A theoretical model for Zika virus transmission

In this paper, we present and analyze an SEIR Zika epidemic model. Firstly, we investigate the model with constant controls. The steady states of the model is found to be locally and globally asymptotically stable. Thereafter, we incorporate time dependent controls into the model in order to investigate the optimal effects of bednets, treatments of infective and spray of insecticides on the disease spread. Furthermore, we used Pontryagin’s Maximum Principle to determine the necessary conditions for effective control of the disease. Also, the numerical results were presented.

[1]  D. Musso,et al.  Rapid spread of emerging Zika virus in the Pacific area. , 2014, Clinical microbiology and infection : the official publication of the European Society of Clinical Microbiology and Infectious Diseases.

[2]  Xinyu Song,et al.  Analysis of an SEIR Epidemic Model with Saturated Incidence and Saturated Treatment Function , 2014, TheScientificWorldJournal.

[3]  Juan Zhang,et al.  Assessing reappearance factors of H7N9 avian influenza in China , 2017, Appl. Math. Comput..

[4]  M. A. Khan,et al.  Mathematical modeling and stability analysis of Pine Wilt Disease with optimal control , 2017, Scientific Reports.

[5]  Zhen Jin,et al.  Transmission dynamics of cholera: Mathematical modeling and control strategies , 2017, Commun. Nonlinear Sci. Numer. Simul..

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

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

[8]  Tao Dong,et al.  Modeling Computer Virus and Its Dynamics , 2013 .

[9]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[10]  M. Javidi,et al.  Stability analysis of a novel VEISV propagation model of computer worm attacks , 2014 .

[11]  Zhen Jin,et al.  Modeling direct and indirect disease transmission using multi-group model , 2017 .

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

[13]  Hui Wan,et al.  Rich Dynamics of an Epidemic Model with Saturation Recovery , 2013, J. Appl. Math..

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

[15]  Li Li,et al.  MONTHLY PERIODIC OUTBREAK OF HEMORRHAGIC FEVER WITH RENAL SYNDROME IN CHINA , 2016 .

[16]  Oluwole Daniel Makinde,et al.  A co-infection model of malaria and cholera diseases with optimal control. , 2014, Mathematical biosciences.

[17]  Complex dynamics in biological systems arising from multiple limit cycle bifurcation , 2016, Journal of biological dynamics.

[18]  Carla Rossi,et al.  A nested-epidemic model for the spread of hepatitis C among injecting drug users. , 2004, Mathematical biosciences.

[19]  Juan Zhang,et al.  Transmission dynamics of a multi-group brucellosis model with mixed cross infection in public farm , 2014, Appl. Math. Comput..

[20]  Zi-Ke Zhang,et al.  Global stability for a sheep brucellosis model with immigration , 2014, Appl. Math. Comput..

[21]  Saeed Islam,et al.  Dynamic Behavior of Leptospirosis Disease with Saturated Incidence Rate , 2016 .

[22]  Michael Y. Li,et al.  A graph-theoretic approach to the method of global Lyapunov functions , 2008 .

[23]  Brian D. Foy,et al.  Probable Non–Vector-borne Transmission of Zika Virus, Colorado, USA , 2011, Emerging infectious diseases.

[24]  Linda J. S. Allen,et al.  An introduction to mathematical biology , 2006 .

[25]  S. Hay,et al.  Anticipating the international spread of Zika virus from Brazil , 2016, The Lancet.

[26]  Eva K. Lee,et al.  A Compartmental Model for Zika Virus with Dynamic Human and Vector Populations , 2016, AMIA.

[27]  Kazeem O. Okosun,et al.  Optimal control strategies and cost-effectiveness analysis of a malaria model , 2013, Biosyst..

[28]  D. Simpson.  ZIKA VIRUS INFECTION IN MAN. , 1964, Transactions of the Royal Society of Tropical Medicine and Hygiene.

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

[30]  J. Janssen,et al.  Deterministic and Stochastic Optimal Control , 2013 .

[31]  R. Ruth,et al.  Stability of dynamical systems , 1988 .

[32]  Ebenezer Bonyah,et al.  A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment , 2014, Comput. Math. Methods Medicine.

[33]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[34]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[35]  Xiangyun Shi.  Analysis of a differential equation model of HIV infection of CD4+ T-cells with saturated reverse function , 2011 .

[36]  Ryo Kinoshita,et al.  Transmission potential of Zika virus infection in the South Pacific. , 2016, International journal of infectious diseases : IJID : official publication of the International Society for Infectious Diseases.

[37]  D. Musso,et al.  Potential for Zika virus transmission through blood transfusion demonstrated during an outbreak in French Polynesia, November 2013 to February 2014. , 2014, Euro surveillance : bulletin Europeen sur les maladies transmissibles = European communicable disease bulletin.

[38]  M. Khan,et al.  Mathematical analysis of typhoid model with saturated incidence rate , 2015 .

[39]  José Roberto Castilho Piqueira,et al.  Dynamic models for computer viruses , 2008, Comput. Secur..