Export file:
Format
- RIS(for EndNote,Reference Manager,ProCite)
- BibTex
- Text
Content
- Citation Only
- Citation and Abstract
Modeling Ebola Virus Disease transmissions with reservoir in a complex virus life ecology
1. Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67 Dschang, Cameroon
2. Department of Mathematics and Computer Science, University of Douala, P.O. Box 24157 Douala, Cameroon
3. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
4. Department of Mathematics, Faculty of Sciences, University of Yaounde 1, P.O. Box 812 Yaounde, Cameroon
Received: , Accepted: , Published:
We propose a new deterministic mathematical model for the transmission dynamics of Ebola Virus Disease (EVD) in a complex Ebola virus life ecology. Our model captures as much as possible the features and patterns of the disease evolution as a three cycle transmission process in the two ways below. Firstly it involves the synergy between the epizootic phase (during which the disease circulates periodically amongst non-human primates populations and decimates them), the enzootic phase (during which the disease always remains in fruit bats population) and the epidemic phase (during which the EVD threatens and decimates human populations). Secondly it takes into account the well-known, the probable/suspected and the hypothetical transmission mechanisms (including direct and indirect routes of contamination) between and within the three different types of populations consisting of humans, animals and fruit bats. The reproduction number $\mathcal R_0$ for the full model with the environmental contamination is derived and the global asymptotic stability of the disease free equilibrium is established when $\mathcal R_0 < 1$. It is conjectured that there exists a unique globally asymptotically stable endemic equilibrium for the full model when $\mathcal R_0>1$. The role of a contaminated environment is assessed by comparing the human infected component for the sub-model without the environment with that of the full model. Similarly, the sub-model without animals on the one hand and the sub-model without bats on the other hand are studied. It is shown that bats influence more the dynamics of EVD than the animals. Global sensitivity analysis shows that the effective contact rate between humans and fruit bats and the mortality rate for bats are the most influential parameters on the latent and infected human individuals. Numerical simulations, apart from supporting the theoretical results and the existence of a unique globally asymptotically stable endemic equilibrium for the full model, suggest further that: (1) fruit bats are more important in the transmission processes and the endemicity level of EVD than animals. This is in line with biological findings which identified bats as reservoir of Ebola viruses; (2) the indirect environmental contamination is detrimental to human beings, while it is almost insignificant for the transmission in bats.
Keywords: Ebola; zoonotic disease; reservoir; environmental transmission; stability; simulation
Citation: Tsanou Berge, Samuel Bowong, Jean Lubuma, Martin Luther Mann Manyombe. Modeling Ebola Virus Disease transmissions with reservoir in a complex virus life ecology. Mathematical Biosciences and Engineering, 2018, 15(1): 21-56. doi: 10.3934/mbe.2018002
References:
- [1] C. Althaus, Estimating the reproduction number of Ebola (EBOV) during outbreak in West Africa, PLOS Currents, 2014.
- [2] R. M. Anderson,R. M. May, null, Infectious Diseases of Humans: Dynamics and Control, , Oxford University Press, Oxford, England, 1991.
- [3] S. Anita,V. Capasso, On the stabilization of reaction-diffusion systems modeling a class of man-environment epidemics: A review, Math. Meth. Appl. Sci., 33 (2010): 1235-1244.
- [4] S. Anita,V. Capasso, Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control, Nonlinear Analysis: Real World Applications, 13 (2012): 725-735.
- [5] A. A. Arata and B. Johnson, Approaches toward studies on potential reservoirs of viral haemorrhagic fever in southern Sudan (1977), In Ebola Virus Haemorrhagic Fever (Pattyn, S. R. S. , ed. ), (1978), 191-200.
- [6] S. Baize,D. Pannetier,L. Oestereich, Emergence of Zaire Ebola Virus Disease in Guinea -Preliminary Report, New England Journal of Medecine, null (2014).
- [7] S. Baize,D. Pannetier,L. Oestereich, Emergence of Zaire Ebola Virus Disease in Guinea -Preliminary Report, New England Journal of Medecine, null (2014).
- [8] M. Bani-Yabhoub, Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn., 6 (2012): 923-940.
- [9] T. Berge,J. Lubuma,G. M. Moremedi,N. Morris,R. K. Shava, A simple mathematical model for Ebola in Africa, J. Biol. Dyn., 11 (2016): 42-74.
- [10] K. Bibby, Ebola virus persistence in the environment: State of the knowledge and research needs, Environ. Sci. Technol. Lett., 2 (2015): 2-6.
- [11] M. C. J. Bootsma,N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in US cities, PNAS, 104 (2007): 7588-7593.
- [12] V. Capasso,S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the european mediterranean region, Revue dépidémiologié et de santé publiqué, 27 (1979): 121-132.
- [13] C. Castillo-Chavez,K. Barley,D. Bichara,D. Chowell,E. Diaz Herrera,B. Espinoza,V. Moreno,S. Towers,K. E. Yong, Modeling ebola at the mathematical and theoretical biology institute (MTBI), Notices of the AMS, 63 (2016): 366-371.
- [14] C. Castillo-Chavez and H. Thieme, Asymptotically autonomous epidemic models, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds. ), Math. Pop. Dyn. : Analysis of Heterogeneity, Springer, Berlin, 1995, p33.
- [15] N. Chitnis,J. M. Hyman,J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of mathematical model, Bull. Math. Biol., 70 (2008): 1272-1296.
- [16] G. Chowell, The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J. Theor. Biol., 229 (2004): 119-126.
- [17] C. T. Codeço, Endemic and epidemic dynamic of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001): p1.
- [18] M.-A. de La Vega,D. Stein,G. P. Kobinger, Ebolavirus evolution: Past and present, PLoS Pathog, 11 (2015): e1005221.
- [19] O. Diekmann,J. A. P. Heesterbeek,M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010): 873-885.
- [20] P. van den Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 29-48.
- [21] A. d'Onofrio,P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Theor. Biol., 256 (2009): 473-478.
- [22] M. Eichner,S. F. Dowell,N. Firese, Incubation period of Ebola Hemorrhagic Virus subtype Zaire, Osong Public Health and Research Perspectives, 2 (2011): 3-7.
- [23] B. Espinoza, V. Moreno, D. Bichara and C. Castillo-Chavez, Assessing the Efficiency of Cordon Sanitaire as a Control Strategy of Ebola, arXiv: 1510.07415v1 [q-bio. PE] 26 Oct 2015.
- [24] F. O. Fasina, A. Shittu, D. Lazarus, O. Tomori, L. Simonsen, C. Viboud and G. Chowell, Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014 Eurosurveill, 19 (2014), 20920, Available online: https://www.ncbi.nlm.nih.gov/pubmed/25323076.
- [25] H. Feldmann, Ebola virus ecology: A continuing mystery, Trends Microbiol, 12 (2004): 433437.
- [26] A. Groseth,H. Feldmann,J. E. Strong, The ecology of ebola virus, TRENDS in Microbiology, 15 (2007): 408-416.
- [27] J. K. Hale, null, Ordinary Differential Equations, Pure and Applied Mathematics, , John Wiley & Sons, New York, 1969.
- [28] A. M. Henao-Restrepo, Efficacy and effectiveness of an rVSV-vectored vaccine expressing Ebola surface glycoprotein: interim results from the Guinea ring vaccination cluster-randomised trial, The Lancet, 386 (1996): 857-866.
- [29] H. W. Hethcote,H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985): 205-227.
- [30] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988): 1-53.
- [31] B. Ivorra,D. Ngom,A. M. Ramos, Be-CoDiS: A mathematical model to predict the risk of human diseases spread between countries-validation and application to the 2014-2015 ebola virus disease epidemic, Bull. Math. Biol., 77 (2015): 1668-1704.
- [32] M. H. Kuniholm, Bat exposure is a risk factor for Ebola virus infection. In Filoviruses: Recent Advances and Future Challenges: An ICID Global Symposium, 2006.
- [33] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems Monographs and Textbooks in Pure and Applied Mathematics, 125. Marcel Dekker, Inc. , New York, 1989.
- [34] J. P. LaSalle, The Stability of Dynamical Systems Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
- [35] J. Legrand,R. F. Grais,P. Y. Boelle,A. J. Valleron,A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007): 610-621.
- [36] P. E. Lekone,B. F. Finkenstädt, Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006): 1170-1177.
- [37] E. M. Leroy, Fruit bats as reservoirs of Ebola virus, Nature, 438 (2005): 575-576.
- [38] E. M. Leroy, Multiple Ebola virus transmission events and rapid decline of central African wildlife, Science, 303 (2004): 387-390.
- [39] M. Y. Li,J. R. Graef,L. Wang,J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999): 191-213.
- [40] M. Y. Li,J. S. Muldowney, A geometrical approach to global-stability problems, SIAM J. Appl. Anal., 27 (1996): 1070-1083.
- [41] P. Manfredi,A. d'Onofrio, null, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, , Springer, 2013.
- [42] M. L. Mann Manyombe,J. Mbang,J. Lubuma,B. Tsanou, Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers, Math. Biosci. Eng, 13 (2016): 813-840.
- [43] S. Marino,I. B. Hogue,C. J. Ray,D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol, 254 (2008): 178-196.
- [44] D. Ndanguza, Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo, Afr. Mat., 24 (2013): 55-68.
- [45] T. J. Oähea, Bat Flight and Zoonotic Viruses, Emerging Infectious Diseases, null (2014).
- [46] T. J. Piercy, The survival of filoviruses in liquids, on solid substrates and in a dynamic aerosol, J. Appl. Microbiol., 109 (2010): 1531-1539.
- [47] X. Pourrut, Spatial and temporal patterns of Zaire Ebola virus antibody prevalence in the possible reservoir bat species, J. Infect. Dis., 15 (2007): 176-183.
- [48] H. L. Smith, Systems of ordinary differential equations which generate an order preserving flow, A survey of results, SIAM Rev., 30 (1988): 87-113.
- [49] J. P. Tian,J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011): 31-41.
- [50] The Centers for Disease Control and Prevention, 2014-2016 Ebola outbreak in West Africa, https://www.cdc.gov/vhf/ebola/outbreaks/2014-west-africa/index.html (Page last reviewed, October 21,2016).
- [51] The Centers for Disease Control and Prevention, https://www.cdc.gov/vhf/ebola, (Page last reviewed, June 22,2016).
- [52] S. Towers, O. Patterson-Lomba and C. Castillo-Chavez, Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak PLOS Currents Outbreaks, Sept 18,2014.
- [53] B. Tsanou,S. Bowong,J. Lubuma,J. Mbang, Assessment the impact of the environmental contamination on the transmission of Ebola Virus Disease (EVD), J. Appl. Math. Comput., null (2016): 1-39.
- [54] M. Vidyasagar, Decomposition techniques for large-scale systems with non-additive interactions: stability and stabilizability, IEEE Trans. Autom. Control., 25 (1980): 773-779.
- [55] WHO, Ebola Response Roadmap Situation Report, 1 October 2014, http://apps.who.int/iris/bitstream/10665/135600/1/roadmapsitrep_1Oct2014_eng.pdf.
- [56] WHO, Ebola virus disease. Fact sheet N^{o} 103, Updated January 2016, http://www.who.int/mediacentre/factsheets/fs103/en/
- [57] WHO Ebola Response Team, Ebola Virus Disease in West Africa -The First 9 Months of the Epidemic and Forward Projections, N. Engl. J. Med, null (2014).
- [58] WHO, Unprecedented number of medical staff infected with Ebola, http://www.who.int/mediacentre/news/ebola/25-august-2014/en/.
- [59] R. E. Wilson,V. Capasso, Analysis of a reaction-diffusion system modeling manenvironment-man epidemics, SIAM J. Appl. Math., 57 (1997): 327-346.
- [60] D. Youkee et al. , Assessment of environmental contamination and environmental decontamination practices within an Ebola holding unit, Freetown, Sierra Leone. PLOS ONE, December 1,2015.
- [61] J. Zhang,Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Math. Biosci., 185 (2003): 15-32.
This article has been cited by:
- 1. T. Berge, A. J. Ouemba Tassé, H. M. Tenkam, J. Lubuma, Mathematical modeling of contact tracing as a control strategy of Ebola virus disease, International Journal of Biomathematics, 2018, 1850093, 10.1142/S1793524518500936
- 2. Leontine Nkague Nkamba, Thomas Timothee Manga, Franklin Agouanet, Martin Luther Mann Manyombe, Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis, Journal of Biological Dynamics, 2019, 13, 1, 26, 10.1080/17513758.2018.1563218
- 3. Felix B. He, Krister Melén, Laura Kakkola, Ilkka Julkunen, , Re-Emerging Filovirus Diseases [Working Title], 2019, 10.5772/intechopen.86749
- 4. Gregory N. Price, Does Productivity in the Formal Food Sector Drive Human Ebola Virus Infections in Sub‐Saharan Africa?, African Development Review, 2019, 31, 2, 167, 10.1111/1467-8268.12375
- 5. A. Mhlanga, Dynamical analysis and control strategies in modelling Ebola virus disease, Advances in Difference Equations, 2019, 2019, 1, 10.1186/s13662-019-2392-x
Reader Comments
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *