The multi-scale coupled epidemiological models(also known as the immuno-epidemiological models)has provided valuable biological insights into the study of infectious diseases by integrating pathogen dynamics within hosts and disease transmission processes between hosts. These models have addressed various research areas, including multi-strain infectious diseases, vector-borne transmission, environmental transmission, and optimal control. This paper reviewed the advancements in immuno-epidemiological models research over the past two decades. The studies not only focused on model analysis and the exploration of key biological factors but also examined the transition from homogeneous to heterogeneous mixing, as well as the extension from unidirectional to bidirectional coupling. Finally, based on the authors own work and understanding of the field, several important questions for future research were proposed.

A predator-prey patchy model with the dispersal delay and the population loss during the dispersal of the predator is established. The stability of the coexistence equilibrium is analyzed. Our results show that the dispersal delay of the predator does not affect the stability of the coexistence equilibrium in most cases. But it can induce stability switches under some conditions of the dispersal rate and population loss. Finally, numerical simulations are presented to demonstrate the correctness of the theoretical results.

A vegetation-water reaction-diffusion model with cross diffusion is established and human activities are chosen as the control function. The sparse optimal control problem of the model is studied which reveals how to improve the robustness of the ecosystem through human activities from the perspective of controlling the formation of vegetation pattern. Firstly, the conditions for generating Turing pattern are given. Furthermore, the first order necessary optimality condition is derived. Finally, the rationality of the control method and the effectiveness of the control strategy are verified in terms of the control effect, control error and control cost by numerical simulations.

This paper considers the stability of bistable traveling wave solutions for a class of system with asymmetric and nonlocal dispersal. On the basis of the existence of bistable waves, the global stability of bistable waves is obtained with the help of the method of super-and sub-solutions combining with the convergence results of monotone semiflows. Then, the uniqueness of bistable waves is established by using the analysis techniques.

This paper studies an age-structured SEIR epidemic model with periodic infection rate and population flows. Firstly, the existence and uniqueness of non-negative solution of the model are proved. Subsequently, by using the operator fixed-point theorem and the periodic renewal theorem, the existence of endemic periodic solution and the global asymptotic stability of the disease-free periodic solution of the model are demonstrated. By introducing the spectral radius r(F )of the Fréchet derivative F of the periodic solution operator at point 0, it is shown that when r(F )>1, the model has an endemic periodic solution(disease outbreak); when r(F )<1, the disease-free periodic solution is globally asymptotically stable(disease extinction).

By considering media publicity and information dissemination among people, we establish an SEIR(susceptible-exposed-infectious-removed,)model with two types of susceptible compartments on complex networks to study the impact of self-protection awareness on the spread of infectious diseases. The basic reproduction number R₀ is obtained by calculation, and it is proved that there is a unique endemic equilibrium point when R₀>1. According to Hurwitz criterion and comparison theorem, the stability of the disease-free equilibrium is analyzed. Then it is proved that the disease is uniformly persistent when R₀>1. Sensitivity analysis determines the importance of parameters for R₀. Numerical simulation shows that improving self-protection awareness can effectively reduce the probability of being infected.

The stability of positive equilibrium and the existence of Hopf bifurcation are studied by analyzing the distribution of eigenvalues. The critical time delay of Hopf bifurcation is obtained. Applying center manifold method and normal form theory, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are discussed. It is shown that there are two types of bistability. The prey-free equilibrium and the positive equilibrium are both stable. The prey-free equilibrium and the periodic solution are both stable. Numerical simulations are presented to support the theoretical results.

This paper is concerned with the coexistence solutions of a predator-prey model with Allee effect and density-dependent diffusion in the predator under homogeneous Dirichlet boundary conditions. Based on a priori estimate of coexistence solutions, the sufficient conditions for the existence of coexistence solutions are established by using the theory of fixed point index in positive cone. The results show that the density-dependent diffusion has a significant effect on the existence of coexistence solutions, and it is also find that the functional response function between the two species has an essential effect on the existence of coexistence solutions.

Vector-borne diseases are infectious diseases transmitted by vectors, and mosquito-borne diseases are the most common. Considering the dual vertical transmission of host and vector and Logistic growth for vector, the authors establish a vector-borne disease transmission model, calculate the basic reproduction number, analyze the existence and global stability of the equilibrium points, and show that when R₀<1, the disease-free equilibrium is globally asymptotically stable, and when R₀>1, the positive equilibrium is globally asymptotically stable. Finally, the numerical simulation verifies the conclusion, and reveals that when the vector is growing with Logistic, if the mosquito is not killed, the vector-borne disease is always present, and when the mosquito killing rate reaches a certain proportion, the vector-borne disease would eventually die out, and improving the killing rate of mosquitoes will have a positive impact on the prevention and control of infectious diseases.

In this paper, we propose a diffusive predator-prey model with cooperative hunting and group defense, and investigate the existence of Turing instability and Turing-Hopf bifurcation induced by diffusion. Taking coefficient of diffusion as bifurcation parameter, we analyze the stability of the coexistence equilibrium, and explore the complex dynamical behaviours of the system by calculating the normal forms near the Turing-Hopf bifurcation points. Moreover, we carry out some numerical simulations to illustrate the theoretical analysis. Our study shows that the system demonstrates complex dynamical behaviours near the Turing-Hopf bifurcation point, including steady-state solutions, homogeneous periodic solutions, non-homogeneous steady-state solutions, and non-homogeneous periodic solutions.

In order to study the spread of Mpox in China of 2023, firstly, the instantaneous regeneration of Mpox in China was estimated based on the Bayesian calculation method of Poisson process, and it was found that the instantaneous regeneration number continued to be lower than 1 after August 7 and decreased significantly after August 19. Secondly, the SEIQR model with isolation measures was established, which estimated that the basic reproduction number at the beginning of the Mpox epidemic in China was 1.433 and the isolation rate of patients in medical institutions was 0.42. Finally, the SEIQ₁Q₂R model was established to evaluate the effect of isolation in medical institutions or home isolation for different infected patients, and the basic reproduction number of Mpox epidemic in China decreased to 0.344, the isolation rate in medical institutions was 0.75, and the home isolation rate was 1.48. The basic reproduction number under this measure was significantly lower than the threshold of 1, indicating that the Mpox epidemic in China has been effectively controlled after adjusting the prevention and control measures.