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The spread of infectious diseases often presents the emergent properties, which leads to more difficulties in prevention and treatment. In this paper, the SIR model with both delay and network is investigated to show the emergent properties of the infectious diseases' spread. The stability of the SIR model with a delay and two delay is analyzed to illustrate the effect of delay on the periodic outbreak of the epidemic. Then the stability conditions of Hopf bifurcation are derived by using central manifold to obtain the direction of bifurcation, which is vital for the generation of emergent behavior. Also, numerical simulation shows that the connection probability can affect the types of the spatio-temporal patterns, further induces the emergent properties. Finally, the emergent properties of COVID-19 are explained by the above results. [ABSTRACT FROM AUTHOR]

The epidemic often spreads along social networks and shows the effect of memorability on the outbreak. But the dynamic mechanism remains to be illustrated in the fractional-order epidemic system with a network. In this paper, Turing instability induced by the network and the memorability of the epidemic are investigated in a fractional-order epidemic model. A method is proposed to analyze the stability of the fractional-order model with a network through the Laplace transform. Meanwhile, the conditions of Turing instability and Hopf bifurcation are obtained to discuss the role of fractional order in the pattern selection and the Hopf bifurcation point. These results prove the fractional-order epidemic model may describe dynamical behavior more accurately than the integer epidemic model, which provides the bridge between Turing instability and the outbreak of infectious diseases. Also, the early warning area is discussed, which can be treated as a controlled area to avoid the spread of infectious diseases. Finally, the numerical simulation of the fractional-order system verifies the academic results is qualitatively consistent with the instances of COVID-19. [ABSTRACT FROM AUTHOR]

Turing instability is a prominent feature of reaction–diffusion systems, which is widely investigated in many fields, such as ecology, neurobiology, chemistry. However, although the inhomogeneous diffusion between prey and predators exist in their network space, there are few considerations on how network diffusion affects the stability of prey–predator models. Therefore, in this paper we study the pattern dynamics of a modified reaction–diffusion Holling–Tanner prey–predator model over a random network. Specifically, we study the relationship between the node degrees of the random network and the eigenvalues of the network Laplacian matrix. Then, we obtain conditions under which the network system instability, Hopf bifurcation as well as Turing bifurcation occur. Also, we find an approximate Turing instability region of the diffusion coefficient and the connection probability of the network. Finally, we apply the mean-field approximation theory with numerical simulation to confirm the correctness of our results. The instability region indicates the random migration of the prey and predators among different communities. [ABSTRACT FROM AUTHOR]

P53 plays a vital role in DNA repair, and several mathematical models of the p53-Mdm2 feedback loop were used to explain the biological mechanism. In this paper, a p53-Mdm2 model described by a delay reaction–diffusion equation is studied both analytically and numerically. This research aims to provide an understanding of the impact of delay and sustained pressure on the p53-Mdm2 dynamics and tries to explain some biological mechanism. It is found that the type of pattern formation is affected by Hopf bifurcation. Also, the amplitude equation in delay diffusive system is derived and it is shown that sustained stress plays an essential role in the function of p53. Finally, simulation is used to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey system with double Allee effects and time delay is considered. On the one hand, by using time delay as bifurcation parameter, the stability of the interior equilibrium point is studied. It is shown that under certain assumptions the equilibrium point of the delay model is locally asymptotically stable for all delay values, otherwise, there is a critical value, such that the equilibrium point is locally stable when time delay is less than the critical value, and Hopf bifurcation occurs when the delay crosses the critical value. Numerical simulations are carried out to validate the analytical results. On the other hand, we investigate the following four submodels respectively: the time-delayed system without Allee effect, the time-delayed system with Allee effect on predator, the time-delayed system with Allee effect on prey and the time-delayed system with double Allee effects. Compared with the situation without Allee effects, Allee effect on predator causes a decline of the equilibrium state of the predator, the equilibrium state of the prey and critical time delay are almost unaltered; Allee effect on prey and Allee effect on prey and predator all lead to the descent of the equilibrium state of the predator and prey, inversely, and critical time delay has an obvious rise. A comparison of the four cases indicates that Allee effect can alter the magnitudes of predator density and prey density at future time, the time that predator and prey cost to keeping steady state is also changed. [ABSTRACT FROM AUTHOR]