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In this paper, a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed. The well-posedness of the solution is firstly established. Then the basic reproduction number R0 is defined and a threshold dynamics is obtained. That is, when R0 < 1, the disease-free steady state is locally stable, which implies that the disease is extinct, when R0 > 1, the disease is permanent, and there exists at least one positive steady state solution. Finally, the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated. Furthermore, as an application of theoretical analysis, a numerical example involving the spread of influenza is discussed. Based on the numerical simulations, we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation, and the increase of diffusion rate, saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation. Therefore, we propose to reduce the flow of people to lower the effect of spatial heterogeneity, increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate, and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.

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In this paper, a reaction-diffusion-chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L∞-boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S0, CTL-inactivated infection steady state S1, and CTL-activated infection steady state S∗. Our analyses indicate that S0 is globally asymptotically stable if the basic reproduction number R0 is less than 1; if R0 is between 1 and a threshold, then S1 is globally asymptotically stable. However, if R0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S∗, and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications.

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COVID-19 has globally spread to over 4 million people and the epidemic situation in Japan is very serious. The purpose of this research was to assess the risk of COVID-19 epidemic dissemination in Japan by estimating the current state of epidemic dissemination and providing some epidemic prevention and control recommendations. Firstly, the period from 6 January to 31 March 2020 was divided into four stages and the relevant parameters were estimated according to the imported cases in Japan. The basic reproduction number of the current stage is 1.954 (95% confidence interval (CI) 1.851-2.025), which means COVID-19 will spread quickly, and the self-healing rate of Japanese is about 0.495 (95% CI 0.437-0.506), with small variations in the four stages. Secondly, the results were applied to the actual reported cases from 1 to 5 April 2020, verifying the reliability of the estimated data using the accumulated reported cases located within the 95% confidence interval and the relative error of forecast data of five days being less than 2 . 5 % . Thirdly, considering the medical resources in Japan, the times the epidemic beds and ventilators become fully occupied are predicted as 5 and 15 May 2020, respectively. Keeping with the current situation, the final death toll in Japan may reach into the millions. Finally, based on experience with COVID-19 prevention and control in China, robust measures such as nationwide shutdown, store closures, citizens isolating themselves at home, and increasing PCR testing would quickly and effectively prevent COVID-19 spread.

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A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

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In this note, we point out that Theorems 5.1 and 6.3 in Biswas et al. (2015) are not true. We present a new result and discuss the existence and local stability of the equilibria for the delay model.

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In this paper, we analyze a general predator-prey model with state feedback impulsive harvesting strategies in which the prey species displays a strong Allee effect. We firstly show the existence of order-1 heteroclinic cycle and order-1 positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-1 positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.

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