Modeling and Dynamics Analysis of Zika Transmission with Limited Medical Resources

Zika virus, a reemerging 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.

[1]  Gui-Hua Li,et al.  Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates , 2017, PloS one.

[2]  R. Ross,et al.  Prevention of malaria. , 2012, BMJ.

[3]  A. Wiratsudakul,et al.  Dynamics of Zika virus outbreaks: an overview of mathematical modeling approaches , 2018, PeerJ.

[4]  Carlos Castillo-Chavez,et al.  Dynamical models of tuberculosis and their applications. , 2004, Mathematical biosciences and engineering : MBE.

[5]  D. Musso,et al.  Potential Sexual Transmission of Zika Virus , 2015, Emerging infectious diseases.

[6]  C. Pardo,et al.  Zika Virus–Associated Neurological Disease in the Adult: Guillain–Barré Syndrome, Encephalitis, and Myelitis , 2016, Seminars in Reproductive Medicine.

[7]  A. Barros,et al.  Catastrophic spending on health care in Brazil: private health insurance does not seem to be the solution. , 2011, Cadernos de saude publica.

[8]  A. L. Mosimann,et al.  First report of autochthonous transmission of Zika virus in Brazil , 2015, Memorias do Instituto Oswaldo Cruz.

[9]  Alessandro Vespignani,et al.  Spread of Zika virus in the Americas , 2017, Proceedings of the National Academy of Sciences.

[10]  Nita H. Shah,et al.  Preventions and Controls on Congenital Transmissions of Zika: Mathematical Analysis , 2017 .

[11]  Huaiping Zhu,et al.  Dynamics of a Filippov epidemic model with limited hospital beds. , 2018, Mathematical biosciences and engineering : MBE.

[12]  Yanni Xiao,et al.  Implication of vaccination against dengue for Zika outbreak , 2016, Scientific Reports.

[13]  D. Kanduc,et al.  Zika virus and autoimmunity: From microcephaly to Guillain-Barré syndrome, and beyond. , 2016, Autoimmunity reviews.

[14]  Mónica Roa.  Zika virus outbreak: reproductive health and rights in Latin America , 2016, The Lancet.

[15]  G. Dick,et al.  Zika virus. I. Isolations and serological specificity. , 1952, Transactions of the Royal Society of Tropical Medicine and Hygiene.

[16]  Zhidong Teng,et al.  Stability analysis in a class of discrete SIRS epidemic models , 2012 .

[17]  C. B. Marcondes,et al.  Zika virus in Brazil and the danger of infestation by Aedes (Stegomyia) mosquitoes. , 2015, Revista da Sociedade Brasileira de Medicina Tropical.

[18]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[19]  Sebastian Funk,et al.  Transmission Dynamics of Zika Virus in Island Populations: A Modelling Analysis of the 2013–14 French Polynesia Outbreak , 2016, bioRxiv.

[20]  W. Fagan,et al.  Mathematical model of Zika virus with vertical transmission , 2017, Infectious Disease Modelling.

[21]  M. G. Roberts,et al.  A new method for estimating the effort required to control an infectious disease , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[22]  M. Hoelscher,et al.  Long-term kinetics of Zika virus RNA and antibodies in body fluids of a vasectomized traveller returning from Martinique: a case report , 2017, BMC Infectious Diseases.

[23]  Sourav Kumar Sasmal,et al.  Modeling the Spread of Zika Virus in a Stage-Structured Population: Effect of Sexual Transmission , 2018, Bulletin of mathematical biology.

[24]  van Dh Dick Campen,et al.  Bifurcation phenomena in non-smooth dynamical systems , 2006 .

[25]  C. Ashby,et al.  Sofosbuvir: an antiviral drug with potential efficacy against Zika infection. , 2017, International journal of infectious diseases : IJID : official publication of the International Society for Infectious Diseases.

[26]  N. Almeida-Filho.  Higher education and health care in Brazil , 2011, The Lancet.

[27]  J A P Heesterbeek,et al.  The type-reproduction number T in models for infectious disease control. , 2007, Mathematical biosciences.

[28]  G. Macdonald,et al.  The analysis of equilibrium in malaria. , 1952, Tropical diseases bulletin.

[29]  C. M. Saad-Roy,et al.  The effect of sexual transmission on Zika virus dynamics , 2018, Journal of Mathematical Biology.

[30]  Pauline van den Driessche,et al.  Global Stability of Infectious Disease Models Using Lyapunov Functions , 2013, SIAM J. Appl. Math..

[31]  Huaiping Zhu,et al.  Modeling the spread and control of dengue with limited public health resources. , 2016, Mathematical biosciences.

[32]  D. Musso,et al.  Emerging arboviruses in the Pacific , 2014, The Lancet.

[33]  Sharon Bewick,et al.  Mathematical model for Zika virus dynamics with sexual transmission route , 2017 .

[34]  Hongyong Zhao,et al.  Dynamics analysis of a Zika–dengue co-infection model with dengue vaccine and antibody-dependent enhancement , 2019, Physica A: Statistical Mechanics and its Applications.

[35]  pNuha Algarzaep.  Zika virus outbreak , 2019 .

[36]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[37]  Ping Liu,et al.  25‐Hydroxycholesterol Protects Host against Zika Virus Infection and Its Associated Microcephaly in a Mouse Model , 2017, Immunity.

[38]  Carolina Q. Sacramento,et al.  The clinically approved antiviral drug sofosbuvir inhibits Zika virus replication , 2017, Scientific Reports.

[39]  Zika Virus Controversies: Epidemics as a Legacy of Mega Events? , 2016 .

[40]  S. Sardi,et al.  Zika Virus Outbreak, Bahia, Brazil , 2015, Emerging infectious diseases.

[41]  Wendi Wang.  Backward bifurcation of an epidemic model with treatment. , 2006, Mathematical biosciences.

[42]  N. Hens,et al.  Dynamic Epidemiological Models for Dengue Transmission: A Systematic Review of Structural Approaches , 2012, PloS one.

[43]  J. Neyts,et al.  The Viral Polymerase Inhibitor 7-Deaza-2’-C-Methyladenosine Is a Potent Inhibitor of In Vitro Zika Virus Replication and Delays Disease Progression in a Robust Mouse Infection Model , 2016, bioRxiv.

[44]  S. Ruan,et al.  Bifurcations in an epidemic model with constant removal rate of the infectives , 2004 .

[45]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[46]  Gerardo Chowell,et al.  Prevention and Control of Zika as a Mosquito-Borne and Sexually Transmitted Disease: A Mathematical Modeling Analysis , 2016, Scientific Reports.

[47]  Sebastian Funk,et al.  Comparative Analysis of Dengue and Zika Outbreaks Reveals Differences by Setting and Virus , 2016, bioRxiv.

[48]  Huaiping Zhu,et al.  Modeling the transmission and control of Zika in Brazil , 2017, Scientific Reports.

[49]  Myrielle Dupont-Rouzeyrol,et al.  Detection of Zika Virus in Urine , 2015, Emerging infectious diseases.

[50]  L. Schuler‐Faccini,et al.  Possible Association Between Zika Virus Infection and Microcephaly - Brazil, 2015. , 2016, MMWR. Morbidity and mortality weekly report.

[51]  Mudassar Imran,et al.  Transmission Dynamics of Zika Fever: A SEIR Based Model , 2017 .

[52]  A Model for the Risk of Microcephaly Induced by the Zika Virus (ZIKV) , 2016 .

[53]  R. Lanciotti,et al.  Zika virus outbreak on Yap Island, Federated States of Micronesia. , 2009, The New England journal of medicine.

[54]  Huaiping Zhu,et al.  Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds , 2014 .