Mathematical models for the study of Zika diffusion with exposed state and delay

Autores

DOI:

https://doi.org/10.17268/sel.mat.2020.02.01

Palavras-chave:

Diffusion, epidemic, model, delay, Zika

Resumo

Zika virus spreads to people primarily through the bite of an infected Aedes aegypti species mosquito. But it Zika can also be passed through sex from an infected to his or her sex partners and it can be spread from a pregnant woman to her fetus. Zika continues to spreading geographically to areas where competent vectors are present. Although a decline in cases of Zika virus infection has been reported in some countries, or in some parts of countries, vigilance needs to remain high. In this work, we present two mathematical models for the Zika diffusion by using (1) ordinary differential equations with exposed state and, (2) ordinary differential equations with delay (discrete), which is the time it takes mosquitoes to develop the virus. We make a comparison between the two modeling variants. Computational simulations is performed for Santa Ana, which is that is prone to develop the epidemic in an endemic manner.

Biografia do Autor

Erick Manuel Delgado Moya, IME-University of Sao Paulo, Rua do Matao, 1010- CEP 05508-090- Sao Paulo-SP, Brazil

IME-USP

Referências

Bonyah E, Okosun KC. Mathematical modeling of Zika virus. Asian Pacific Journal of Tropical Disease. 2016; 6(9):673-679.DOI: 0.1016/S2222-1808(16)61108.

Chicone C. Ordinary differential equation with application. Missouri: Springer; 1999.

Delgado EM, Marrero A. Mathematical models for Zika with exposed variables and delay. Comparison and experimentation in Suriname and El Salvador. Selecciones Matemáticas. 2019; 6(1):1-13. DOI: 10.17268/sel.mat.2019.01.01.

Delgado EM, Marrero A. Mathematical model for the study of the diffusion of Zika. Computational experimentation in Paramaribo and Santa Ana. Selecciones Matemáticas. 2019; 6(2):196-203. DOI:10.17268/sel.mat.2019.02.06.

Dick GW, Kitchen SF, Haddow AJ. Zika Virus (I). Isolations and serogical specifity. Trans R Soc Trop Med Hyg. 1952; 46(5):509-520. DOI: 10.1016/0035-9203(52)90042-4.

Dick GW, Kitchen SF, Haddow AJ. Zika virus (II). Pathogenicity and physical properties. Trans R Soc Trop Med Hyg. 1952; 46:521-534. DOI: 10.1016/0035-9203(52)90043-6.

Diekmam O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemical models. J. Royal Society Interface. 2010; 7:873-885. DOI: 10.1098/rsif.2009.0386.

Driver RD. Ordinary and delay differential equations. New York: Springer Verlag; 1977.

Esteva L, Vargas C. Analysis of a Dengue disease transmission model. Mathematical Biosciences. 1998; 150:131-151. DOI:10.1016/S0025-5564(98)10003-2.

Kassen TG, Garba EJD. A mathematical model for the spatial spread oh HIV in heterogeneous population. Mathematical Theory and Modeling. 2016; 6(4):95-104.

Lin H, Wang F. On a reaction-diffusion system modeling the Dengue transmission with nonlocal infections and crowding effects. Applied Mathematics and Computation. 2014; 248:184-194. DOI: 10.1016/j.amc.2014.09.101.

Lotfi EM, Maziane M, Hattaf K, Yousti N. Partial differential equations of an epidemic model with spatial diffusion. International Journal of Partial Differential Equations. 2014: article ID 186437, 6 pages. DOI: 10.1155/2014/186437.

Macufa MM, Mayer JF, Krindges A. Diffusion of Malaria in Mozambique. Modeling with computational simulations. Biomatemática. 2015; 25:161-184.

Maidana NA, Yang HM. A spatial model to describe the Dengue propagation. TEMA, Tend. Mat. Apli. Comp. 2007; 8(1):83-92.

Mattheij R, Molenar J. Ordinary differential equations in theory and practice. Philadelphia: Society for Industrial and Applied Mathematic. 2002.

Oluyo TO, Adeyemi MO. Mathematical analysis of Zika epidemic model. IOSR-JM. 2017; 12(6):21-33.

Peng R, Zhao XQ. A reaction-diffusion SIS epidemic model in a time periodic environment. Nonlinearity. 2012; 25:1451-1471. DOI: 10.1088/0951-7715/25/5/1451.

Ren X, Tian Y, Liu L, Liu X. A reaction diffusion withen-host HIV model with cell-to-cell transmission. J. Math. Biol. 2018, 76; 1831-1872. DOI:10.1007/s00285-017-1202-x.

Shutt DP, Manore CA, Pankavich S, Porter AT, Del Valle SY. Estimating the reproductive number, total outbreak size, and reporting rates for Zika epidemics in South and Central America. Epidemics. 2017; 21:63-79. DOI:10.1016/j.epidem.2017.06.005.

Tiemi TT, Maidan NA, Ferreira WC, Paulino P, Yang H.Mathematical Models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind. Bulletin of Mathematical Biology. 2005; 67:509-528. DOI: 10.1016/j.bulm.2004.08.005.

Van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 2002; 180:29-48. DOI: 10.1016/S0025-5564(02)00108-6.

Zhang G, Xiao A, Zhou J. Implicit–explicit multistep finite-element methods for nonlinear convection-diffusion-reaction equations with time delay. Int. J. of Computer Mathematics. 2018; 95(12): 2496-2510. DOI: 10.1080/00207160.2017.1408802.

Downloads

Publicado

2020-12-25

Como Citar

Delgado Moya, E. M. (2020). Mathematical models for the study of Zika diffusion with exposed state and delay. Selecciones Matemáticas, 7(02), 192-201. https://doi.org/10.17268/sel.mat.2020.02.01

Edição

Seção

Articles