Dynamical analysis and control strategies in modelling Ebola virus disease. (English) Zbl 1485.92144
Summary: Ebola virus disease (EVD) is a severe infection with an extremely high fatality rate spread through direct and indirect contacts. Recently, an outbreak of EVD in West Africa brought public attention to this deadly disease. We study the spread of EVD through a two-patch model. We determine the basic reproduction number, the disease-free equilibrium, two boundary equilibria and the endemic equilibrium when the disease persists in the two sub-populations for specific conditions. Further, we introduce time-dependent controls into our proposed model. We analyse the optimal control problem where the control system is a mathematical model for EVD that incorporates educational campaigns. The control functions represent educational campaigns in their respective patches, with one patch having more effective controls than the other. We aim to study how these control measures would be implemented for a certain time period, in order to reduce or eliminate EVD in the respective communities, while minimising the intervention implementation costs. Numerical simulations results are provided to illustrate the dynamics of the disease in the presence of controls.
MSC:
| 92D30 | Epidemiology |
| 92C60 | Medical epidemiology |
| 49N90 | Applications of optimal control and differential games |
Software:
Be-CoDiSReferences:
| [1] | Leroy, E.M., Kumulungui, B., Pourrut, X., Rouquet, P., Hassanin, A., Yaba, P., Delicat, A., Paweska, J.T., Gonzalez, J.P., Swanepoel, R.: Fruit bats as reservoirs of Ebola virus. Nature 438, 575-576 (2015) · doi:10.1038/438575a |
| [2] | Takada, A., Robison, C., Goto, H., Sanchez, A., Murti, K.G., Whitt, M.A., et al.: A system for functional analysis of Ebola virus glycoprotein. Proc. Natl. Acad. Sci. 94(26), 14764-14769 (1997) · doi:10.1073/pnas.94.26.14764 |
| [3] | Wool-Lewis, R.J., Bates, P.: Characterization of Ebola virus entry by using pseudo typed viruses: identification of receptor-deficient cell lines. J. Virol. 72, 3155-3160 (1998) |
| [4] | Lough, S.: Lessons from Ebola bring WHO reforms. CMAJ, Can. Med. Assoc. J. 187(12), E377-E378 (2015) · doi:10.1503/cmaj.109-5125 |
| [5] | WHO: Ebola response road-map situation report. World Health Organization. http://www.who.int/csr/disease/Ebola/situation-reports/en/ (2015). Accessed 5 Feb 2015 |
| [6] | Lewnard, J.A., Ndeffo Mbah, M.L., Alfaro-Murillo, J.A., Altice, F.L., Bawo, L., Nyenswah, T.G., et al.: Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysis. Lancet Infect. Dis. 14(12), 1189-1195 (2014) · doi:10.1016/S1473-3099(14)70995-8 |
| [7] | NPR.org: Ebola in a conflict zone. Retrieved 2 August 2018 |
| [8] | Relief Web: Congo Ebola outbreak compounds already dire humanitarian crisis. Retrieved 3 August 2018 |
| [9] | Reuters: Editorial, Reuters (2018-09-04). Rebels ambush South African peacekeepers in Congo Ebola zone. Retrieved 4 September 2018 |
| [10] | VOA: Rebel attack in Congo Ebola zone kills at least 14 civilians. Retrieved 23 September 2018 |
| [11] | Peters, C., Peters, J.: An introduction to Ebola: the virus and the disease. J. Infect. Dis. 179, 9-16 (1999). https://doi.org/10.1086/514322 · doi:10.1086/514322 |
| [12] | CDC: CDC report to Ebola virus disease 2014. Technical report (2014) |
| [13] | Bibby, K., Casson, L.W., Stachler, E., Haas, C.N.: Ebola virus persistence in the environment: state of the knowledge and research needs. Environ. Sci. Technol. Lett. 2, 2-6 (2015) · doi:10.1021/ez5003715 |
| [14] | Piercy, T.J., Smither, S.J., Steward, J.A., Eastaugh, L., Lever, M.S.: The survival of filoviruses in liquids, on solid substrates and in a dynamic aerosol. J. Appl. Microbiol. 109(5), 1531-1539 (2010) |
| [15] | Leroy, E.M., Rouquet, P., Formenty, P., Souquière, S., Kilbourne, A., Froment, J.M., Bermejo, M., Smit, S., Karesh, W., Swanepoel, R., Zaki, S.R., Rollin, P.E.: Multiple Ebola virus transmission events and rapid decline of central African wildlife. Science 303(5656), 387-390 (2006) · doi:10.1126/science.1092528 |
| [16] | Leroy, E.M., Kumulungui, B., Pourrut, X., Rouquet, P., Hassanin, A., Yaba, P., Délicat, A., Paweska, J.T., Gonzalez, J.P., Swanepoel, R.: Fruit bats as reservoirs of Ebola virus. Nature 438, 575-576 (2005) · doi:10.1038/438575a |
| [17] | Butler, D.: Six challenges to stamping out Ebola. http://www.nature.com/ (2015) |
| [18] | Chan, M.: Ebola virus disease in West Africa—no early end to the outbreak. N. Engl. J. Med. 371, 1183-1185 (2014) · doi:10.1056/NEJMp1409859 |
| [19] | Sharomi, O., Malik, T.: Optimal control in epidemiology. Ann. Oper. Res. 251(1-2), 55-71 (2015). https://doi.org/10.1007/s10479-015-1834-4 · Zbl 1373.92140 · doi:10.1007/s10479-015-1834-4 |
| [20] | Groseth, A., Feldmann, H., Strong, J.E.: The ecology of Ebola virus. Trends Microbiol. 15, 408-416 (2007) · doi:10.1016/j.tim.2007.08.001 |
| [21] | Judson, S.D., Fischer, R., Judson, A., Munster, V.J.: Ecological contexts of index cases and spillover events of different Ebola viruses. PLoS Pathog. 12(8), e1005780 (2016) · doi:10.1371/journal.ppat.1005780 |
| [22] | Area, I., Losada, J., Ndaïrou, F., Nieto, J.J., Tcheutia, D.D.: Mathematical modeling of 2014 Ebola outbreak. Math. Methods Appl. Sci. 40, 6114-6122 (2017) · Zbl 1381.34066 · doi:10.1002/mma.3794 |
| [23] | Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Torres, A.: On a fractional order Ebola epidemic model. Adv. Differ. Equ. 2015, 278 (2015). https://doi.org/10.1186/s13662-015-0613-5 · Zbl 1344.92150 · doi:10.1186/s13662-015-0613-5 |
| [24] | Imran, M., Khan, A., Ansari, A., Shah, S.: Modeling transmission dynamics of Ebola virus disease. Int. J. Biomath. 10(4), 1750057 (2015). https://doi.org/10.1142/S1793524517500577 · Zbl 1373.92123 · doi:10.1142/S1793524517500577 |
| [25] | Ivorra, B., Ngom, D., Ramos, A.M.: 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(9), 1668-1704 (2015) · Zbl 1339.92085 · doi:10.1007/s11538-015-0100-x |
| [26] | Berge, T., Lubuma, J.M.S., Moremedi, G.M., Morris, N., Kondera-Shava, R.: A simple mathematical model for Ebola in Africa. J. Biol. Dyn. 11(1), 42-74 (2017). https://doi.org/10.1080/17513758.2016.1229817 · Zbl 1448.92276 · doi:10.1080/17513758.2016.1229817 |
| [27] | Chavez, C., Barley, K., Bichara, D., Chowell, D., Diaz Herrera, E., Espinoza, B., Moreno, V., Towers, S., Yong, K.E.: Modeling Ebola at the Mathematical and Theoretical Biology Institute (MTBI). Not. Am. Math. Soc. 63(4), 367-371 (2016) · Zbl 1338.92122 |
| [28] | Agusto, F.B.: Mathematical model of Ebola transmission dynamics with relapse and reinfection. Math. Biosci. 283, 48-59 (2017) · Zbl 1367.92114 · doi:10.1016/j.mbs.2016.11.002 |
| [29] | Berge, T., Bowong, S., Lubuma, J., Manyombe, M.L.M.: Modeling Ebola virus disease transmissions with reservoir in a complex virus life ecology. Math. Biosci. Eng. 15(1), 21-56 (2018). https://doi.org/10.3934/mbe.2018002 · Zbl 1378.34071 · doi:10.3934/mbe.2018002 |
| [30] | Funk, S., Camacho, A., Kucharski, A.J., Eggo, R.M., Edmunds, W.J.: Real-time forecasting of infectious disease dynamics with a stochastic semi-mechanistic model. Epidemics 22, 56-61 (2018) · doi:10.1016/j.epidem.2016.11.003 |
| [31] | Berge, T., Ouemba Tasse, A.J., Tenkam, H.M., Lubuma, J.: Mathematical modelling of contact tracing as a control strategy of Ebola virus disease. Int. J. Biomath. 11(7), 1850093 (2018) · Zbl 1400.92315 · doi:10.1142/S1793524518500936 |
| [32] | Guo, Z., Xiao, D., Li, D., Wang, X., Wang, Y., Yan, T., Wang, Z.: Predicting and evaluating the epidemic trend of Ebola virus disease in the 2014-2015 outbreak and the effects of intervention measures. PLoS ONE 11(4), e0152438 (2016). https://doi.org/10.1371/journal.pone.0152438 · doi:10.1371/journal.pone.0152438 |
| [33] | Salem, D.; Smith, R., A mathematical model of Ebola virus disease: using sensitivity analysis to determine effective intervention targets, Montreal, Quebec, Canada, July 24-27 2016 |
| [34] | Weitz, J.S., Dushoff, J.: Modeling post-death transmission of Ebola: challenges for inference and opportunities for control. Sci. Rep. 5, 8751 (2015). https://doi.org/10.1038/srep08751 · doi:10.1038/srep08751 |
| [35] | Tulu, T.W., Tian, B., Wu, Z.: Modeling the effect of quarantine and vaccination on Ebola disease. Adv. Differ. Equ. 2017, 178 (2017). https://doi.org/10.1186/s13662-017-1225-z · Zbl 1422.92170 · doi:10.1186/s13662-017-1225-z |
| [36] | Berge, T., Chapwanya, M., Lubuma, J., Terefe, Y.A.: A mathematical model for Ebola epidemic with self protection measures. J. Biol. Syst. 26(1), 107-131 (2017). https://doi.org/10.1142/S0218339018500067 · Zbl 1409.92226 · doi:10.1142/S0218339018500067 |
| [37] | Denes, A., Gumel, A.B.: Modeling the impact of quarantine during an outbreak of Ebola virus disease. Infect. Dis. Model. 4, 12-27 (2019) |
| [38] | Kucharski, A.J., Eggo, R.M., Watson, C.H., Camacho, A., Funk, S., Edmunds, W.J.: Effectiveness of ring vaccination as control strategy for Ebola virus disease. Emerg. Infect. Dis. 22(1), 105-108 (2016) · doi:10.3201/eid2201.151410 |
| [39] | Bodine, E.N., Cook, C., Shorten, M.: The potential impact of a prophylactic vaccine for Ebola in Sierra Leone. Math. Biosci. Eng. 15(2), 337-359 (2018) · Zbl 1375.92062 · doi:10.3934/mbe.2018015 |
| [40] | Kelly, J., et al.: Projections of Ebola outbreak size and duration with and without vaccine use in Équateur, Democratic Republic of Congo, as of May 27, 2018. PLoS ONE 14, e0213190 (2019) · doi:10.1371/journal.pone.0213190 |
| [41] | Jiang, S., Wang, K., Li, C., Hong, G., Zhang, X., Shan, M., Li, H., Wang, J.: Mathematical models for devising the optimal Ebola virus disease eradication. J. Transl. Med. 15, 124 (2017). https://doi.org/10.1186/s12967-017-1224-6 · doi:10.1186/s12967-017-1224-6 |
| [42] | Diane, S., Njakou, D., Nyabadza, F.: An optimal control model for Ebola virus disease. J. Biol. Syst. 24(1), 1-21 (2016) |
| [43] | Muhammad, D.A., Muhammad, U., Adnan, K., Mudassar, I.: Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination. Infect. Dis. Poverty 5, 72 (2016). https://doi.org/10.1186/s40249-016-0161-6 · doi:10.1186/s40249-016-0161-6 |
| [44] | Area, I., et al.: Ebola model and optimal control with vaccination constraints. J. Ind. Manag. Optim. 14(2), 427-446 (2018) · Zbl 1412.49005 |
| [45] | Rachah, A., Torres, D.F.M.: Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa. Discrete Dyn. Nat. Soc. 2015, Article ID 842792 (2015). https://doi.org/10.1155/2015/842792. · Zbl 1418.92188 · doi:10.1155/2015/842792 |
| [46] | Takaidza, I., Makinde, O.D., Okosun, O.K.: Computational modelling and optimal control of Ebola virus disease with non-linear incidence rate. J. Phys. Conf. Ser. 818(1), 012003 (2017) · doi:10.1088/1742-6596/818/1/012003 |
| [47] | Meakin, S., Tildesley, M., Davis, E., Keeling, M.: A meta-population model for the 2018 Ebola outbreak in Equateur province in the Democratic Republic of the Congo. Cold Spring Harbor Laboratory: bioRxiv 465062. https://doi.org/10.1101/465062 (2018) |
| [48] | Ivorra, B., Ngom, D., Ramos, A.M.: Version 4: Be-CoDiS: an epidemiological model to predict the risk of human diseases spread between countries. Validation and application to the 2014 Ebola Virus Disease epidemic. Preprint arXiv.org; Cornell Universiy Library, Date:. 1410. 1-32. (2014). http://arxiv.org/abs/1410.6153 · Zbl 1339.92085 |
| [49] | Njagarah, J.B., Nyabadza, F.: A meta-population model for cholera transmission dynamics between communities linked by migration. Appl. Math. Comput. 241, 317-331 (2014). https://doi.org/10.1016/j.amc.2014.05.036 · Zbl 1334.92237 · doi:10.1016/j.amc.2014.05.036 |
| [50] | Castillo, C.: Optimal control of an epidemic through educational campaigns. Electron. J. Differ. Equ. 2006, 125 (2006) · Zbl 1108.92035 |
| [51] | Piercy, T.J., Smither, S.J., Steward, J.A., Eastaugh, L., Lever, M.T.: The survival of filo-viruses in liquids, on solid substrates and in a dynamic aerosol. J. Appl. Microbiol. 109(5), 1531-1539 (2010) |
| [52] | Francesconi, P., Yoti, Z., Declich, S., Onek, P.A., Fabiani, M., Olango, J., Andraghetti, R., Rollin, P.E., Opira, C., Greco, D., Salmaso, S.: Ebola hemorrhagic fever transmission and risk factors of contacts, Uganda. Emerg. Infect. Dis. 9(11), 1430-1437 (2003) · doi:10.3201/eid0911.030339 |
| [53] | Chowell, G., Nishiura, H.: Transmission dynamics and control of Ebola virus disease (EVD): a review. BMC Med. 12, 196 (2014) · doi:10.1186/s12916-014-0196-0 |
| [54] | Youkee, D., Brown, C.S., Lilburn, P., Shetty, N., Brooks, T., Simpson, A., Bentley, N., Lado, M., Kamara, T.B., Walker, N.F., Johnson, O.: Assessment of environmental contamination and environmental decontamination practices within an Ebola holding unit, Freetown, Sierra Leone. PLoS ONE (2015). https://doi.org/10.1371/journal.pone.0145167 · doi:10.1371/journal.pone.0145167 |
| [55] | 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) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 |
| [56] | Castillo-Chavez, C.; Feng, Z.; Huang, W., On the computation of R \(0R_0\) and its role on global stability, Minneapolis, MN, 1999, New York · Zbl 1021.92032 · doi:10.1007/978-1-4757-3667-0_13 |
| [57] | Korobeinikov, A.: Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. 21, 75-83 (2004) · Zbl 1055.92051 · doi:10.1093/imammb/21.2.75 |
| [58] | McCluskey, C.C.: Lyapunov functions for tuberculosis models with fast and slow progression. Math. Biosci. Eng. 3, 603-614 (2006) · Zbl 1113.92057 · doi:10.3934/mbe.2006.3.603 |
| [59] | Shuai, Z., Heesterbeek, J.A.P., Van den Driessche, P.: Extending the type reproduction number to infectious disease control targeting contact between types. J. Math. Biol. 67, 1067-1082 (2013) · Zbl 1278.92031 · doi:10.1007/s00285-012-0579-9 |
| [60] | Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) · Zbl 0576.15001 · doi:10.1017/CBO9780511810817 |
| [61] | La Salle, J.P.: The Stability of Dynamical Systems. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 12 (1976) · Zbl 0364.93002 · doi:10.1137/1.9781611970432 |
| [62] | Thieme, H.R.: Persistence under relaxed point-dissipativity with an application to en epidemic model. SIAM J. Math. Anal. 24, 407-435 (1993) · Zbl 0774.34030 · doi:10.1137/0524026 |
| [63] | Zhao, X.Q.: Uniform persistence and periodic coexistence states in infinite-dimensional periodic semi flows with applications. Can. Appl. Math. Q. 3, 473-495 (1995) · Zbl 0849.34034 |
| [64] | Wang, W.D., Zhao, X.Q.: An epidemic model in a patchy environment. Math. Biosci. 190, 97-112 (2004) · Zbl 1048.92030 · doi:10.1016/j.mbs.2002.11.001 |
| [65] | Wang, W.D., Fergola, P., Tenneriello, C.: Innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51-67 (2003) · Zbl 1052.91076 |
| [66] | Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361-404 (2004) · Zbl 1060.92041 · doi:10.3934/mbe.2004.1.361 |
| [67] | Levy, B., Edholm, C., Gaoue, O., Kaondera-Shava, R., Kgosimore, M., Lenhart, S., Lephodisa, B., Lungu, E., Marijani, T., Nyabadza, F.: Modeling the role of public health education in Ebola virus disease outbreaks in Sudan. Infect. Dis. Model. 2(3), 323-340 (2017) |
| [68] | Mallela, A., Lenhart, S., Vaidya, N.K.: HIV-TB co-infection treatment: modelling and optimal control theory perspectives. J. Comput. Appl. Math. 307, 143-161 (2016) · Zbl 1345.92150 · doi:10.1016/j.cam.2016.02.051 |
| [69] | Kirschner, D., Lenhart, S., Serbin, S.: Optimal control of the chemotherapy of HIV. J. Math. Biol. 35(7), 775-792 (1997) · Zbl 0876.92016 · doi:10.1007/s002850050076 |
| [70] | WHO: Ebola and Marburg disease epidemics: preparedness, alert, control and evaluation. World Health Organization, WHO/HSE/PED/CED/2014.05 (2014) |
| [71] | Pontryagin, L.S., Boltyanskii, V.T., Gamkrelidze, R.V., Mishchevko, E.F.: The Mathematical Theory of Optimal Processes. Gordon & Breach, New York, 4 (1985) |
| [72] | Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975) · Zbl 0323.49001 · doi:10.1007/978-1-4612-6380-7 |
| [73] | Lenhart, S., Workman, J.T.: Optimal Controls Applied to Biological Models. Chapman & Hall/CRC, London (1997) · Zbl 1291.92010 |
| [74] | Rivers, C.M., Lofgren, E.T., Marathe, M., Eubank, S., Lewis, B.L.: Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. PLOS Curr. Outbreaks. Edition 1. https://doi.org/10.1371/currents.outbreaks.fd38dd85078565450b0be3fcd78f5ccf (2014) |
| [75] | Ndanguza, D., Tchuenche, J.M., Haario, H.: Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo. Afr. Math. 24, 55-68 (2013) · Zbl 1271.62264 · doi:10.1007/s13370-011-0039-5 |
| [76] | WHO Ebola Response Team: Ebola virus disease in West Africa-the first 9 months of the epidemic and forward projections. N. Engl. J. Med. 371(16), 1481-1495 (2014). https://doi.org/10.1056/NEJMoa1411100 · doi:10.1056/NEJMoa1411100 |
| [77] | Bibby, K., Casson, L.W., Stachler, E., Haas, C.N.: Ebola virus persistence in the environment: state of the knowledge and research needs. Environ. Sci. Technol. Lett. 2, 2-6 (2015) · doi:10.1021/ez5003715 |
| [78] | The Centers for Disease Control and Prevention: Ebola (Ebola virus disease). http://www.cdc.gov/Ebola/resources/virus-ecology.html (Page last reviewed August 1, 2014) |
| [79] | Fasina, F.O., Shittu, A., Lazarus, D., Tomori, O., Simonsen, L., Viboud, C., Chowell, G.: Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014. Euro Surveill. 19(40), 20920 (2014). http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=20920 · doi:10.2807/1560-7917.ES2014.19.40.20920 |
| [80] | Towers, S., Patterson-Lomba, O., Castillo-Chavez, C.: Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak. PLOS Curr. September 18 (2014) |
| [81] | Li, M.Y., Graef, J.R., Wang, L., Karsai, J.: Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160, 191-213 (1999) · Zbl 0974.92029 · doi:10.1016/S0025-5564(99)00030-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
