Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination

Leopard C. Mpande; Damian Kajunguri; Emmanuel A. Mpolya

doi:doi:10.11648/j.acm.20150406.16

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Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination

Leopard C. Mpande, Damian Kajunguri, Emmanuel A. Mpolya

Published in Applied and Computational Mathematics (Volume 4, Issue 6)

Received: 11 September 2015 Accepted: 26 September 2015 Published: 23 October 2015

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Abstract

In this paper, a metapopulation model is formulated as a system of ordinary differential equations to study the impact of vaccination on the spread of measles. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if R_C <1 and unstable if R_C >1. We show that when there are no movements between the two patches, there exists at least one endemic equilibrium for all R_Ci >1 and bifurcation analysis of endemic equilibrium point proves that forward (supercritical) bifurcation occurs in each patch. Numerical simulation results are also presented to validate analytical results and to show the impact of vaccination on the incidence and prevalence of measles in a metapopulation.

Published in	Applied and Computational Mathematics (Volume 4, Issue 6)
DOI	10.11648/j.acm.20150406.16
Page(s)	431-444
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Vaccination, Metapopulation, Measles, Bifurcation Analysis

References

[1]	Abubakar, S., Akinwande, N. I., Oguntolu, S. A. F. A., Bifurcation Analysis on the Mathematical Model of Measles Disease Dynamics. Universal Journal of Applied Mathematics, 1(4) (2003), 212-216.
[2]	Adewale, S. O., Mohammed, I. T., Olopade, I. A., Mathematical Analysis of Effect of area on The Dynamical Spread of Measles, IOSR Journal of Engineering, 4 (3) (2014), 43-57.
[3]	Anderson, R. M., May, R. M., Infectious Diseases of Humans, Oxford Univesity Press, London, New York, 1991.
[4]	Arino, J., Diseases in metapopulations, Modeling and dynamics of infectious diseases, 11 (2009), 65-123.
[5]	Arino, J., Van den Driessche, P., Disease spread in metapopulations, Fields Institute Communications, 48 (2006), 1-12.
[6]	Barbalat, I., Syste`me d’e´quations diffe´rentielles d’oscillation nonline`aires. Rev Roumaine Math Pures Appl 4 (1959): 267–270
[7]	Bhunu, C. P., Garira, W., Mukandavire, Z., Magombedze, G., Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control. Journal of theoretical biology, 254(3) (2008), 633-649.
[8]	Buonomo, B., Lacitignola, D., On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Analysis: Modelling and Control, 16(1) (2011), 30-46.
[9]	Carr, J., 1981. Applications Centre Manifold Theory. Springer, New York.
[10]	Castillo-Chavez, C., Song, B., Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1 (2) (2004), 361–404.
[11]	Castillo-Chavez, C., Z. Feng and W. Huang., On the Computation of R0 and Its Role on Global Stability, In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York (2002): 229- 250.
[12]	Chitnis, N., Hyman, J., Cushing, J., Determining important parameters in the spread of malaria through the sensitivity analysis of a malaria model, Bull. Math. Biology. 70 (2008), 1272-1296.
[13]	Diekman, O., Heesterbeek, J.A.P. and Metz, J.A.P. (1990). On the definition and Computation of the basic reproduction ratio R₀ in the model of infectious disease in Heterogeneous populations. Journal of Mathematical Biology. 2(1): 265-382.
[14]	Doungmo Goufo, E. F., Oukouomi Noutchie, S. C., Mugisha, S., A Fractional SEIR Epidemic Model for Spatial and Temporal Spread of Measles in Metapopulations, In Abstract and Applied Analysis, Hindawi Publishing Corporation, 2014 (2014).
[15]	Edward, S., Kuznetsov, D., Mirau, S., Modeling andS Stability Analysis for a Varicella Zoster Virus Model with Vaccination, Applied and Computational Mathematics, 3 (4) (2014), 150- 162.
[16]	Ejima, K., Omori, R., Aihara, K., Nishiura, H., Real-time investigation of measles epidemics with estimate of vaccine efficacy, International journal of biological sciences, 8(5) (2012), 620.
[17]	Elbasha, E. H., Gumel, A. B., Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits. Bulletin of mathematical biology, 68(3) (2006), 577-614.
[18]	Fred, M. O., Sigey, J. K., Okello, J. O., Okwoyo, J. M., Kang'ethe, G. J., (2014), Mathematical Modeling on the Control of Measles by Vaccination: Case Study of KISII County, Kenya, The SIJ Transactions on Computer Science Engineering & its Applications (CSEA), The Standard International Journals (The SIJ), 2 (3) (2014), 61-69.
[19]	Gahr, P., DeVries, A. S., Wallace, G., Miller, C., Kenyon, C., Sweet, K., Lynfield, R., An outbreak of measles in an under vaccinated community, Pediatrics, 134 (1) (2014), e220-e228.
[20]	Hethcote, H. W., The mathematics of infectious diseases, SIAM review, 42 (4) (2000), 599-653.
[21]	Korobeinikov, A., Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology. 21(2004): 75–83.
[22]	LaSalle, J. P., (1976) The stability of dynamical systems, CBMS-NSF regional conference series in applied mathematics 25. SIAM, Philadelphia.
[23]	Liao, S., Yang, W., On the dynamics of a vaccination model with multiple transmission ways. International Journal of Applied Mathematics and Computer Science, 23(4) (2013), 761-772.
[24]	Massawe, L. N., Massawe, E. S., Makinde, O. D., Temporal Model for Dengue Disease with Treatment, Advances in Infectious Diseases, 5 (1) (2015), 21-36.
[25]	McCluskey, C. C., Lyapunov functions for tuberculosis models with fast and slow progression. MathBiosci Eng 3(4) (2006), 603–614.
[26]	Mlay, G. M., Luboobi, L. S., Kuznetsov, D., Shahada, F., the Role of Re-Infection in Modeling the Dynamics of One-Strain Tuberculosis Involving Vaccination and Treatment, Asian Journal of Mathematics and Applications, 2014 (2014).
[27]	Momoh, A. A., Ibrahim, M. O., Uwanta, I. J., Manga, S.B., Mathematical model for control of Measles epidemiology. International Journal of Pure and Applied Mathematics, 87 (5) (2013), 707-717.
[28]	Mossong, J., Muller, C. P., Modelling measles re-emergence as a result of waning of immunity in vaccinated populations. Vaccine, 21(31) (2003), 4597-4603.
[29]	Mossong, J., Muller, C. P., Estimation of the basic reproduction number of measles during an outbreak in a partially vaccinated population, Epidemiology and infection, 124 (02) (2000), 273-278.
[30]	Mpeshe, S. C., Tchuenche, J. M., Haario, H., (2009): “A Mathematical Model of Rift Valley Fever with Human Host”. MSc (Mathematics) Dissertation Unpublished. UDSM: Dar-es-salaam.
[31]	Mukandavire, Z., Garira, W., Tchuenche, J. M., Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics. Appl Math Model 33 (2009), 2084–2095.
[32]	Mukandavire, Z., Chiyaka, C., Magombedze, G., Musuka, G., Malunguza, N. J., Assessing the effects of homosexuals and bisexuals on the intrinsic dynamics of HIV/AIDS in heterosexual settings. Mathematical and computer modelling, 49(9) (2009), 1869-1882.
[33]	Ngwenya, O., The Role of Incidence Functions on the dynamics of SEIR Model (Doctoral dissertation, University of Manitoba, Canada), 2009.
[34]	Ochoche, J. M., Gweryina, R. I., A Mathematical Model of Measles with Vaccination and Two Phases of Infectiousness, IOSR Journal of Mathematics 10 (1) (2014), 95-105.
[35]	Onyejekwe, O. O., Kebede, E. Z., Epidemiological Modeling of Measles Infection with Optimal Control of Vaccination and Supportive Treatment, Applied and Computational Mathematics, 4 (4) (2015), 264-274.
[36]	Perko, L., Differential Equations and Dynamical Systems, Springer 2000.
[37]	Salmani, M., Van den Driessche, P., A model for Disease Transmission in a Patchy Environment, Discrete and Continuous Dynamical Systems Series B, 6 (1) (2006), 185.
[38]	World Health Organization, Technical guidelines for integrated disease surveillance and response in the African region. Regional Office for Africa, Division of Communicable Disease Prevention and Control, 2001.
[39]	Tessa, O. M., Mathematical model for control of measles by vaccination. In Proceedings of Mali Symposium on Applied Sciences, (2006), 31-36.
[40]	Tumwiine, J., Mugisha, J. Y. T., Luboobi, L. S., A Mathematical Model for the Dynamics of Malaria in a Human Host and Mosquito Vector with Temporary Immunity. Applie Mathematics and Computation, 189 (2007), 1953-1965.
[41]	Van den Driessche, P., Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences,180 (1) (2002), 29-48.
[42]	World Health Organization (2014). Fact sheet measles.
[43]	Xia, Y., Bjørnstad, O. N., Grenfell, B. T., Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics. The American Naturalist, 164 (2) (2004), 267-281.

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APA Style

Leopard C. Mpande, Damian Kajunguri, Emmanuel A. Mpolya. (2015). Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination. Applied and Computational Mathematics, 4(6), 431-444. https://doi.org/10.11648/j.acm.20150406.16

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ACS Style

Leopard C. Mpande; Damian Kajunguri; Emmanuel A. Mpolya. Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination. Appl. Comput. Math. 2015, 4(6), 431-444. doi: 10.11648/j.acm.20150406.16

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AMA Style

Leopard C. Mpande, Damian Kajunguri, Emmanuel A. Mpolya. Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination. Appl Comput Math. 2015;4(6):431-444. doi: 10.11648/j.acm.20150406.16

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@article{10.11648/j.acm.20150406.16,
  author = {Leopard C. Mpande and Damian Kajunguri and Emmanuel A. Mpolya},
  title = {Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {6},
  pages = {431-444},
  doi = {10.11648/j.acm.20150406.16},
  url = {https://doi.org/10.11648/j.acm.20150406.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150406.16},
  abstract = {In this paper, a metapopulation model is formulated as a system of ordinary differential equations to study the impact of vaccination on the spread of measles. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if RC <1 and unstable if RC >1. We show that when there are no movements between the two patches, there exists at least one endemic equilibrium for all RCi >1 and bifurcation analysis of endemic equilibrium point proves that forward (supercritical) bifurcation occurs in each patch. Numerical simulation results are also presented to validate analytical results and to show the impact of vaccination on the incidence and prevalence of measles in a metapopulation.},
 year = {2015}
}

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TY  - JOUR
T1  - Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination
AU  - Leopard C. Mpande
AU  - Damian Kajunguri
AU  - Emmanuel A. Mpolya
Y1  - 2015/10/23
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150406.16
DO  - 10.11648/j.acm.20150406.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 431
EP  - 444
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150406.16
AB  - In this paper, a metapopulation model is formulated as a system of ordinary differential equations to study the impact of vaccination on the spread of measles. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if RC <1 and unstable if RC >1. We show that when there are no movements between the two patches, there exists at least one endemic equilibrium for all RCi >1 and bifurcation analysis of endemic equilibrium point proves that forward (supercritical) bifurcation occurs in each patch. Numerical simulation results are also presented to validate analytical results and to show the impact of vaccination on the incidence and prevalence of measles in a metapopulation.
VL  - 4
IS  - 6
ER  -

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Author Information

Leopard C. Mpande

Nelson Mandela African Institution of Science and Technology, School of CoCSE, Arusha, Tanzania
Damian Kajunguri

Nelson Mandela African Institution of Science and Technology, School of CoCSE, Arusha, Tanzania
Emmanuel A. Mpolya

Nelson Mandela African Institution of Science and Technology, School of LiSBE, Arusha, Tanzania

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APA Style

Leopard C. Mpande, Damian Kajunguri, Emmanuel A. Mpolya. (2015). Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination. Applied and Computational Mathematics, 4(6), 431-444. https://doi.org/10.11648/j.acm.20150406.16

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ACS Style

Leopard C. Mpande; Damian Kajunguri; Emmanuel A. Mpolya. Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination. Appl. Comput. Math. 2015, 4(6), 431-444. doi: 10.11648/j.acm.20150406.16

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AMA Style

Leopard C. Mpande, Damian Kajunguri, Emmanuel A. Mpolya. Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination. Appl Comput Math. 2015;4(6):431-444. doi: 10.11648/j.acm.20150406.16

Copy | Download

@article{10.11648/j.acm.20150406.16,
  author = {Leopard C. Mpande and Damian Kajunguri and Emmanuel A. Mpolya},
  title = {Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {6},
  pages = {431-444},
  doi = {10.11648/j.acm.20150406.16},
  url = {https://doi.org/10.11648/j.acm.20150406.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150406.16},
  abstract = {In this paper, a metapopulation model is formulated as a system of ordinary differential equations to study the impact of vaccination on the spread of measles. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if RC <1 and unstable if RC >1. We show that when there are no movements between the two patches, there exists at least one endemic equilibrium for all RCi >1 and bifurcation analysis of endemic equilibrium point proves that forward (supercritical) bifurcation occurs in each patch. Numerical simulation results are also presented to validate analytical results and to show the impact of vaccination on the incidence and prevalence of measles in a metapopulation.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination
AU  - Leopard C. Mpande
AU  - Damian Kajunguri
AU  - Emmanuel A. Mpolya
Y1  - 2015/10/23
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150406.16
DO  - 10.11648/j.acm.20150406.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 431
EP  - 444
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150406.16
AB  - In this paper, a metapopulation model is formulated as a system of ordinary differential equations to study the impact of vaccination on the spread of measles. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if RC <1 and unstable if RC >1. We show that when there are no movements between the two patches, there exists at least one endemic equilibrium for all RCi >1 and bifurcation analysis of endemic equilibrium point proves that forward (supercritical) bifurcation occurs in each patch. Numerical simulation results are also presented to validate analytical results and to show the impact of vaccination on the incidence and prevalence of measles in a metapopulation.
VL  - 4
IS  - 6
ER  -

Copy | Download