Modeling the recent outbreak of COVID-19 in India and its control strategies

The recent emergence of COVID-19 has drawn attention to the various methods of disease control. Since no proper treatment is available till date and the vaccination is restricted to certain age groups, also vaccine efficacy is still under progress, the emphasis has been given to the method of isolation and quarantine. This control is induced by tracing the contacts of the infectious individuals, putting them to the quarantine class and based on their symptoms, classifying them either as the susceptible or sick individuals and moving the sick individuals to the isolated class. To track the current pandemic situation of COVID-19 in India, we consider an extended Susceptible-Exposed-Quarantine-Infected-Isolated-Recovered (SEQ1IQ2R) compartmental model along with calculating its control reproductive number Rc. The disease can be kept in control if the value of Rc remains below one. This “threshold” value of Rc is used to optimize the period of quarantine, and isolation and have been calculated in order to eradicate the disease. The sensitivity analysis of Rc with respect to the quarantine and isolation period has also been done. Partial rank correlation coefficient method is applied to identify the most significant parameters involved in Rc. Based on the observed data, 7-days moving average curves are plotted for prelockdown, lockdown and unlock 1 phases. Following the trend of the curves for the infection, a generalized exponential function is used to estimate the data, and corresponding 95% confidence intervals are simulated to estimate the parameters. The effect of control measures such as quarantine and isolation are discussed. Following various mathematical and statistical tools, we systematically explore the impact of lockdown strategy in order to control the recent outbreak of COVID-19 transmission in India

Spatiotemporal pattern induced by self and cross-diffusion in a spatial Holling-Tanner model

In this paper, we have made an attempt to provide a unified framework to understand the complex spatiotemporal patterns induced by self and cross diffusion in a spatial Holling-Tanner model forphytoplankton-zooplankton-fish interaction. The effect of critical wave length which can drive the system to instability is investigated. We have examined the criterion between two cross-diffusivity (constant and timevarying)on the stability of the model system and for diffusive instability to occur. Based on these conditions and by performing a series of extensive simulations, we observed the irregular patterns, stationary strips, spots, and strips-spots mixture patterns. Numerical simulation results reveal that the regular strip-spot mixture patterns prevail over the whole domain on increasing the values of self- diffusion coefficients of phytoplankton and zooplankton and the dynamics of the system do not undergo any further changes

Complex dynamics of a three species food-chain model with Holling type IV functional response

In this paper, dynamical complexities of a three species food chain model with Holling type IV predator response is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence criterion of the food chain model is obtained. Numerical bifurcation analysis reveals the chaotic behavior in a narrow region of the bifurcation parameter space for biologically realistic parameter values of the model system. Transition to chaotic behavior is established via period-doubling bifurcation and some sequences of distinctive period-halving bifurcation leading to limit cycles are observed

Fuzzy modeling for the spread of influenza virus and its possible control

In this paper, we analyze a model of Influenza spread with an asymptotic transmission rate, wherein the disease transmission rate and death rate are considered as fuzzy sets. Comparative studies of the equilibrium points of the disease for the classical and fuzzy models are performed. Using the concept of probability measure and fuzzy expected value, we obtain the fuzzy basic reproduction number for groups of infected individuals with different virus loads. Further, a basic reproduction number for the classical and the fuzzy model are compared. Finally, a program based on the basic reproduction value of disease control is suggested and the numerical simulations are carried out to illustrate the analytical results

Deterministic chaos versus stochastic oscillation in a prey-predator-top predator model

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings

Spatiotemporal characteristics in systems of diffusively coupled excitable slow-fast FitzHugh-Rinzel dynamical neurons

In this paper, we study an excitable, biophysical system that supports wave propagation of nerve impulses. We con- sider a slow-fast, FitzHugh-Rinzel neuron model where only the membrane voltage interacts diffusively, giving rise to the formation of spatiotemporal patterns. We focus on local, nonlinear excitations and diverse neural responses in an excitable 1- and 2-dimensional configuration of diffusively coupled FitzHugh-Rinzel neurons. The study of the emerg- ing spatiotemporal patterns is essential in understanding the working mechanism in different brain areas. We derive analytically the coefficients of the amplitude equations in the vicinity of Hopf bifurcations and characterize various patterns, including spirals exhibiting complex geometric substructures. Further, we derive analytically the condition for the development of antispirals in the neighborhood of the bifurcation point. The emergence of broken target waves can be observed to form spiral-like profiles. The spatial dynamics of the excitable system exhibits 2- and multi-arm spirals for small diffusive couplings. Our results reveal a multitude of neural excitabilities and possible conditions for the emergence of spiral-wave formation. Finally, we show that the coupled excitable systems with different firing characteristics, participate in a collective behavior that may contribute significantly to irregular neural dynamics

Ecological system with fear induced group defence and prey refuge

In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction model that includes the following: (i) fear effect incorporated in prey birth rate; (ii) group defence of prey against predators; and (iii) prey refuge. We provide comprehensive mathematical analysis of extinction and persistence scenarios for both prey and predator species. To better explore the dynamics of the system, a thorough investigation of bifurcation analysis has been performed using fear level, prey birth rate, and prey death rate caused by intra-prey competition as bifurcation parameter. All potential occurrences of bi-stability dynamics have also been investigated for some relevant sets of parametric values. Our numerical evaluations show that high levels of fear can stabilize the prey-predator system by ruling out the possibility of periodic solutions. Also, our model Hopf bifurcation is subcritical in contrast to traditional prey-predator models, which ignore the cost of fear and have supercritical Hopf bifurcations in general. In contrast to the general trend, predator species go extinct at higher values of prey birth rates. We have also found that, contrary to the typical tendency for prey species to go extinct, both prey and predator populations may coexist in the system as intra-prey competition level grows noticeably. The stability and Turing instability of associated spatial model have also been investigated analytically. We also perform the numerical simulation to observe the effect of different parameters on the density distribution of species. Different types of spatiotemporal patterns like spot, mixture of spots and stripes have been observed via variation of time evolution, diffusion coefficient of predator population, level of fear factor and prey refuge. The fear level parameter (k) has a great impact on the spatial dynamics of model system

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics

In this paper, we report on the generation and propagation of traveling pulses in a homogeneous network of diffusively coupled, excitable, slow-fast dynamical neurons. The spatially extended system is modelled using the nearest neighbor coupling theory, in which the diffusion part measures the spatial distribution of the coupling topology. We derive analytically the conditions for traveling wave profiles that allow the construction of the shape of traveling nerve impulses. The analytical and numerical results are used to explore the nature of the propagating pulses. The symmetric or asymmetric nature of the traveling pulses is characterized and the wave velocity is derived as a function of system parameters. Moreover, we present our results for an extended excitable medium by considering a slow-fast biophysical model with a homogeneous, diffusive coupling that can exhibit various traveling pulses. The appearance of series of pulses is an interesting phenomenon from biophysical and dynamical perspective. Varying the perturbation and coupling parameters, we observe the propagation of activities with various amplitude modulations and transition phases of different wave profiles that affect the speed of the pulses in certain parameter regimes. We observe different types of traveling pulses, such as envelope solitons and multi-bump solutions and show how system parameters and the coupling play a major role in the formation of different traveling pulses. Finally, we obtain the conditions for stable and unstable plane waves

Introduction to mathematical modeling and chaotic dynamics

The book first covers the necessary mathematical preliminaries, including testing of stability. It then describes the modeling of systems from natural science, focusing on one- and two-dimensional continuous and discrete time models. Moving on to chaotic dynamics, the authors discuss ways to study chaos, types of chaos, and methods for detecting chaos. They also explore chaotic dynamics in single and multiple species systems. The text concludes with a brief discussion on models of mechanical systems and electronic circuits