Modeling and dynamics analysis of Zika transmission with limited medical resources. (English) Zbl 1508.92154
Summary: Zika virus, a re-emerging mosquito-borne flavivirus, posed a global public health emergency in 2016. Brazil is the most seriously affected country. Some measures have been implemented to control the Zika transmission, such as spraying mosquitoes, developing vaccines and drugs. However, because of the limited medical resources (LMRs) in the country, not every infected patient can be treated in time when infected with Zika virus. We aim to build a deterministic Zika model by introducing a piecewise smooth treatment recovery rate to research the effect of LMRs on the transmission and control of Zika. For the model without treatment, we analyze the global stability of equilibria. For the model with treatment, the model exhibits complex dynamics. We prove that the model with treatment undergoes backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2. It means that the model with LMRs is sensitive to parameters and initial conditions, which has important significance for control of Zika. We also apply the model to estimate the basic and control reproduction numbers for the Zika transmission by using the data on weekly reported accumulated Zika cases from March 25, 2016, to April 14, 2018, in Brazil.
MSC:
| 92C60 | Medical epidemiology |
Keywords:
Zika virus; medical resources; mosquito-borne and sexual transmission; backward bifurcation; Hopf bifurcation; Bogdanov-Takens bifurcationReferences:
| [1] | Abdelrazec, A.; Bélair, J.; Shan, C.; Zhu, H., Modeling the spread and control of dengue with limited public health resources, Math Biosci, 271, 136-145 (2016) · Zbl 1364.92036 |
| [2] | Agusto, FB; Bewick, S.; Fagan, WF, Mathematical model for Zika virus dynamics with sexual transmission route, Ecol Complex, 29, 61-81 (2017) |
| [3] | Agusto, F.; Bewick, S.; Fagan, W., Mathematical model of Zika virus with vertical transmission, Infect Dis Model, 2, 2, 244-267 (2017) |
| [4] | Almeida-Filho, N., Higher education and health care in Brazil, The Lancet, 377, 9781, 1898-1900 (2011) |
| [5] | Andraud, M.; Hens, N.; Marais, C.; Beutels, P., Dynamic epidemiological models for dengue transmission: a systematic review of structural approaches, PLoS ONE, 7, 3161-3164 (2011) |
| [6] | Arbex, AK; Bizarro, VR; Paletti, MT, Zika Virus controversies: epidemics as a legacy of mega events?, Health, 8, 7, 711-722 (2016) |
| [7] | Barros AJ, Bastos JL, D \({\hat{a}}\) maso AH (2011) Catastrophic spending on health care in Brazil: private health insurance does not seem to be the solution. Cadernos de Saúde \(P{\hat{u}}\) blica 27:s254-s262 |
| [8] | Brazil Ministry of Health, Zika cases from the Brazil Ministry of Health. http://portalms.saude.gov.br/boletins-epidemiologicos |
| [9] | Camila, Z.; De, MVCA; Pamplona, MAL, First report of autochthonous transmission of Zika virus in Brazil, Memórias do Inst Oswaldo Cruz, 110, 4, 569-572 (2015) |
| [10] | Campos, GS; Bandeira, AC; Sardi, SI, Zika virus outbreak, Bahia, Brazil, Emerg Infect Dis, 21, 1885-1886 (2015) |
| [11] | Cao-Lormeau, VM; Musso, D., Emerging arboviruses in the Pacific, Lancet, 384, 9954, 1571-1572 (2014) |
| [12] | Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math Biosci Eng, 1, 2, 361-404 (2004) · Zbl 1060.92041 |
| [13] | Clarke, FH; Ledyaev, YS; Stern, RJ; Wolenski, PR, Nonsmooth analysis and control theory, 178 (2008), Berlin: Springer, Berlin |
| [14] | Dick, GWA; Kitchen, SF; Haddow, AJ, Zika virus (I). Isolations and serological specificity, Trans R Soc Trop Med Hyg, 46, 5, 509-520 (1952) |
| [15] | Diekmann, O.; Heesterbeek, JAP; Metz, JA, On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J Math Biol, 28, 4, 365-382 (1990) · Zbl 0726.92018 |
| [16] | Duffy, MR, Zika virus outbreak on Yap Island, federated states of Micronesia, N Engl J Med, 360, 2536-2543 (2009) |
| [17] | Froeschl, G., Long-term kinetics of Zika virus RNA and antibodies in body fluids of a vasectomized traveller returning from Martinique: a case report, BMC Infect Dis, 17, 1, 55 (2017) |
| [18] | Funk, S., Comparative analysis of dengue and Zika outbreaks reveals differences by setting and virus, PLoS Negl Trop Dis, 10, 12, e0005173 (2016) |
| [19] | Gao, D., Prevention and control of Zika fever as a mosquito-borne and sexually transmitted disease, Sci Rep, 6, 28070 (2016) |
| [20] | Gourinat AC, \(O^{\prime }\) Connor O, Calvez E, Goarant C, Dupont-Rouzeyrol M (2015) Detection of Zika virus in urine. Emerg Infect Dis 21(1):84 |
| [21] | Heesterbeek, JAP; Roberts, MG, The type-reproduction number T in models for infectious disease control, Math Biosci, 206, 1, 3-10 (2007) · Zbl 1124.92043 |
| [22] | Hu, Z.; Teng, Z.; Jiang, H., Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal Real World Appl, 13, 5, 2017-2033 (2012) · Zbl 1254.92082 |
| [23] | Imran M, Usman M, Dur-e-Ahmad M, Khan A (2017) Transmission dynamics of Zika fever: a SEIR based model. Differ Equ Dyn Syst. 10.1007/s12591-017-0374-6 · Zbl 1479.34082 |
| [24] | Kucharski, AJ, Transmission dynamics of Zika virus in island populations: a modelling analysis of the 2013-14 French Polynesia outbreak, PloS Negl Trop Dis, 10, 5, e0004726 (2016) |
| [25] | Kuznetsov, YA, Elements of applied bifurcation theory, applied mathematical scienses (1995), New York: Springer, New York · Zbl 0829.58029 |
| [26] | Leine RI, Nijmeijer H (2004) Bifurcations of equilibria in non-smooth continuous systems. In: Dynamics and bifurcations of non-smooth mechanical systems. Springer, Berlin, pp 125-176 · Zbl 1068.70003 |
| [27] | Leine, RI; Van Campen, DH, Bifurcation phenomena in non-smooth dynamical systems, Eur J Mech A/Solids, 25, 4, 595-616 (2006) · Zbl 1187.70041 |
| [28] | Li, G.; Zhang, Y., Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, PLoS ONE, 12, 4, e0175789 (2017) |
| [29] | Li, C., 25-Hydroxycholesterol protects host against Zika virus infection and its associated microcephaly in a mouse model, Immunity, 46, 3, 446-456 (2017) |
| [30] | Lucchese, G.; Kanduc, D., Zika virus and autoimmunity: from microcephaly to Guillain-Barré syndrome, and beyond, Autoimmun Rev, 15, 8, 801-808 (2016) |
| [31] | Macdonald, G., The analysis of equilibrium in malaria, Trop Dis Bull, 49, 813-829 (1952) |
| [32] | Marcondes, CB; Ximenes, MFF, Zika virus in Brazil and the danger of infestation by \(\mathit{Aedes (Stegomyia)}\) mosquitoes, Rev Soc Bras Med Trop, 49, 4-10 (2016) |
| [33] | Munoz, LS; Barreras, P.; Pardo, CA, Zika virus-associated neurological disease in the adult: Guillain-Barré syndrome, encephalitis, and myelitis, Semin Reprod Med, 34, 5, 273-279 (2016) |
| [34] | Musso, D.; Roche, C.; Robin, E., Potential sexual transmission of Zika virus, Emerg Infect Dis, 21, 2, 359 (2015) |
| [35] | Petersen, LR; Jamieson, DJ; Powers, AM, Zika Virus, N Engl J Med, 374, 16, 1552-1563 (2016) |
| [36] | Pizza, D., A model for the risk of microcephaly induced by the Zika virus (ZIKV), Open J Model Simul, 4, 3, 109 (2016) |
| [37] | Reznik, SE; Ashby, CR, Sofosbuvir: an antiviral drug with potential efficacy against Zika infection, Int J Infect Dis, 55, 29-30 (2017) |
| [38] | Roa, M., Zika virus outbreak: reproductive health and rights in Latin America, Lancet, 387, 843-843 (2016) |
| [39] | Roberts, MG; Heesterbeek, JAP, A new method for estimating the effort required to control an infectious disease, Proc R Soc Lond B Biol Sci, 270, 1522, 1359-1364 (2003) |
| [40] | Ross, R., The prevention of malaria (1911), London: John Murray, London |
| [41] | Saad-Roy, CM; Ma, J.; van den Driessche, P., The effect of sexual transmission on Zika virus dynamics, J Math Biol, 77, 6-7, 1917-1941 (2018) · Zbl 1404.92202 |
| [42] | Sacramento, CQ; De Melo, GR; De Freitas, CS, The clinically approved antiviral drug sofosbuvir inhibits Zika virus replication, Sci Rep, 7, 40920 (2017) |
| [43] | Sasmal, SK; Ghosh, I.; Huppert, A.; Chattopadhyay, J., Modeling the spread of Zika virus in a stage-structured population: effect of sexual transmission, Bull Math Biol, 80, 11, 3038-3067 (2018) · Zbl 1401.92204 |
| [44] | Schuler-Faccini, L., Possible association between Zika virus infection and microcephaly-Brazil, 2015, Morb Mort Wkly Rep, 65, 59-62 (2016) |
| [45] | Shah, NH; Patel, ZA; Yeolekar, BM, Preventions and controls on congenital transmissions of Zika: mathematical analysis, Appl Math, 8, 4, 500 (2017) |
| [46] | Shan, C.; Zhu, H., Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J Differ Equ, 257, 5, 1662-1688 (2014) · Zbl 1300.34113 |
| [47] | Shuai, Z.; Van den Driessche, P., Global stability of infectious disease models using Lyapunov functions, SIAM J App Math, 73, 4, 1513-1532 (2013) · Zbl 1308.34072 |
| [48] | Tang, B.; Xiao, Y.; Wu, J., Implication of vaccination against dengue for Zika outbreak, Sci Rep, 6, 35623 (2016) |
| [49] | Van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 1, 29-48 (2002) · Zbl 1015.92036 |
| [50] | Wang, W., Backward bifurcation of an epidemic model with treatment, Math Biosci, 201, 1, 58-71 (2006) · Zbl 1093.92054 |
| [51] | Wang, W.; Ruan, S., Bifurcations in an epidemic model with constant removal rate of the infectives, J Math Anal Appl, 291, 2, 775-793 (2004) · Zbl 1054.34071 |
| [52] | Wang, L.; Zhao, H., Dynamics analysis of a Zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement, Phys A Stat Mech Appl, 522, 248-273 (2019) · Zbl 07561951 |
| [53] | Wang, L.; Zhao, H.; Oliva, SM; Zhu, H., Modeling the transmission and control of Zika in Brazil, Sci Rep, 7, 1, 7721 (2017) |
| [54] | Wang, A.; Xiao, Y.; Zhu, H., Dynamics of a Filippov epidemic model with limited hospital beds, Math Biosci Eng, 15, 3, 739-764 (2018) · Zbl 1406.92625 |
| [55] | Wiratsudakul, A.; Suparit, P.; Modchang, C., Dynamics of Zika virus outbreaks: an overview of mathematical modeling approaches, PeerJ, 6, e4526 (2018) |
| [56] | World Health Organization (2005-2017) World health statistics |
| [57] | World Health Organization (2016) Essential medicines and health products. Annual Report |
| [58] | World Health Organization (WHO) (2016) WHO statement on the first meeting of the International Health Regulations (2005) Emergency Committee on Zika virus and observed increase in neurological disorders and neonatal malformations |
| [59] | World Health Organization, Case definitions of Zika virus. http://www.paho.org/hq/index. php?option=com_content&view=article& id=11117 |
| [60] | Zhang, Q., Spread of Zika virus in the Americas, Proc Natl Acad Sci, 114, 22, E4334-E4343 (2017) |
| [61] | Zmurko, J., The viral polymerase inhibitor 7-deaza-\(2^{\prime }-C\)-methyladenosine is a potent inhibitor of in vitro Zika virus replication and delays disease progression in a robust mouse infection model, PLoS Negl Trop Dis, 10, 5, e0004695 (2016) |
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.
