Research

1. Nonstandard finite difference methods: We examine mostly higher-order nonstandard finite difference methods, and apply these schemes to solve several differential equations.
   
   • B. T. "An insight on some properties of high order nonstandard linear multistep methods." 2025, submitted. preprint.
   • I. Faragó, G. Svantnerné Sebestyén, B. T. "A Nonstandard Finite Difference Method for a General Epidemic Model." International Conference on Large-Scale Scientific Computing. Cham: Springer Nature Switzerland, 2023. conference proceeding.
2. Epidemic models with general incidence functions: In the classic epidemic models, the transmission of illness is represented by the product of the number of susceptibles (S) and infectious (I).
   In these works, this term is replaced by a general g(S,I) function.
   We observe the analytic solution of these equations, and construct such numerical schemes, which preserve the properties of the continuous model.
   • B. T., G. Svantnerné Sebestyén, I. Faragó "High-order reliable numerical methods for epidemic models with non-constant recruitment rate." Appl. Num. Math. 206 (2024): 75-93. paper.
3. Space-dependent epidemic models: We examine the transmission of some property (e.g. illness, information, fire) in a continuous domain.
   We examine the properties of the analytic solution of the corresponding integro differential equations (existence, positivity), and then construct such numerical schemes that preserve the properties of the continuous model.
   An exception is the paper written with Daniel Keliger and Illes Horvath, in which the differential equation defined on the limit of graphs has a very similar structure to those mentioned before.
   • B. T., Y. Hadjimichael. "High order discretization methods for spatial-dependent epidemic models." Math. Comp. Sim. 198 (2022): 211-236. paper.
   • B. T., I. Faragó, R. Horváth, D. Repovš "Qualitative properties of space-dependent SIR models with constant delay and their numerical solutions." Comp. Meth. Appl. Math. 22.3 (2022): 713-728. paper.
   • D. Keliger, I. Horváth, B. T. "Local-density dependent Markov processes on graphons with epidemiological applications." Stoch. Proc. Appl. 148 (2022): 324-352. paper.
   • P. Csomós, B. T. "Operator splitting for space-dependent epidemic model." Appl. Num. Math. 159 (2021): 259-280. paper.
   • B. T., R. Horváth, I. Faragó "Space dependent models for studying the spread of some diseases." Comp. Math. Appl. 80.2 (2020): 395-404. paper.
4. The mathematical models of the ecological collapse of Easter Island: The civilization of Easter Island collapsed before the arrival of European settlers, while all the trees of the island disappeared.
   In our model, we not only take into account the effect of humans, but also the rats brought to the island by the original polynesian settlers.
   The island is modeled in some papers as an annulus, and in others as a two-dimensional object.
   We examine the effect of diffusion on the stability of the equilibrium of the system.
   • B. T., R. Horváth, I. Faragó "The effect of tree diffusion in a two-dimensional continuous model for Easter Island." European J. Math. 5.3 (2019): 845-857. paper.
   • B. T. "A Continuous Model for the Ecological Collapse of Easter Island." International Conference on Finite Difference Methods. Cham: Springer International Publishing, 2018. conference proceeding.
   • B. T., R. Horváth, I. Faragó "The effect of tree-diffusion in a mathematical model of Easter Island's population." Electr. J. Qualit. Theory of Diff. Eq. 84. (2016): 1-11. paper.
   • B. T. "Analysis of some characteristic parameters in an invasive species model." Ann. Univ. Sci. Budapest. Sect. Comput. 45 (2016): 119-133. paper.

Conferences and workshops: