Diffusive instabilities in a hyperbolic activator-inhibitor system with superdiffusion

Dynamics of a cross-superdiffusive SIRS model with delay effects in transmission and treatment

We investigate the dynamics of a SIRS epidemiological model taking into account cross-superdiffusion and delays in transmission, Beddington-DeAngelis incidence rate and Holling type II treatment. The superdiffusion is induced by inter-country and inter-urban exchange. The linear stability analysis for the steady-state solutions is performed, and the basic reproductive number is calculated. The sensitivity analysis of the basic reproductive number is presented, and we show that some parameters strongly influence the dynamics of the system. A bifurcation analysis to determine the direction and stability of the model is carried out using the normal form and center manifold theorem. The results reveal a proportionality between the transmission delay and the diffusion rate. The numerical results show the formation of patterns in the model, and their epidemiological implications are discussed.

Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models

In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions.

Computer simulation of the dynamics of a spatial susceptible-infected-recovered epidemic model with time delays in transmission and treatment

BACKGROUND AND OBJECTIVE

In this work, we analyze the spatial-temporal dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively.

METHODS

We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established.

RESULTS

We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain.

CONCLUSIONS

The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.

Base pair opening in a damped helicoidal Joyeux-Buyukdagli model of DNA in an external force field

Upon the Joyeux-Buyukdagli model of DNA, the helicoidal interactions are introduced, and their effects on the dynamical behaviors of the molecule investigated. A theoretical framework for the analysis is presented in an external force field, taking into account Stokes and hydrodynamics viscous forces. In the semi-discrete approximation, the dynamics of the molecule is found governed by the cubic complex Ginzburg-Landau (CGL) equation. By choosing an appropriate decoupling ansatz, the cubic CGL equation is transformed into a nonlinear differential equation whose analytical solitary wave-like solutions can be explored by means of the direct method, which is more tractable in case where the form of soliton solutions is known. Based on this, a dissipative bright-like soliton solution is obtained. Numerical experiments have been done, and relevant results were brought out, such as the quantitative and qualitative influences of the helical interactions on the parameters of the traveling bubble. The important role-played by these interactions in the DNA biological processes is brought out, showing that depending on the wave number, their effects can increase, decrease, or keep constant the bubble angular frequency, velocity, amplitude, and width, as well as the energy involved by enzymes in the initiation of DNA biological processes. This can prevent some coding or reading errors and resulting genetic damages. Analytical predictions and numerical experiments were in good agreement.