Mathematics Days in Sofia – 2026
Section “Differential Equations and Mathematical Physics”
Participants
Invited Speaker
- Mirko Tarulli, American University in Bulgaria and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
Contributors
- Bogdan Djordjevic, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia
- Dumitru Cozma, Ion Creanga State Pedagogical University of Chisinau, Moldova
- Georgi Boyadzhiev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
- Kaloian Stoilov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
- Ognyan Christov, Sofia University “St. Kliment Ohridski”, Bulgaria
- Omer Korat, Deloitte Israel, Israel
- Tsvetana Stoyanova, Sofia University “St. Kliment Ohridski”, Bulgaria
- Vladimira Suvandjieva, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
Program and Abstracts
We investigate the problem of distinguishing between a center and a focus (the center focus problem) for planar polynomial differential systems having a singular point with purely imaginary eigenvalues (a weak focus). This problem is one of the classical unresolved problems in the qualitative theory of differential equations and arose as part of investigation of the local 16th Hilbert problem which deals with the estimation of the number of small amplitude limit cycles that can bifurcate from a singular point a weak focus.
In this work, we study the center-focus problem for planar polynomial differential systems with invariant algebraic curves and a weak focus singular point, which is equivalent to the problem of local integrability of such systems in a neighborhood of a weak focus. The relationship between the existence of invariant algebraic curves, Lyapunov quantities and the integrability of polynomial differential systems is established. The center-focus problem is solved for several families of cubic differential systems with singular points of weak focus type having invariant algebraic curves (invariant straight lines, invariant conics, and invariant cubics). Finally, we show that the existence of invariant algebraic curves influences the number limit cycles.
This work was supported by the National Agency for Research and Development of the Republic of Moldova under project number “25.80012.5007.76SE Qualitative and algebraic investigation of differential models”.
In this talk is construct a spatial SEIR model of disease spread in bounded domain. In this model incubational and recovery periods are implemented in the principal term of the system and the result is a quasi-linear reaction-diffusion system with delays.
The spatial SEIR models allow more precise modelling of epidemic than the classical ones since the spread of the disease is described not only in time, but in space as well. Solving the inverse problem one can determine the entering points of the infection in the area in study.
Some important qualitative proper es of quasi-linear reaction-diffusion systems with delays are given as well, in between them the validity of the comparison principle, existence and uniqueness of classical solutions.
Axial Spondyloarthritis (axSpA) is a chronic inflammatory disease of the intervertebral discs and sacroiliac joints. Initially it manifests in bone erosion of the vertebrae, caused by immune disruption originating from the gut. In some patients axSpA progresses into ankylosing spondylitis, the ossification of tender tissue near the vertebrae, ultimately resulting in fusion of the vertebrae, and lack of mobility. Most often the disease is initiated around the sacroiliac joints, in the lowest parts of the spine, and it spreads upwards.
Currently no mathematical model of axSpA pathology exists. We present two models for the intervertebral disc pathology: one for the initial disease stage of bone resorption and another for the second stage of soft tissue ossification. Both models are based on systems of PDEs in 1-D with moving boundary that describe the spatio-temporal dynamics for the pro-inflammatory cytokines, immune cells and bone cells.
