman, “Optimal Control Applied to Biological Models,” Champion and Hall/CRC, London, 2007.

[18] G. Zaman, Y. H. Kang and I. H. Jung, “Optimal Treatment of an SIR Epedemic Model with Time Delay,” Biosystems, Vol. 98, No. 1, 2009, pp. 43-50. doi:10.1016/j.biosystems.2009.05.006

[19] G. Zaman, M. A. Khan, et al., “Mathematical Formulation and Dynamical Interaction of Leptospirosis Disease,” Submitted 2012.

[20] O. Sharomi, C. N. Podder, A. B. Gumel, E. H. Elbasha and J. Watmough, “Role of Incidence Function in VaccineInduced Backward Bifurcation in Some HIV Models,” Mathematical Biosciences, Vol. 210, No. 2, 2007, pp. 436-463. doi:10.1016/j.mbs.2007.05.012

[21] K. W. Blayneh, Y. Cao and H. D. Kwon, “Optimal Control of Vector-Born Disease: Treatment and Prevention,” Discrete and Continuous Dynamical Systems Series B, Vol. 11, No. 3, 2009, pp. 587-611.

[22] G. Birkhoff and G.-C. Rota, “Ordinary Differential Equations,” 4th Edition, John Wiley and Sons, New York, 1989.

[23] D. L. Lukes, “Differential Equations: Classical to Controlled,” Academic Press, New York, 1982.

[24] M. I. Kamien and N. L. Schwartz, “Dynamics Optimization: The Calculus of Variations and Optimal Control in Economics and Management,” 2nd Edition, Elsevier Science, Amsterdam, 1991.