Abstract
This chapter discusses a mathematical model for the spread of an infectious disease with transmission through a pathogen in an environment, including the effects of human contact with the environment. The model assumes a structured susceptible population consisting of both “low-risk” and “high-risk” individuals. It also includes the effects of shedding the pathogen by the infected population into the environment. The model has a disease-free equilibrium state, and a linear stability analysis shows three possible transmission routes. The model is applied to Buruli ulcer disease, a debilitating disease induced by Mycobacterium ulcerans. There is some uncertainty about the exact transmission path, but the bacteria is known to live in natural water environments. The model parameters are estimated from data on Buruli ulcer disease in Ghana. This chapter includes a sensitivity analysis of the total number of infected individuals to the parameters in the model.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Altizer, S., Ostfeld, R.S., Johnson, P.T., et al.: Climate change and infectious diseases: from evidence to a predictive framework. Science 341(6145), 514–519 (2013)
Amofah, G., Bonsu, F., Tetteh, C., et al.: Buruli Ulcer in Ghana: results of a national case search. Emerg. Infect. Dis. 2, 167–170 (2002)
Benbow, M.E., Williamson, H., Kimbirauskas, R., et al.: Aquatic invertebrates as unlikely vectors of Buruli ulcer disease. Emerg. Infect. Dis. 14(8), 1247 (2008)
Benbow, M.E., Kimbirauskas, R., McIntosh, M.D., et al.: Aquatic macroinvertebrate assemblages of Ghana, West Africa: understanding the ecology of a neglected tropical disease. Ecohealth 11(2), 168–183 (2014)
Benjamini, Y., Hochberg, Y.: Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. R. Stat. Soc. Ser. B 57(1), 289–300 (1995)
Blower, S.M., Dowlatabadi, H.: Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int. Stat. Rev. 62(2), 229–243 (1994)
Bonyah, E., Dontwi, I., Nyabadza, F.: A theoretical model for the transmission dynamics of the Buruli ulcer with saturated treatment. Comput. Math. Methods Med. 2014, 576039 (2014)
Breban, R.: Role of environmental persistence in pathogen transmission: a mathematical modeling approach. J. Math. Biol. 66, 535–546 (2013)
Cook, A.: The Mengo Hospital Notes. Makerere College Medical School Library, Kampala (1897)
De Silva, M.T., Portaels, F., Pedrosa, J.: Aquatic insects and mycobacterium ulcerans: an association relevant to Buruli ulcer control. PLoS Med. 4, e63 (2007). https://doi.org/10.1371/journal.pmed.0040063
De Silva, K.R., Eda, S., Lenhart, S.: Modeling environmental transmission of MAP infection in dairy cows. Math. Biosci. Eng. 4, 1001–1017 (2017)
Diekmann, O., Heesterbeek, J.P.: Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester (2000)
Diekmann, O., Heesterbeek, H., Britton, T.: Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press, Princeton (2012)
Duker, A.A., Portaels, F., Hale, M.: Pathways of Mycobacterium ulcerans infection: a review. Environ. Int. 32, 567–573 (2006)
Feller, E.C., Pearson, E.S.: Tests for rank correlation coefficients: II. Biometrika 48, 29–40 (1961)
Garchitorena, A., Ngonghala, C.N., Guegan, J.F., et al.: Economic inequality caused by feedbacks between poverty and the dynamics of a rare tropical disease: the case of Buruli ulcer in sub-Saharan Africa. Proc. R. Soc. B 282, 20151426 (2015)
Garchitorena, A., Ngonghala, C.N., Texier, G., et al.: Environmental transmission of Mycobacterium ulcerans drives dynamics of Buruli ulcer in endemic regions of Cameroon. Sci. Rep. 5, 18055 (2015)
Ghana Health Service: https://www.ghanahealthservice.org/
Janssens, P.G., Quertinmont, M.J., Sieniawski, J., et al.: Necrotic tropical ulcers and Mycobacterial causative agents. Trop. Geogr. Med. 11, 293–312 (1959)
Kelly Jr, M.R., Tien, J.H., Eisenberg, M.C., et al.: The impact of spatial arrangements on epidemic disease dynamics and intervention strategies. J. Biol. Dyn. 10, 222–249 (2016)
Kenu, E., Nyarko, K.M., Seefeld, L., et al.: Risk factors for Buruli ulcer in Ghana–a case control study in the Suhum-Kraboa-Coaltar and Akuapem South Districts of the eastern region. PLoS Negl. Trop. Dis. 8(11), e3279 (2014)
LaSalle, J.P.: The Stability of Dynamical Systems, vol. 25. SIAM, Philadelphia (1976)
MacCallum, P., Tolhurst, J.C.: A new Mycobacterial infection in man. J. Pathol. Bacteriol. 60, 93–122 (1948)
Macklin, J.T.: An investigation of the properties of double radio sources using the Spearman partial rank correlation coefficient. Mon. Not. R. Astron. Soc. 199, 1119–1136 (1982)
Marino, S., Hogue, I.B., Ray, C.J., et al.: A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254, 178–196 (2008)
Marion, E., Eyangoh, S., Yeramian, E., et al.: Seasonal and regional dynamics of M. ulcerans transmission in environmental context: deciphering the role of water bugs as hosts and vectors. PLoS Negl. Trop. Dis. 4, e731 (2010)
Marsollier, L., Robert, R., Aubry, J., et al.: Aquatic insects as a vector for Mycobacterium ulcerans. Appl. Environ. Microbiol. 68(9), 4623–4628 (2002)
Martcheva, M., Lenhart, S., Eda, S., et al.: An immuno-epidemiological model for Johne’s disease in cattle. Vet. Res. 46, 69 (2015)
McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979)
Merritt, R.W., Walker, E.D., Small, P.L., et al.: Ecology and transmission of Buruli ulcer disease: a systematic review. PLoS Negl. Trop. Dis. 4, e911 (2016)
Miller Neilan, R.L., Schaefer, E., Gaff, H., et al.: Modeling optimal intervention strategies for cholera. Bull. Math. Biol. 72, 2004–2018 (2010)
Morris, A., Gozlan, R.E., Hassani, H., et al.: Complex temporal climate signals drive the emergence of human water-borne disease. Emerg. Microbes Infect. 3, e56 (2014)
Morris, A., Guégan, J.F., Benbow, M.E., et al.: Functional diversity as a new framework for understanding the ecology of an emerging generalist pathogen. EcoHealth 13, 570–581 (2016)
Nyabadza, F., Bonyah, E.: On the transmission dynamics of Buruli ulcer in Ghana: insights through a mathematical model. BMC Res. Notes 8, 656 (2015)
Pascual, M., Bouma, M.J., Dobson, A.P.: Cholera and climate: revisiting the quantitative evidence. Microbes Infect. 4, 237–245 (2002)
Portaels, F., Elsen, P., Guimaraes-Peres, A., et al.: Insects in the transmission of Mycobacterium ulcerans infection. Lancet 353, 986 (1999)
Röltgen, K., Pluschke, G.: Epidemiology and disease burden of Buruli ulcer: a review. Res. Rep. Trop. Med. 6, 59–73 (2016)
Shuai, Z., van den Driessche, P.: Global stability of infectious disease models using Lyapunov functions. SIAM J. Appl. Math. 73, 1513–1532 (2013)
Siewe, N., Yakubu, A.A., Satoskar, A.R., et al.: Immune response to infection by Leishmania: a mathematical model. Math. Biosci. 276, 28–43 (2016)
Sopoh, G.E., Johnson, R.C., Chauty, A., et al.: Buruli ulcer surveillance, Benin, 2003–2005. Emerg. Infect. Dis. 9, 1374–1376 (2007)
The World Bank: Ghana data. Technical report, The World Bank (2016). Retrieved from http://data.worldbank.org/country/ghana
Thomas, C.D., Cameron, A., Green, R.E., et al.: Extinction risk from climate change. Nature 427, 145–148 (2004)
Tien, J.H., Earn, D.J.: Multiple transmission pathways and disease dynamics in a waterborne pathogen model. Bull. Math. Biol. 72, 1506–1533 (2010)
van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
van den Driessche, P., Watmough, J.: Mathematical Epidemiology: Further Notes on the Basic Reproduction Number. Springer, Berlin (2008)
van Ravensway, J., Benbow, M.E., Tsonis, A.A., et al.: Climate and landscape factors associated with Buruli ulcer incidence in Victoria, Australia. PLoS One 7, e51074 (2012)
Wansbrough-Jones, M., Phillips, R.: Buruli ulcer: emerging from obscurity. Lancet 367(9525), 1849–1858 (2006)
Williamson, H.R., Benbow, M.E., Nguyen, K.D., et al.: Distribution of Mycobacterium ulcerans in Buruli ulcer endemic and non-endemic aquatic sites in Ghana. PLoS Negl. Trop. Dis. 2 (2008)
Williamson, H.R., Benbow, M.E., Campbell, L.P., et al.: Detection of Mycobacterium ulcerans in the environment predicts prevalence of Buruli ulcer in Benin. PLoS Negl. Trop. Dis. 6, e1506 (2012)
Williamson, H., Mosi, L., Donnell, R., et al.: Mycobacterium ulcerans fails to infect through skin abrasions in a guinea pig infection model: implications for transmission. PLoS Negl. Trop. Dis. 8, e2770 (2014)
World Health Organization (WHO): Weekly epidemiological record. Technical report, World Health Organization (2002). http://www.who.int/wer/2002/en/wer7732.pdf
World Health Organization (WHO): Buruli ulcer (Mycobacterium ulcerans infection), Fact sheet. Technical report, World Health Organization (2016). http://www.who.int/mediacentre/factsheets/fs199/en/
Worldometers: Population (2016). Retrieved from http://www.worldometers.info/world-population/ghana-population/
Acknowledgements
The authors acknowledge partial support from the National Science Foundation (NSF) under Grant No. 1343651 through the Southern Africa Mathematical Sciences Association (SAMSA) Masamu Program—a program that aims to enhance research in the mathematical sciences by serving as a platform for US–Africa research collaborations. This work was also partially supported by the National Institute for Mathematical and Biological Synthesis (NIMBioS)—one of the several Mathematical Sciences Institutes sponsored by the NSF Division of Mathematical Sciences—through NSF Award DBI-1300426, with additional support from The University of Tennessee, Knoxville. The authors appreciate the support for travel expenses from the Society of Mathematical Biology. They acknowledge an invaluable conversation with Pam Small and Heather Williamson about transmission mechanisms.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Edholm, C. et al. (2019). A Risk-Structured Mathematical Model of Buruli Ulcer Disease in Ghana. In: Kaper, H., Roberts, F. (eds) Mathematics of Planet Earth. Mathematics of Planet Earth, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-22044-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-22044-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22043-3
Online ISBN: 978-3-030-22044-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)