## Pattern formation in a reaction-diffusion system

Dispersal-induced pattern formation is important both from fundamental and application points of view. Spatial patterns in an ecological system can strongly depend on exogenous factors like arrangement of the natural and artificial physical features of the habitat (topography) and distribution of resources. It is also influenced by endogenous factors (intrinsic biological forces) such as the ecological interactions of individuals. Here we consider various ecological models (both continuous and discrete) with self-diffusion (due to endogenous factors) and cross-diffusion (due to exogeneous factors). We observe a wide range of complex spatiotemporal patterns (like periodic, quasi periodic, chaotic) are observed with respect to the variation of diffusion coefficients and other local interacting parameters of the model.

## Systems Biology

Diseases are manifested through changes in metabolism. It is therefore highly important and relevant to elucidate mechanistic explanation of the disease to gain further insights. Any dysfunction in the secretion of insulin and/or in its use is critical in the development of type 2 diabetes. Mycobacterium tuberculosis (Mtb) inhibits Phosphatidylinositol 3-phosphate (PI3P) formation on phagosomal membrane and disrupts the process of phagosome maturation, the known hostâ€™s innate defense mechanism, to spread in the host body. Cancer is caused due to various disorders in the complex intra-and-inter cellular networks that link tissues and organs. This project will emphasize understanding of complex biological processes and open up new treatment avenues for three diseases: Diabetic, Tuberculosis and Cancer. The objectives are: (i) to study the dynamic interaction of glucose induced insulin secretion mechanism through glucose metabolism and ATP-dependent calcium influx, (ii) to investigate whether modulation of PI3P oscillations through cytosolic calcium is a potential therapeutic intervention strategy for Mycobacterium tuberculosis (Mtb), (iii) to find the predictive value of cancer cell-specific genome-scale metabolic models in ranking cancer drugs.

## Stochastic models in biological systems

The proposal focuses on emergent collective behaviors in complex networks of dynamical systems where each node of the networks is represented by ecological models. We emphasize networks of different topologies, like globally coupled, non-locally coupled and scale-free networks. We consider different parameter situations, viz. identical nodes, nodes with distributed parameters and, then locate the parameter space of the onset of synchrony in ecological networks. We explore whether chimera-like states originate in the networks of identical ecological patches, when the whole population of the network divides into two coexisting synchronous and non-synchronous subpopulations. This is to develop an understanding whether such coexistence of synchronous and non-synchronous states is a better prescription for the survival of species. For this we explore both the steady state behaviors and oscillatory regime of the dynamical units in the networks. We particularly search for multi-clustered inhomogeneous steady states. Furthermore, we consider eco- epidemiological models (where both ecological and epidemiological issues are considered simultaneously) in the network and try to understand the process of spreading of epidemics in an ecological network. Finally, we search for a control strategy to stop the spreading of disease in the network.

## Emergent dynamics in ecological networks: Spreading of disease and its control

Deterministic models assume that the environment is unvarying and therefore, birth rate, death rate, etc. are constant through time and space. However, any environment is bound to be subject to random fluctuations of one sort or another. Particularly, human immunodeficiency virus interacts with CD4+T cells and immune system within human body where the environment is completely uncertain and stochastic. Therefore, one has to consider the system parameters which characterize populations as random or fluctuating, giving rise to a multidimensional stochastic system of differential equations. Emerging data reveal that aging, nutrition, physical activity and other physiological factors have large impact on viral replication and immune system in HIV infected individuals. Due to these factors, virus replication may vary from 10-2500 virons per cell lysis. Thus, incorporation of fluctuating environment is important to make the model biologically close to realistic and to predict the future dynamics of the system accurately. The main objective of this project is to compare the results of deterministic system with its stochastic counter part and to verify whether introduction of noise modifies the deterministic stability threshold of the disease-free and endemic equilibrium.

## COVID-19

A highly infectious disease caused by novel coronavirus spread in Wuhan city of Hubei province in China at the end of 2019. Through population migration, Covid-19 disease became pandemic and spread over more than 211 countries/territories across the globe, causing 12,14,466 confirmed cases of infected individuals and 67,767 confirmed deaths as of April 4, 2020. Mathematical models and computation techniques may play an important role in understanding this epidemic and contribute a lot in policy making to curb disease spreading in a more systematic and effective way. Here we want to propose a mathematical model taking into consideration the epidemiological features of Covid-19 and essential preventive measures. The outcomes will benefit the frontline health professionals and policy makers to define effective control mechanisms.

## NSFD

Discretization of nonlinear ordinary differential equations is important because it, in general, cannot be solved and therefore discretization is inevitable for good approximation of the solutions, and secondly discrete models permits arbitrary time-step units. However, conventional discretization schemes, such as Euler method, Runge-Kutta method, produces spurious solutions which are not observed in its parent model and its dynamics depend on the step-size. For example, the dynamics of logistic differential equation in continuous system is very simple: any solution starting with positive initial value reaches to nontrivial equilibrium point, implying that the nontrivial equilibrium point is stable. Dynamics of the corresponding discrete logistic model formulated by Euler forward method is, however, very complex. The nontrivial equilibrium point is stable if the step-size is relatively small. The system shows very complicated period doubling bifurcation to chaos as the step-size is gradually increased. Thus the dynamics of the discrete model depends on the step-size. Another important drawback of the conventional discrete model is that the positivity of its solutions does not hold for all positive initial values. Any finite-difference scheme that allows negative solutions will have numerical instabilities. So the following questions arise: (i) Can you construct a discrete system which will show the exact behavior of the corresponding continuous system? (ii) Are the solutions of the proposed discrete model remain positive for all positive initial conditions? (ii) Are the dynamics of the system independent of the step-size?

For such dynamic preserving discretization of the continuous-time system, one can go for non-standard finite difference (NSFD) scheme introduced by Mickens in 1989. A finite difference scheme is called non-standard finite difference (NSFD) method if at least one of the following conditions is met: (1) denominator functions for the discrete derivatives must, in general, be expressed in terms of more complicated functions of the step-sizes than those conventionally used, (2) nonlinear terms should, in general, be replaced by non-local discrete representation, i.e. by a suitable function of several points of mesh.

In this project, I would like to propose some dynamics preserving NSFD models (dimension) of nonlinear biological phenomena. It will be shown that all the dynamic properties of NSFD models are identical with the corresponding continuous models. It will also be shown that the proposed discrete models are robust even when the discretization time-step is large.