Dynamical Systems
I have worked on the bifurcations and dynamics of high dimensional dynamical systems, both within the realm of smooth and piecewise smooth dynamical systems. Dimensionality of a dynamical system increases due to introduction of time delay, coupling two or more lower dimensional systems or interacting systems in general on a complex network.
The questions that I have been interested in are as follows:
In a high dimensional dynamical system how does the bifurcations unfold as parameters of the system change?
What is unique characteristics that emerge due to high dimensional characteristics of the system that is not readily observed in a lower dimensional system?
Are certain phenomenon observed in real physical problems of interests readily explainable in terms of a high dimensional dynamical system framework?
I have investigated some of these questions in my research using toy models that include Henon mapping, Lozi mapping, Standard map (Kicked rotator), Rossler oscillator, Lorenz oscillator, and complex networks.
Examples of high - dimensional systems formed by coupling individual systems can exhibit phenomenon such as synchronization. A general formalism to study synchronization in such systems is the Master Stability Function (MSF) approach. I have worked on one aspect of the master stability analysis for coupled flows.
Coupled and forced/driven/perturbed dynamical systems:
As a detour to high dimensional systems, I have investigated quasiperiodically forced maps in the weakly dissipation regime. The problem is of practical interest since many practical systems are weakly dissipative and it would be interesting to know what happens to attractors of the system when forced. The investigation of this problem is made challenging/interesting due to multistability and chaotic transients.
Piecewise smooth dynamical systems are smooth but not differentiable at a set of boundary points. These systems occur in the study of electrical circuits due to switches and mechanical system that have impacting parts such as gears, or social systems where continuous change can induce discrete actions. Well known examples of piecewise smooth systems are tent map (in one dimension) and Lozi map (two dimensions). I studied the Lozi map by generalizing it to higher dimensions. Noteworthy were the emergence of hyperchaotic orbits, and convergence of fixed point dynamics in the parameter space as a function of dimensions (see . https://doi.org/10.1080/10236198.2022.2041625)
Disease Dynamics
Infectious diseases are transmitted either via direct contact between humans or through an intermediary vector (mosquitoes, flies, snails etc).
The questions I have been interested in how threshold behavior is influenced by factors that include seasonality and climate change, complex interactions between multiple hosts and vectors, emergent disease properties on a complex interaction networks that represent spatio-temporal heterogeneity. I am also interested in resilience of infectious diseases to perturbations that could be caused by a change in environmental or behavioral properties of hosts/vectors.
The impact of climate change on transmission dynamics.
Resilience of disease free or endemic state as factors including transmission parameter among others change.
Spatial heterogeneity and disease control.
I studied these questions on models of Flu, Dengue, Visceral Leishmaniasis, Covid19, and Onchocerciasis. We reported for example non-chaoitc attractors in models of Flu assuming a quasi-periodic change in seasonality.
Resilience
I also investigated the resilience of the disease-free and endemic states, i.e. their ability to persist under perturbation in epidemiological and ecological conditions i.e. changing vector biting preferences, additional host availability that are either competitive or dead end in the transmission dynamics.
Spatial Heterogeneity
Complex interactions patterns arise due to interactions subject to constraints of space. Typically such systems are studied through the framework of couple systems, where the coupling could represent two or more interacting subsystems. The tools and tricks of complex network analysis are readily applied to quantify the emergent dynamical properties and their implication for disease control.
We developed a model to study dengue transmission dynamics in a set of cities in Malaysia, that are frequented by commuters for work and other activities. An initial assessment of travel patterns gave us the following network of human interactions. We found that typical the node with more connections or the hubs are key to control with minimal effort.
Agent-Based models
Ordinary differential equations (ODE) based models assume a homogeneous mixing between individuals and are therefore il - equipped to account for spatial heterogeneities that exists in the real world. An individual based model or an agent based model addresses this issue by considering individuals as agents who interact with other agents via a set of rules and in the process transmit the disease.
I worked in a collaborative effort to develop on an agent-based model for Covid19 in Hillsborough County, Florida, USA. This involved creating a realistic network where agents interact, a movement model based on origin-destination matrix, and a disease model. This work was made part of a dashboard available at this link: http://www.seir-abm.online:8050/ and a peer reviewed paper published here.