Hopf bifurcation and patterns in a modified SIR model

Infectious diseases have constantly threatened human safety because the diffusion of the susceptible and infected may make more individuals infected and even die. In this paper, a modified SIR model with both external stimulus and diffusion is considered to illustrate the dynamical mechanism of the periodic outbreak and pattern formation. Firstly, we propose a modified SIR model based on the propagation behaviour of infectious diseases to show the effects of the different parameters and diffusion on the outbreak. The Hopf bifurcation and multiscale methods are performed to analyze the stability of this model, which explains the dynamical mechanism of the periodic outbreak. Then, the pattern formation and Turing instability are discussed through comparison principles to reveal the role of periodic disturbances and diffusion in selecting pattern formation. Also, we find rich patterns that may occur when the frequency modulation is close to the intrinsic frequency. Finally, our theoretical results are verified by numerical simulation

Bifurcation and pattern dynamics in the nutrient-plankton network

This paper used a Holling-IV nutrient-plankton model with a network to describe algae's spatial and temporal distribution and variation in a specific sea area. The stability and bifurcation of the nonlinear dynamic model of harmful algal blooms (HABs) were analyzed using the nonlinear dynamic theory and de-eutrophication's effect on algae's nonlinear dynamic behavior. The conditions for equilibrium points (local and global), saddle-node, transcritical, Hopf-Andronov and Bogdanov-Takens (B-T) bifurcation were obtained. The stability of the limit cycle was then judged and the rich and complex phenomenon was obtained by numerical simulations, which revealed the robustness of the nutrient-plankton system by switching between nodes. Also, these results show the relationship between HABs and bifurcation, which has important guiding significance for solving the environmental problems of HABs caused by the abnormal increase of phytoplankton

Data_Sheet_2_Dynamics of antiphase bursting modulated by the inhibitory synaptic and hyperpolarization-activated cation currents.DOCX

Antiphase bursting related to the rhythmic motor behavior exhibits complex dynamics modulated by the inhibitory synaptic current (Isyn), especially in the presence of the hyperpolarization-activated cation current (Ih). In the present paper, the dynamics of antiphase bursting modulated by the Ih and Isyn is studied in three aspects with a theoretical model. Firstly, the Isyn and the slow Ih with strong strength are the identified to be the necessary conditions for the antiphase bursting. The dependence of the antiphase bursting on the two currents is different for low (escape mode) and high (release mode) threshold voltages (Vth) of the inhibitory synapse. Secondly, more detailed co-regulations of the two currents to induce opposite changes of the bursting period are obtained. For the escape mode, increase of the Ih induces elevated membrane potential of the silence inhibited by a strong Isyn and shortened silence duration to go beyond Vth, resulting in reduced bursting period. For the release mode, increase of the Ih induces elevated tough value of the former part of the burst modulated by a nearly zero Isyn and lengthen burst duration to fall below Vth, resulting in prolonged bursting period. Finally, the fast-slow dynamics of the antiphase bursting are acquired. Using one-and two-parameter bifurcations of the fast subsystem of a single neuron, the burst of the antiphase bursting is related to the stable limit cycle, and the silence modulated by a strong Isyn to the stable equilibrium to a certain extent. The Ih mainly modulates the dynamics within the burst and quiescent state. Furthermore, with the fast subsystem of the coupled neurons, the silence is associated with the unstable equilibrium point. The results present theoretical explanations to the changes in the bursting period and fast-slow dynamics of the antiphase bursting modulated by the Isyn and Ih, which is helpful for understanding the antiphase bursting and modulating rhythmic motor patterns.</p

Data_Sheet_3_Dynamics of antiphase bursting modulated by the inhibitory synaptic and hyperpolarization-activated cation currents.ZIP

Antiphase bursting related to the rhythmic motor behavior exhibits complex dynamics modulated by the inhibitory synaptic current (Isyn), especially in the presence of the hyperpolarization-activated cation current (Ih). In the present paper, the dynamics of antiphase bursting modulated by the Ih and Isyn is studied in three aspects with a theoretical model. Firstly, the Isyn and the slow Ih with strong strength are the identified to be the necessary conditions for the antiphase bursting. The dependence of the antiphase bursting on the two currents is different for low (escape mode) and high (release mode) threshold voltages (Vth) of the inhibitory synapse. Secondly, more detailed co-regulations of the two currents to induce opposite changes of the bursting period are obtained. For the escape mode, increase of the Ih induces elevated membrane potential of the silence inhibited by a strong Isyn and shortened silence duration to go beyond Vth, resulting in reduced bursting period. For the release mode, increase of the Ih induces elevated tough value of the former part of the burst modulated by a nearly zero Isyn and lengthen burst duration to fall below Vth, resulting in prolonged bursting period. Finally, the fast-slow dynamics of the antiphase bursting are acquired. Using one-and two-parameter bifurcations of the fast subsystem of a single neuron, the burst of the antiphase bursting is related to the stable limit cycle, and the silence modulated by a strong Isyn to the stable equilibrium to a certain extent. The Ih mainly modulates the dynamics within the burst and quiescent state. Furthermore, with the fast subsystem of the coupled neurons, the silence is associated with the unstable equilibrium point. The results present theoretical explanations to the changes in the bursting period and fast-slow dynamics of the antiphase bursting modulated by the Isyn and Ih, which is helpful for understanding the antiphase bursting and modulating rhythmic motor patterns.</p

Data_Sheet_1_Dynamics of antiphase bursting modulated by the inhibitory synaptic and hyperpolarization-activated cation currents.ZIP

Antiphase bursting related to the rhythmic motor behavior exhibits complex dynamics modulated by the inhibitory synaptic current (Isyn), especially in the presence of the hyperpolarization-activated cation current (Ih). In the present paper, the dynamics of antiphase bursting modulated by the Ih and Isyn is studied in three aspects with a theoretical model. Firstly, the Isyn and the slow Ih with strong strength are the identified to be the necessary conditions for the antiphase bursting. The dependence of the antiphase bursting on the two currents is different for low (escape mode) and high (release mode) threshold voltages (Vth) of the inhibitory synapse. Secondly, more detailed co-regulations of the two currents to induce opposite changes of the bursting period are obtained. For the escape mode, increase of the Ih induces elevated membrane potential of the silence inhibited by a strong Isyn and shortened silence duration to go beyond Vth, resulting in reduced bursting period. For the release mode, increase of the Ih induces elevated tough value of the former part of the burst modulated by a nearly zero Isyn and lengthen burst duration to fall below Vth, resulting in prolonged bursting period. Finally, the fast-slow dynamics of the antiphase bursting are acquired. Using one-and two-parameter bifurcations of the fast subsystem of a single neuron, the burst of the antiphase bursting is related to the stable limit cycle, and the silence modulated by a strong Isyn to the stable equilibrium to a certain extent. The Ih mainly modulates the dynamics within the burst and quiescent state. Furthermore, with the fast subsystem of the coupled neurons, the silence is associated with the unstable equilibrium point. The results present theoretical explanations to the changes in the bursting period and fast-slow dynamics of the antiphase bursting modulated by the Isyn and Ih, which is helpful for understanding the antiphase bursting and modulating rhythmic motor patterns.</p

Pattern dynamics in the epidemic model with diffusion network

It is well known that the outbreak of infectious diseases is affected by the diffusion of the infected. However, the diffusion network is seldom considered in the network-organized SIR model. In this work, we investigate the effect of the maximum eigenvalue on Turing instability and show the role of network parameters (the network connection rate, the network's infection, etc.) on the outbreak of infectious diseases. Meanwhile, stability of network-organized SIR is given by using the maximum eigenvalue of the network matrix which is proportional to the network connection rate and the networks infection rate. The bridge between the two rates and Turing instability was also revealed which can explain the spread mechanism of infectious diseases. In the end, some measures to mitigate the spread of infectious diseases are proposed and the feasible strategies for prevention and control can be provided in our paper, the data from COVID-19 validated the above results

Experimental Study of Spray Characteristics of Kerosene-Ethanol Blends from a Pressure-Swirl Nozzle

Partial replacement of kerosene by ethanol in a gas turbine is regarded as a good way to improve the spray quality and reduce the fossil energy consumption. The present work is aimed at studying the spray characteristics of kerosene-ethanol blends discharging from a pressure-swirl nozzle. The spray cone angle, discharge coefficient, breakup length, and velocity distribution are obtained by particle image velocimetry, while droplet size is acquired by particle/droplet imaging analysis. Kerosene, E10 (10% ethanol, 90% kerosene), E20 (20% ethanol, 80% kerosene), and E30 (30% ethanol, 70% kerosene) have been considered under the injection pressure of 0.1–1 MPa. The results show that as injection pressure is increased, the discharge coefficient and breakup length decrease, while the spray cone angle, drop size, and spray velocity increase. Meanwhile, the drop size decreases and the spray velocity increases with ethanol concentration when the injection pressure is lower than 0.8 MPa. However, the spray characteristics are not affected obviously by the ethanol concentration when the injection pressure exceeds 0.8 MPa. A relation to breakup length for kerosene-ethanol blends is obtained. The findings demonstrate that the adding of ethanol into kerosene can promote atomization performance

Analysis of Prognostic Factors of Esophageal and Gastric Cardiac Carcinoma Patients after Radical Surgery Using Cox Proportional Hazard Model-A Random Sampling Study from the Fourth Hospital of Hebei Medical University during the Period of 1996-2004

OBJECTIVE To retrospectively analyze clinical data of patients from our hospital who underwent radical surgery for esophageal carcinoma and for adenocarcinoma of the gastric cardia, as well as to investigate prognostic factors affecting the long-term survival of the patients. METHODS Data from the patients eligible for our study, admitted to the 4th Hospital of Hebei Medical University from January 1996 to December 2004, were randomized, and 12 distinctive clinicopathologic factors influencing the survival rate of those who underwent radical surgery for esophageal carcinoma or carcinoma of the gastric cardia were collected. Univariate and multivariate analysis of these individual variables were performed using the Cox proportional hazard model. RESULTS It was shown by univariate analysis that age, tumor size, pathologic type, lymph node status, TNM staging, depth of infi ltration and encroachment into local organs, etc., were the factors that markedly influenced the prognosis of patients ( P < 0.01). Multivariate analysis showed that pathologic type, number of the lymph node metastases, involvement of local organs, and TNM staging were independent prognostic factors ( P < 0.05). CONCLUSION The independent factors influencing the prognosis of patients with esophageal cancer and carcinoma of the gastric cardia include pathologic type, number of lymph node metastases, involvement of local organs and TNM staging. The main prognostic factors aff ecting the patient’s survival are patient age, tumor size and depth of in fi ltration. In addition, patients with involvement of the local organs have a worse prognosis, and they should be closely followed up