Spatiotemporal Patterns in Hantavirus Infection: A Review of Mathematical Models
Hantaviruses are a group of RNA viruses that cause hantavirus pulmonary syndrome (HPS) and hemorrhagic fever with renal syndrome (HFRS). These diseases are transmitted to humans through contact with infected rodents or their excretions. Hantavirus infection is a significant public health problem in many parts of the world, and mathematical models have been developed to help understand and predict the spatiotemporal patterns of hantavirus epidemics.
In 2002, Abramson and Kenkre published a paper in the journal Physical Review E, in which they proposed a mathematical model for the spatiotemporal patterns of hantavirus infection. Their model considered the dynamics of the virus in a population of rodents and humans, and included parameters such as the rate of transmission, the incubation period, and the recovery rate. The authors used numerical simulations to study the effects of these parameters on the spread of the virus.
In a follow-up paper published in 2003 in the Bulletin of Mathematical Biology, Abramson and Kenkre extended their model to include traveling waves of infection. They showed that the waves could be caused by the migration of infected rodents or by the seasonal fluctuation in rodent populations. The authors also compared the predictions of their model to field data from a hantavirus outbreak in New Mexico.
Another group of researchers, Allen, Langlais, and Phillips, developed a model in 2003 that considered the co-infection of two different hantaviruses in a single host population. Their model included parameters such as the transmission rates of each virus and the cross-immunity between the two viruses. The authors used numerical simulations to study the effects of these parameters on the dynamics of the co-infection.
In 2006, Allen, McCormack, and Jonsson developed a mathematical model for hantavirus infection in rodents. Their model considered the dynamics of the virus in a population of rodents, including parameters such as the birth rate, death rate, and transmission rate. The authors used numerical simulations to study the effects of these parameters on the spread of the virus.
Barbera, Curro, and Valenti developed a hyperbolic reaction-diffusion model for hantavirus infection in 2008. Their model considered the dynamics of the virus in a population of rodents and included parameters such as the rate of transmission, the incubation period, and the recovery rate. The authors used numerical simulations to study the effects of these parameters on the spread of the virus.
Several other researchers have also developed mathematical models for hantavirus infection, using a variety of numerical methods. Aydin, Sezer, and Kocayigit used Bernstein polynomials to determine timelike curves of constant breadth in Minkowski 3-space in 2018. Bildik and Deniz used Taylor collocation and Adomian decomposition method for systems of ordinary differential equations in 2015. D’Ambrosio, Ferro, Jackiewicz, and Paternoster used two-step almost collocation methods for ordinary differential equations in 2010.
Goh, Ismail, Noorani, and Hashim used the variational iteration method to study the dynamics of hantavirus infection in 2009. Gökdoğan, Merdan, and Yildirim used a multistage differential transformation method for approximate solution of the hantavirus infection model in 2012. Guo and Wang used a spectral collocation method for solving initial value problems of first order ordinary differential equations in 2010.
Işik, Güney, and Sezer used Bernstein series solutions of pantograph equations using polynomial interpolation in 2012. Işik and Sezer used Bernstein series solution of a class of Lane-Emden type equations in 2013. They also used a rational approximation based on Bernstein polynomials for high order initial and boundary values problems in 2011, and Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel in 2011. Işik, Sezer, and Güney used Bernstein series solution of linear second-order partial differential equations with mixed conditions in 2014.
Sezer, Gülsu, and Tanay used a rational Chebyshev collocation method for solving higher-order linear ordinary differential equations in 2011. Wang, Meng, and Fang used efficient implementation of RKN-type Fourier collocation methods for second-order differential equations in 2017. Wang and Guo used Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations in 2012. Wu and Wang used exponential Fourier collocation methods for first-order differential equations in 2018.
Yap, Ismail, and Senu used an accurate block hybrid collocation method for third order ordinary differential equations in 2014. Yüzbaşı used a collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations in 2016. Yüzbaşı and Sezer used an exponential matrix method for numerical solutions of hantavirus infection model in 2013, and for solving systems of linear differential equations in 2013. Yüzbaşı used numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials in 2013. Yüzbaşı and Yıldırım used a Laguerre approach for solving of the systems of linear differential equations and residual improvement in 2021, and Laguerre collocation method for solutions of systems of first order linear differential equations in 2018.
In conclusion, mathematical models have been developed to help understand and predict the spatiotemporal patterns of hantavirus epidemics. These models have used a variety of numerical methods, including Bernstein polynomials, collocation methods, and numerical simulations. The insights gained from these models can help public health officials develop effective strategies for controlling the spread of hantaviruses.
- Hantavirus infection model
- Numerical solutions
- Bernstein polynomials
- Mathematical modeling of infectious diseases
- Computational methods for epidemiology
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