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February  2018, 15(1): 21-56. doi: 10.3934/mbe.2018002

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

* Corresponding author: Tsanou Berge

1IRD UMI 209 UMMISCO, University of Yaounde I, P.O. Box 337 Yaounde and LIRIMAGRIMCAPE Team Project, University of Yaounde I, P.O. Box 812 Yaounde, Cameroon

Received  September 14, 2016 Accepted  March 31, 2017 Published  May 2017

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.

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 & Engineering, 2018, 15 (1) : 21-56. doi: 10.3934/mbe.2018002
References:
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S. Anita and 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. doi: 10.1002/mma.1267. Google Scholar

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M. Bani-Yabhoub, Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn., 6 (2012), 923-940. Google Scholar

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T. BergeJ. LubumaG. M. MoremediN. Morris and R. K. Shava, A simple mathematical model for Ebola in Africa, J. Biol. Dyn., 11 (2016), 42-74. Google Scholar

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K. Bibby, Ebola virus persistence in the environment: State of the knowledge and research needs, Environ. Sci. Technol. Lett., 2 (2015), 2-6. doi: 10.1021/ez5003715. Google Scholar

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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. doi: 10.2807/1560-7917.ES2014.19.40.20920. Google Scholar

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H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205-227. doi: 10.1016/0025-5564(85)90038-0. Google Scholar

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B. IvorraD. Ngom and 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. doi: 10.1007/s11538-015-0100-x. Google Scholar

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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.Google Scholar

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P. E. Lekone and B. F. Finkenstädt, Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006), 1170-1177. doi: 10.1111/j.1541-0420.2006.00609.x. Google Scholar

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E. M. Leroy, Fruit bats as reservoirs of Ebola virus, Nature, 438 (2005), 575-576. Google Scholar

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E. M. Leroy, Multiple Ebola virus transmission events and rapid decline of central African wildlife, Science, 303 (2004), 387-390. Google Scholar

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M. Y. LiJ. R. GraefL. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar

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M. L. Mann ManyombeJ. MbangJ. Lubuma and B. Tsanou, Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers, Math. Biosci. Eng, 13 (2016), 813-840. doi: 10.3934/mbe.2016019. Google Scholar

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S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol, 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

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D. Ndanguza, Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo, Afr. Mat., 24 (2013), 55-68. doi: 10.1007/s13370-011-0039-5. Google Scholar

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T. J. Oähea, Bat Flight and Zoonotic Viruses, Emerging Infectious Diseases, (2014). Google Scholar

[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. Google Scholar

[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. Google Scholar

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H. L. Smith, Systems of ordinary differential equations which generate an order preserving flow, A survey of results, SIAM Rev., 30 (1988), 87-113. doi: 10.1137/1030003. Google Scholar

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show all references

References:
[1]

C. Althaus, Estimating the reproduction number of Ebola (EBOV) during outbreak in West Africa, PLOS Currents, 2014.Google Scholar

[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, England, 1991. Google Scholar
[3]

S. Anita and 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. doi: 10.1002/mma.1267. Google Scholar

[4]

S. Anita and 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. doi: 10.1016/j.nonrwa.2011.08.012. Google Scholar

[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.Google Scholar

[6]

S. BaizeD. Pannetier and L. Oestereich, Emergence of Zaire Ebola Virus Disease in Guinea -Preliminary Report, New England Journal of Medecine, (2014). Google Scholar

[7]

S. BaizeD. Pannetier and L. Oestereich, Emergence of Zaire Ebola Virus Disease in Guinea -Preliminary Report, New England Journal of Medecine, (2014). Google Scholar

[8]

M. Bani-Yabhoub, Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn., 6 (2012), 923-940. Google Scholar

[9]

T. BergeJ. LubumaG. M. MoremediN. Morris and R. K. Shava, A simple mathematical model for Ebola in Africa, J. Biol. Dyn., 11 (2016), 42-74. Google Scholar

[10]

K. Bibby, Ebola virus persistence in the environment: State of the knowledge and research needs, Environ. Sci. Technol. Lett., 2 (2015), 2-6. doi: 10.1021/ez5003715. Google Scholar

[11]

M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in US cities, PNAS, 104 (2007), 7588-7593. Google Scholar

[12]

V. Capasso and 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. Google Scholar

[13]

C. Castillo-ChavezK. BarleyD. BicharaD. ChowellE. Diaz HerreraB. EspinozaV. MorenoS. Towers and K. E. Yong, Modeling ebola at the mathematical and theoretical biology institute (MTBI), Notices of the AMS, 63 (2016), 366-371. doi: 10.1090/noti1364. Google Scholar

[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.Google Scholar

[15]

N. ChitnisJ. M. Hyman and 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. doi: 10.1007/s11538-008-9299-0. Google Scholar

[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. doi: 10.1016/j.jtbi.2004.03.006. Google Scholar

[17]

C. T. Codeço, Endemic and epidemic dynamic of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001), p1. Google Scholar

[18]

M.-A. de La VegaD. Stein and G. P. Kobinger, Ebolavirus evolution: Past and present, PLoS Pathog, 11 (2015), e1005221. doi: 10.1371/journal. Google Scholar

[19]

O. DiekmannJ. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873-885. doi: 10.1098/rsif.2009.0386. Google Scholar

[20]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[21]

A. d'Onofrio and 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. doi: 10.1016/j.jtbi.2008.10.005. Google Scholar

[22]

M. EichnerS. F. Dowell and N. Firese, Incubation period of Ebola Hemorrhagic Virus subtype Zaire, Osong Public Health and Research Perspectives, 2 (2011), 3-7. doi: 10.1016/j.phrp.2011.04.001. Google Scholar

[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.Google Scholar

[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. doi: 10.2807/1560-7917.ES2014.19.40.20920. Google Scholar

[25]

H. Feldmann, Ebola virus ecology: A continuing mystery, Trends Microbiol, 12 (2004), 433437. Google Scholar

[26]

A. GrosethH. Feldmann and J. E. Strong, The ecology of ebola virus, TRENDS in Microbiology, 15 (2007), 408-416. doi: 10.1016/j.tim.2007.08.001. Google Scholar

[27] J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, John Wiley & Sons, New York, 1969. Google Scholar
[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. Google Scholar

[29]

H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205-227. doi: 10.1016/0025-5564(85)90038-0. Google Scholar

[30]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53. doi: 10.1515/crll.1988.383.1. Google Scholar

[31]

B. IvorraD. Ngom and 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. doi: 10.1007/s11538-015-0100-x. Google Scholar

[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.Google Scholar

[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. Google Scholar

[34]

J. P. LaSalle, The Stability of Dynamical Systems Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. Google Scholar

[35]

J. LegrandR. F. GraisP. Y. BoelleA. J. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621. Google Scholar

[36]

P. E. Lekone and B. F. Finkenstädt, Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006), 1170-1177. doi: 10.1111/j.1541-0420.2006.00609.x. Google Scholar

[37]

E. M. Leroy, Fruit bats as reservoirs of Ebola virus, Nature, 438 (2005), 575-576. Google Scholar

[38]

E. M. Leroy, Multiple Ebola virus transmission events and rapid decline of central African wildlife, Science, 303 (2004), 387-390. Google Scholar

[39]

M. Y. LiJ. R. GraefL. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar

[40]

M. Y. Li and J. S. Muldowney, A geometrical approach to global-stability problems, SIAM J. Appl. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449. Google Scholar

[41] P. Manfredi and A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer, 2013. doi: 10.1007/978-1-4614-5474-8. Google Scholar
[42]

M. L. Mann ManyombeJ. MbangJ. Lubuma and B. Tsanou, Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers, Math. Biosci. Eng, 13 (2016), 813-840. doi: 10.3934/mbe.2016019. Google Scholar

[43]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol, 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[44]

D. Ndanguza, Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo, Afr. Mat., 24 (2013), 55-68. doi: 10.1007/s13370-011-0039-5. Google Scholar

[45]

T. J. Oähea, Bat Flight and Zoonotic Viruses, Emerging Infectious Diseases, (2014). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/1030003. Google Scholar

[49]

J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001. Google Scholar

[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).Google Scholar

[51]

The Centers for Disease Control and Prevention, https://www.cdc.gov/vhf/ebola, (Page last reviewed, June 22,2016).Google Scholar

[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. doi: 10.1371/currents.outbreaks.9e4c4294ec8ce1adad283172b16bc908. Google Scholar

[53]

B. TsanouS. BowongJ. Lubuma and J. Mbang, Assessment the impact of the environmental contamination on the transmission of Ebola Virus Disease (EVD), J. Appl. Math. Comput., (2016), 1-39. doi: 10.1007/s12190-016-1033-8. Google Scholar

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Figure 1.  Ebola Virus Disease transmission flow diagram
Figure 2.  GAS of the full model disease-free equilibrium when $\Lambda_h=500$, $\mu_h=0.033$, $\mu_a=0.04$, $\mu_b=0.05$, $\mu_v=0.85$, $\tau_h=4$, $\delta_a=0.05$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{hh}=0.006$, $\beta_{hv}=\beta_{bv}= \beta_{av}=\beta_{ab}=0.0005$, $\beta_{hb}=\beta_{ha}=10^{-8}$, $\beta_{bb}= \beta_{aa}=0.0002$ (so that $\mathcal R_0=0.8<1$)
Figure 3.  Stability of the full model endemic equilibrium when $\Lambda_h=100$, $\mu_h=0.033$, $\mu_a=0.04$, $\mu_b=0.05$, $\mu_v=0.85$, $\tau_h=4$, $\delta_a=0.05$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{hh}=0.3$, $\beta_{hv}=0.5$, $\beta_{bv}=0.5$, $\beta_{bb}=0.0005$, $\beta_{hb}=\beta_{ha}=10^{-8}$, $\beta_{ab}=0.005$, $\beta_{aa}=0.02$, $\beta_{av}=0.5$ (so that $\mathcal R_0=2.0024>1$)
Figure 4.  Stability of $(\overline{P}_h, P^0_a, P^0_b)$ when $\Lambda_h=10$, $\Lambda_a=3$, $\Lambda_b=1.5$, $\mu_h=0.033$, $\mu_a=0.2$, $\mu_b=0.29$, $\delta_a=0.05$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{hh}=0.3$, $\beta_{bb}=0.05$, $\beta_{hb}=\beta_{ha}=10^{-8}$, $\beta_{ab}=0.05$, $\beta_{aa}=0.2$ (so that $\mathcal R_{0, h}=1.7269$, $\mathcal R_{0, a}=0.8074$, $\mathcal R_{0, b}=0.8918$)
Figure 5.  Stability of $(P^0_h, P^0_a, P^0_b)$ when $\Lambda_h=10$, $\Lambda_a=3$, $\Lambda_b=1.5$, $\mu_h=0.033$, $\mu_a=0.2$, $\mu_b=0.29$, $\delta_a=0.05$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{hh}=0.03$, $\beta_{bb}=0.05$, $\beta_{hb}=\beta_{ha}=10^{-8}$, $\beta_{ab}=0.05$, $\beta_{aa}=0.2$ (so that $\mathcal R_{0, h}=0.1727$, $\mathcal R_{0, a}=0.8074$, $\mathcal R_{0, b}=0.8918$)
Figure 6.  Stability of $(E_h^{**}, \overline{P}_a, P^0_b)$ when $\Lambda_h=10$, $\Lambda_a=10$, $\Lambda_b=1.5$, $\mu_h=0.033$, $\mu_a=0.04$, $\mu_b=0.29$, $\delta_a=0.05$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{hh}=0.3$, $\beta_{bb}=0.05$, $\beta_{hb}=\beta_{ha}=10^{-8}$, $\beta_{ab}=0.05$, $\beta_{aa}=0.2$ (so that $\mathcal R_{0, h}=1.7269$, $\mathcal R_{0, a}=2.2429$, $\mathcal R_{0, b}=0.8918$)
Figure 7.  Stability of $(E_h^{***}, \widehat{E}_a, \overline{P}_b)$ when $\Lambda_h=10$, $\Lambda_a=10$, $\Lambda_b=10$, $\mu_h=0.033$, $\mu_a=0.2$, $\mu_b=0.29$, $\delta_a=0.05$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{hh}=0.3$, $\beta_{bb}=0.05$, $\beta_{hb}=\beta_{ha}=10^{-8}$, $\beta_{ab}=0.05$, $\beta_{aa}=0.2$ (so that $\mathcal R_{0, h}=1.7269$, $\mathcal R_{0, a}=0.8074$, $\mathcal R_{0, b}=3.1250$)
Figure 8.  Infected population with and without environment when $\Lambda_h=400$, $\Lambda_a=100$, $\Lambda_b=80$, $\mu_h=0.033$, $\mu_a=0.04$, $\mu_b=0.09$, $\mu_v=0.85$, $\tau_h=4$, $\delta_a=0.5$, $\alpha_h=\alpha_a=\alpha_b=0.95$, $f=0.50$, $\beta_{aa}=0.5$, $\beta_{bb}=\beta_{ab}=0.0005$, $\beta_{hb}=\beta_{ha}=10^{-8}$. (A) $\beta_{hh}=0.3$, $\beta_{hv}=\beta_{bv}=\beta_{av}=0.25$. (B) $\beta_{hh}=0.2$, $\beta_{hv}=\beta_{bv}=\beta_{av}=0.4$
Figure 9.  (A) Infected population with and without bats when $\Lambda_a=100$, $\mu_a=0.04$, $\delta_a=0.5$, $\nu_a=0.04$, $\alpha_a=0.95$, $\beta_{aa}=0.5$, $\beta_{ha}=10^{-8}$, $\beta_{av}=0.4$. (B) Infected population with and without animals when $\Lambda_b=80$, $\mu_b=0.09$, $\nu_b=0.09$, $\alpha_b=0.95$, $\beta_{hb}=10^{-8}$, $\beta_{bb}=0.0005$, $\beta_{bv}=0.4$. With $\Lambda_h=400$, $\mu_h=0.033$, $\mu_v=0.85$, $\tau_h=4$, $\alpha_h=0.95$, $f=0.50$, $\beta_{hh}=0.3$, $\beta_{hv}=0.4$
Table 1.  Routes of transmission for index case in some known Ebola virus outbreaks
Year Country Species Starting dateSource of infection
1976 DRC Zaire September Unknown. Index case was a mission school teacher.
1976 Sudan Sudan June Worker in a cotton factory.
Evidence of bats at site.
1977 DRC Zaire June Unknown (retrospective).
1979 Sudan Sudan July Worker in cotton factory.
Evidence of bats at site.
1994 Gabon Zaire December Gold-mining camps.
Evidence of bats at site.
1994 Ivory Coast Ivory Coast November Scientist performing autopsy on a dead wild chimpanzee.
1995 Liberia Ivory Coast December Unknown. Refugee from civil war.
1995 DRC Zaire January Index case worked in a forest adjoining the city.
1996 Gabon Zaire January People involved in the butchering of a dead chimpanzee.
1996-1997 Gabon Zaire July Index case was a hunter living in a forest camp.
2000-2001 Uganda Sudan September Unknown.
2001-2002 Gabon Zaire October Contact with dead or butchered apes or other wildlife.
2001-2002 DRC Zaire October Contact with dead or butchered apes or other wildlife.
2002-2003 DRC Zaire December Contact with dead or butchered apes or other wildlife.
2003 DRC Zaire November Contact with dead or butchered apes or other wildlife.
2004 Sudan Sudan May Unknown.
2005 DRC Zaire April unknown.
2007 DRC Zaire December Contact with dead or butchered apes or other wildlife.
2007 Uganda Bundibugyo December Unknown.
2008 DRC Zaire December Index case was a village chief and a hunter.
2012 Uganda Bundibugyo June Index case was a secondary school teacher in Ibanda district.
2012 DRC Zaire June Index case was a hunter living in a forest camp.
2013-2015 Guinea Zaire December Contact with bats or fruits contaminated by bat droppings.
2014-2015 Liberia Zaire April Index case was transported from Guinea.
2014-2015 Sierra Leone Zaire April A traditional healer, treating Ebola patients from Guinea.
2014 DRC Zaire August Pregnant women who butchered a bush animal.
Year Country Species Starting dateSource of infection
1976 DRC Zaire September Unknown. Index case was a mission school teacher.
1976 Sudan Sudan June Worker in a cotton factory.
Evidence of bats at site.
1977 DRC Zaire June Unknown (retrospective).
1979 Sudan Sudan July Worker in cotton factory.
Evidence of bats at site.
1994 Gabon Zaire December Gold-mining camps.
Evidence of bats at site.
1994 Ivory Coast Ivory Coast November Scientist performing autopsy on a dead wild chimpanzee.
1995 Liberia Ivory Coast December Unknown. Refugee from civil war.
1995 DRC Zaire January Index case worked in a forest adjoining the city.
1996 Gabon Zaire January People involved in the butchering of a dead chimpanzee.
1996-1997 Gabon Zaire July Index case was a hunter living in a forest camp.
2000-2001 Uganda Sudan September Unknown.
2001-2002 Gabon Zaire October Contact with dead or butchered apes or other wildlife.
2001-2002 DRC Zaire October Contact with dead or butchered apes or other wildlife.
2002-2003 DRC Zaire December Contact with dead or butchered apes or other wildlife.
2003 DRC Zaire November Contact with dead or butchered apes or other wildlife.
2004 Sudan Sudan May Unknown.
2005 DRC Zaire April unknown.
2007 DRC Zaire December Contact with dead or butchered apes or other wildlife.
2007 Uganda Bundibugyo December Unknown.
2008 DRC Zaire December Index case was a village chief and a hunter.
2012 Uganda Bundibugyo June Index case was a secondary school teacher in Ibanda district.
2012 DRC Zaire June Index case was a hunter living in a forest camp.
2013-2015 Guinea Zaire December Contact with bats or fruits contaminated by bat droppings.
2014-2015 Liberia Zaire April Index case was transported from Guinea.
2014-2015 Sierra Leone Zaire April A traditional healer, treating Ebola patients from Guinea.
2014 DRC Zaire August Pregnant women who butchered a bush animal.
Table 2.  Model constant parameters and their biological interpretation
Symbols Biological interpretations
$\Lambda_{h}, \Lambda_a, \Lambda_b$ Recruitment rate of susceptible humans, animals and bats, respectively.
$\mu_{h}, \mu_{a}, \mu_{b}$ Natural mortality rate of humans, animals and bats, respectively.
$\nu_h$ Virulence of Ebola virus in the corpse of the dead humans.
$\tau_h$ Mean duration of time that elapse after death before a human cadaver is completely buried.
$\xi_{h}= 1/\tau_h$ Modification parameter of infectiousness due to dead human individuals.
$\tau_a$ Mean duration of time that elapse after death before an animal's cadaver is completely cleared out.
$\xi_{a} =1/\tau_a$ Modification parameter of infectiousness due to dead animals individuals.
$\nu_a$ Virulence of Ebola virus in the corpse of dead animals.
$\omega $ Incubation rate of human individuals.
$\gamma$ Removal rate from infectious compartment due to either to disease induced death, or by recovery.
$\delta_a$ Death rate of infected animals.
$\alpha_h, \alpha_a, \alpha_b$ Shedding rates of Ebola virus in the environment by humans, animals and bats, respectively.
$r_h$ Mean duration of time that elapse before the complete clearance of Ebola virus in humans.
$\theta_h = 1/r_h$ Modification parameter of contact rate of recovered humans (sexual activity of recovered).
in the semen/breast milk of a recovered man/woman.
$f$ Proortion of removed human individuals who die due EVD (i.e. case fatality rate).
$K$ Virus 50 % infectious dose, sufficient to cause EVD.
$\beta_{hh}$ Contact rate between susceptible humans and infected humans.
$\beta_{hb}$ Contact rate between susceptible humans and bats.
$\beta_{hv}$ Contact rate between susceptible humans and Ebola viruses.
$\beta_{ha}$ Contact rate between susceptible humans and infected animals.
$\beta_{bb}$ Contact rate between susceptible bats and infectious bats.
$\beta_{ab}$ Contact rate between susceptible animals and infectious bats.
$\beta_{bv}$ Contact rate between susceptible bats and and Ebola viruses.
$\beta_{aa}$ Contact rate between susceptible and infected animals.
$\beta_{av}$ Contact rate between susceptible animals and Ebola viruses.
Symbols Biological interpretations
$\Lambda_{h}, \Lambda_a, \Lambda_b$ Recruitment rate of susceptible humans, animals and bats, respectively.
$\mu_{h}, \mu_{a}, \mu_{b}$ Natural mortality rate of humans, animals and bats, respectively.
$\nu_h$ Virulence of Ebola virus in the corpse of the dead humans.
$\tau_h$ Mean duration of time that elapse after death before a human cadaver is completely buried.
$\xi_{h}= 1/\tau_h$ Modification parameter of infectiousness due to dead human individuals.
$\tau_a$ Mean duration of time that elapse after death before an animal's cadaver is completely cleared out.
$\xi_{a} =1/\tau_a$ Modification parameter of infectiousness due to dead animals individuals.
$\nu_a$ Virulence of Ebola virus in the corpse of dead animals.
$\omega $ Incubation rate of human individuals.
$\gamma$ Removal rate from infectious compartment due to either to disease induced death, or by recovery.
$\delta_a$ Death rate of infected animals.
$\alpha_h, \alpha_a, \alpha_b$ Shedding rates of Ebola virus in the environment by humans, animals and bats, respectively.
$r_h$ Mean duration of time that elapse before the complete clearance of Ebola virus in humans.
$\theta_h = 1/r_h$ Modification parameter of contact rate of recovered humans (sexual activity of recovered).
in the semen/breast milk of a recovered man/woman.
$f$ Proortion of removed human individuals who die due EVD (i.e. case fatality rate).
$K$ Virus 50 % infectious dose, sufficient to cause EVD.
$\beta_{hh}$ Contact rate between susceptible humans and infected humans.
$\beta_{hb}$ Contact rate between susceptible humans and bats.
$\beta_{hv}$ Contact rate between susceptible humans and Ebola viruses.
$\beta_{ha}$ Contact rate between susceptible humans and infected animals.
$\beta_{bb}$ Contact rate between susceptible bats and infectious bats.
$\beta_{ab}$ Contact rate between susceptible animals and infectious bats.
$\beta_{bv}$ Contact rate between susceptible bats and and Ebola viruses.
$\beta_{aa}$ Contact rate between susceptible and infected animals.
$\beta_{av}$ Contact rate between susceptible animals and Ebola viruses.
Table 3.  Existence, conditions for existence and stability of equilibria
Equilibria Conditions of existence Stability
$\left(P^0_h, P^0_a, P^0_b \right)$ $ \mathcal R_{0, h}>1, \mathcal R_{0, a}\leq 1, \mathcal R_{0, b} \leq 1 $ GAS
$\left(\overline{E}_h, P^0_a, P^0_b \right)$ $ \mathcal R_{0, h} \leq 1, \mathcal R_{0, a}\leq 1, \mathcal R_{0, b} \leq 1 $ GAS
$\left(E^{**}_h, \overline{P}_a, P^0_b \right)$ $ \mathcal R_{0, a}>1, \mathcal R_{0, b} \leq 1 $ GAS
$\left(E^{***}_h, \widehat{E}_a, \overline{P}_b\right)$ $ \mathcal R_{0, a}\leq 1, \mathcal R_{0, b} > 1 $ GAS
Equilibria Conditions of existence Stability
$\left(P^0_h, P^0_a, P^0_b \right)$ $ \mathcal R_{0, h}>1, \mathcal R_{0, a}\leq 1, \mathcal R_{0, b} \leq 1 $ GAS
$\left(\overline{E}_h, P^0_a, P^0_b \right)$ $ \mathcal R_{0, h} \leq 1, \mathcal R_{0, a}\leq 1, \mathcal R_{0, b} \leq 1 $ GAS
$\left(E^{**}_h, \overline{P}_a, P^0_b \right)$ $ \mathcal R_{0, a}>1, \mathcal R_{0, b} \leq 1 $ GAS
$\left(E^{***}_h, \widehat{E}_a, \overline{P}_b\right)$ $ \mathcal R_{0, a}\leq 1, \mathcal R_{0, b} > 1 $ GAS
Table 4.  PRCCs of full model's parameters
Parameters $E_h$ $I_h$ $V$ $I_a$ $I_b$
$\Lambda_{h}$ $0.7624^{**}$0.23430.19220.01240.0172
$\Lambda_a$-0.18220.20050.1610 $0.8914^{**}$0.0180
$\Lambda_b$-0.3116 $0.4407^*$0.3008 $-0.5346^{**}$0.0132
$\mu_{h}$ $-0.8657^{**}$ $-0.8588^{**}$ $-0.9438^{**}$-0.03410.0329
$\mu_{a}$0.1060-0.1786-0.1134 $-0.4854^*$-0.0148
$\mu_{b}$ $0.5677^{**}$ $-0.6054^{**}$ $-0.4335^*$ $0.7106^{**}$ $0.8966^{**}$
$\mu_{v}$-0.0143-0.0493-0.0453-0.05300.0202
$\xi_{h}$0.0030-0.00990.284-0.0491-0.0250
$\xi_{a}$-0.01070.06300.0010-0.13810.0356
$\nu_{h}$-0.02180.05720.0200-0.0509-0.0518
$\nu_{a}$-0.12130.11490.0410-0.15300.0256
$\omega $-0.1299-0.2465 $0.5385^{**}$0.0513-0.0613
$\gamma$0.0463-0.06230.17350.01080.0092
$\delta_a$0.0239-0.0450-0.0185-0.325-0.0044
$\alpha_h$0.01430.0490-0.01250.00670.0154
$\alpha_a$0.0430.01770.1003-0.0653-0.0434
$\alpha_b$0.0078-0.0041-0.0254-0.0113-0.0506
$\theta_h$0.01330.00730.0845-0.0410-0.0025
$f$0.0142-0.0065 $-0.4980^{*}$-0.0320-0.0106
$K$0.0375-0.05810.01410.00030.0263
$\beta_{hh}$-0.26820.32050.12170.02200.0038
$\beta_{hb}$-0.3700 $0.5287^{**}$0.3747-0.01140.0125
$\beta_{hv}$0.07850.01060.0022-0.0824-0.0129
$\beta_{ha}$-0.18160.23990.1559-0.0395-0.0448
$\beta_{bb}$0.01960.07570.1389-0.0976 $-0.8883^{**}$
$\beta_{ab}$-0.02420.09840.0080 $-0.6039^{**}$-0.0030
$\beta_{bv}$-0.0214-0.0310-0.0280-0.00710.0391
$\beta_{aa}$-0.01450.1266-0.0036 $-0.4099^{*}$-0.0596
$\beta_{av}$-0.02140.01500.07180.07370.0339
Parameters $E_h$ $I_h$ $V$ $I_a$ $I_b$
$\Lambda_{h}$ $0.7624^{**}$0.23430.19220.01240.0172
$\Lambda_a$-0.18220.20050.1610 $0.8914^{**}$0.0180
$\Lambda_b$-0.3116 $0.4407^*$0.3008 $-0.5346^{**}$0.0132
$\mu_{h}$ $-0.8657^{**}$ $-0.8588^{**}$ $-0.9438^{**}$-0.03410.0329
$\mu_{a}$0.1060-0.1786-0.1134 $-0.4854^*$-0.0148
$\mu_{b}$ $0.5677^{**}$ $-0.6054^{**}$ $-0.4335^*$ $0.7106^{**}$ $0.8966^{**}$
$\mu_{v}$-0.0143-0.0493-0.0453-0.05300.0202
$\xi_{h}$0.0030-0.00990.284-0.0491-0.0250
$\xi_{a}$-0.01070.06300.0010-0.13810.0356
$\nu_{h}$-0.02180.05720.0200-0.0509-0.0518
$\nu_{a}$-0.12130.11490.0410-0.15300.0256
$\omega $-0.1299-0.2465 $0.5385^{**}$0.0513-0.0613
$\gamma$0.0463-0.06230.17350.01080.0092
$\delta_a$0.0239-0.0450-0.0185-0.325-0.0044
$\alpha_h$0.01430.0490-0.01250.00670.0154
$\alpha_a$0.0430.01770.1003-0.0653-0.0434
$\alpha_b$0.0078-0.0041-0.0254-0.0113-0.0506
$\theta_h$0.01330.00730.0845-0.0410-0.0025
$f$0.0142-0.0065 $-0.4980^{*}$-0.0320-0.0106
$K$0.0375-0.05810.01410.00030.0263
$\beta_{hh}$-0.26820.32050.12170.02200.0038
$\beta_{hb}$-0.3700 $0.5287^{**}$0.3747-0.01140.0125
$\beta_{hv}$0.07850.01060.0022-0.0824-0.0129
$\beta_{ha}$-0.18160.23990.1559-0.0395-0.0448
$\beta_{bb}$0.01960.07570.1389-0.0976 $-0.8883^{**}$
$\beta_{ab}$-0.02420.09840.0080 $-0.6039^{**}$-0.0030
$\beta_{bv}$-0.0214-0.0310-0.0280-0.00710.0391
$\beta_{aa}$-0.01450.1266-0.0036 $-0.4099^{*}$-0.0596
$\beta_{av}$-0.02140.01500.07180.07370.0339
Table 5.  PRCCs of model's parameters without environment
Parameters $E_h$ $I_h$ $I_a$ $I_b$
$\Lambda_{h}$ $0.7897^{**}$0.33410.06020.0406
$\Lambda_a$-0.14120.22060.8767-0.0122
$\Lambda_b$-0.3108 $0.4231^*$ $-0.4185^*$-0.0214
$\mu_{h}$ $-0.8755^{**}$ $-0.8466^{***}$0.01800.0341
$\mu_{a}$0.0936-0.2108 $-0.4814^{*}$-0.0046
$\mu_{b}$ $0.5727^{**}$ $-0.6096^{**}$ $0.7117^{***}$ $0.9040^{***}$
$\xi_{h}$0.00550.0391-0.0337-0.0270
$\xi_{a}$-0.09130.1327-0.0923-0.0041
$\nu_{h}$-0.01830.01840.0608-0.0183
$\nu_{a}$-0.04830.0745-0.0953-0.0253
$\omega $-0.1496-0.22330.01160.0046
$\gamma$0.0124-0.04100.0286-0.0295
$\delta_a$0.0690-0.0647-0.38690.0175
$\theta_h$-0.02210.03080.00990.0539
$f$0.0035-0.00570.00420.0170
$\beta_{hh}$-0.28650.3144-0.0275-0.0105
$\beta_{hb}$-0.3649 $0.4837^{*}$-0.0063-0.0131
$\beta_{ha}$-0.20570.31680.03720.0050
$\beta_{bb}$-0.06860.0757-0.2270 $-0.8988^{***}$
$\beta_{ab}$-0.07190.0684 $-0.5291^{**}$-0.0245
$\beta_{aa}$0.00630.0049-0.29360.0053
Parameters $E_h$ $I_h$ $I_a$ $I_b$
$\Lambda_{h}$ $0.7897^{**}$0.33410.06020.0406
$\Lambda_a$-0.14120.22060.8767-0.0122
$\Lambda_b$-0.3108 $0.4231^*$ $-0.4185^*$-0.0214
$\mu_{h}$ $-0.8755^{**}$ $-0.8466^{***}$0.01800.0341
$\mu_{a}$0.0936-0.2108 $-0.4814^{*}$-0.0046
$\mu_{b}$ $0.5727^{**}$ $-0.6096^{**}$ $0.7117^{***}$ $0.9040^{***}$
$\xi_{h}$0.00550.0391-0.0337-0.0270
$\xi_{a}$-0.09130.1327-0.0923-0.0041
$\nu_{h}$-0.01830.01840.0608-0.0183
$\nu_{a}$-0.04830.0745-0.0953-0.0253
$\omega $-0.1496-0.22330.01160.0046
$\gamma$0.0124-0.04100.0286-0.0295
$\delta_a$0.0690-0.0647-0.38690.0175
$\theta_h$-0.02210.03080.00990.0539
$f$0.0035-0.00570.00420.0170
$\beta_{hh}$-0.28650.3144-0.0275-0.0105
$\beta_{hb}$-0.3649 $0.4837^{*}$-0.0063-0.0131
$\beta_{ha}$-0.20570.31680.03720.0050
$\beta_{bb}$-0.06860.0757-0.2270 $-0.8988^{***}$
$\beta_{ab}$-0.07190.0684 $-0.5291^{**}$-0.0245
$\beta_{aa}$0.00630.0049-0.29360.0053
Table 6.  PRCCs of model's parameters without animals
Parameters $E_h$ $I_h$ $V$ $I_b$
$\Lambda_{h}$ $0.7853^{**}$0.20460.10340.0202
$\Lambda_b$-0.3295 $0.4674^*$0.3423-0.0096
$\mu_{h}$ $-0.8726^{**}$ $-0.8067^{**}$ $-0.9046^{**}$0.0203
$\mu_{b}$ $0.6098^{**}$ $-0.6607^{**}$ $-0.5215^{**}$ $0.8990^{**}$
$\mu_{v}$0.010.00660.0254-0.0085
$\xi_{h}$-0.00470.04700.0421-0.0097
$\nu_{h}$-0.01100.0116-0.00520.0244
$\omega $-0.1750-0.1661 $0.4079^{*}$-0.0014
$\gamma$0.04040.01960.1127-0.0412
$\alpha_h$-0.0375-0.01050.00710.0263
$\alpha_b$0.0091-0.0128-0.01470.0408
$\theta_h$-0.01820.03160.0038-0.0090
$f$0.00370.0187 $-0.4368^{*}$-0.0041
$K$-0.0096-0.01770.0319-0.0294
$\beta_{hh}$-0.26460.30930.21300.0209
$\beta_{hb}$-0.3794 $0.5955^{**}$ $0.4528^*$0.0162
$\beta_{hv}$0.00550.0171-0.0102-0.0538
$\beta_{bb}$-0.08030.05560.0875 $-0.8952^{***}$
$\beta_{bv}$-0.00940.0178-0.01780.0804
Parameters $E_h$ $I_h$ $V$ $I_b$
$\Lambda_{h}$ $0.7853^{**}$0.20460.10340.0202
$\Lambda_b$-0.3295 $0.4674^*$0.3423-0.0096
$\mu_{h}$ $-0.8726^{**}$ $-0.8067^{**}$ $-0.9046^{**}$0.0203
$\mu_{b}$ $0.6098^{**}$ $-0.6607^{**}$ $-0.5215^{**}$ $0.8990^{**}$
$\mu_{v}$0.010.00660.0254-0.0085
$\xi_{h}$-0.00470.04700.0421-0.0097
$\nu_{h}$-0.01100.0116-0.00520.0244
$\omega $-0.1750-0.1661 $0.4079^{*}$-0.0014
$\gamma$0.04040.01960.1127-0.0412
$\alpha_h$-0.0375-0.01050.00710.0263
$\alpha_b$0.0091-0.0128-0.01470.0408
$\theta_h$-0.01820.03160.0038-0.0090
$f$0.00370.0187 $-0.4368^{*}$-0.0041
$K$-0.0096-0.01770.0319-0.0294
$\beta_{hh}$-0.26460.30930.21300.0209
$\beta_{hb}$-0.3794 $0.5955^{**}$ $0.4528^*$0.0162
$\beta_{hv}$0.00550.0171-0.0102-0.0538
$\beta_{bb}$-0.08030.05560.0875 $-0.8952^{***}$
$\beta_{bv}$-0.00940.0178-0.01780.0804
Table 7.  Baseline numerical values for the parameters of system (1)
Parameters Range Values Units Source
$\Lambda_{h}$Variable100 $indiv.day^{-1}$N/A
$\Lambda_a$Variable5 $indiv.day^{-1}$N/A
$\Lambda_b$Variable10 $indiv.day^{-1}$N/A
$\mu_{h}$0-10.33/365 $day^{-1}$[57]
$\mu_{a}$0-10.4/365 $day^{-1}$Assumed
$\mu_{b}$0-10.5/365 $day^{-1}$Assumed
$\mu_{v}$0-10.85/30 $day^{-1}$Assumed [10,46]
$\xi_{h}= 1/\tau_h$0-11/2.5 $day^{-1}$[50,57]
$\tau_h$1-72.5 $day$[50,57]
$\xi_{a} =1/\tau_a$0-11/7 $day^{-1}$Assumed
$\tau_a$1-147 $day$Assumed
$\nu_{h}$1-51.2 $day^{-2}$Assumed
$\nu_{a}$1-51.3 $day^{-2}$Assumed
$\omega $1/2-1/211/21 $day^{-1}$[22,50]
$\gamma$1/7-1/141/14 $day^{-1}$[57]
$\delta_a$0-10.5/365 $day^{-1}$Assumed
$\alpha_h$10-10050 $cells.(ml.day.indiv)^{-1}$[8]
$\alpha_a$20-200100 $cells.(ml.day.indiv)^{-1}$Assumed
$\alpha_b$50-400200 $cells.(ml.day.indiv)^{-1}$Assumed
$\theta_h = 1/r_h$1/81-11/61 $day^{-1}$[50]
$r_h$1-8161 $day$[50]
$f$0.4-0.90.70dimensionless[50,52,57]
$K$ $10^6$-$10^9$$10^6$ $cells.ml^{-1}$[8]
$\beta_{hh}$0-1 $day^{-1}$Variable
$\beta_{hb}$0-1 $day^{-1}$Variable
$\beta_{hv}$0-1 $day^{-1}$Variable
$\beta_{ha}$0-1 $day^{-1}$Variable
$\beta_{bb}$0-1 $day^{-1}$Variable
$\beta_{ab}$0-1 $day^{-1}$Variable
$\beta_{bv}$0-1 $day^{-1}$Variable
$\beta_{aa}$0-1 $day^{-1}$Variable
$\beta_{av}$0-1 $day^{-1}$Variable
Parameters Range Values Units Source
$\Lambda_{h}$Variable100 $indiv.day^{-1}$N/A
$\Lambda_a$Variable5 $indiv.day^{-1}$N/A
$\Lambda_b$Variable10 $indiv.day^{-1}$N/A
$\mu_{h}$0-10.33/365 $day^{-1}$[57]
$\mu_{a}$0-10.4/365 $day^{-1}$Assumed
$\mu_{b}$0-10.5/365 $day^{-1}$Assumed
$\mu_{v}$0-10.85/30 $day^{-1}$Assumed [10,46]
$\xi_{h}= 1/\tau_h$0-11/2.5 $day^{-1}$[50,57]
$\tau_h$1-72.5 $day$[50,57]
$\xi_{a} =1/\tau_a$0-11/7 $day^{-1}$Assumed
$\tau_a$1-147 $day$Assumed
$\nu_{h}$1-51.2 $day^{-2}$Assumed
$\nu_{a}$1-51.3 $day^{-2}$Assumed
$\omega $1/2-1/211/21 $day^{-1}$[22,50]
$\gamma$1/7-1/141/14 $day^{-1}$[57]
$\delta_a$0-10.5/365 $day^{-1}$Assumed
$\alpha_h$10-10050 $cells.(ml.day.indiv)^{-1}$[8]
$\alpha_a$20-200100 $cells.(ml.day.indiv)^{-1}$Assumed
$\alpha_b$50-400200 $cells.(ml.day.indiv)^{-1}$Assumed
$\theta_h = 1/r_h$1/81-11/61 $day^{-1}$[50]
$r_h$1-8161 $day$[50]
$f$0.4-0.90.70dimensionless[50,52,57]
$K$ $10^6$-$10^9$$10^6$ $cells.ml^{-1}$[8]
$\beta_{hh}$0-1 $day^{-1}$Variable
$\beta_{hb}$0-1 $day^{-1}$Variable
$\beta_{hv}$0-1 $day^{-1}$Variable
$\beta_{ha}$0-1 $day^{-1}$Variable
$\beta_{bb}$0-1 $day^{-1}$Variable
$\beta_{ab}$0-1 $day^{-1}$Variable
$\beta_{bv}$0-1 $day^{-1}$Variable
$\beta_{aa}$0-1 $day^{-1}$Variable
$\beta_{av}$0-1 $day^{-1}$Variable
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