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Accurate prediction of epidemics is pivotal for making well-informed decisions for the control of infectious diseases, but addressing heterogeneity in the system poses a challenge. In this study, we propose a novel modeling framework integrating the spatio-temporal heterogeneity of susceptible individuals into homogeneous models, by introducing a continuous recruitment process for the susceptibles. A neural network approximates the recruitment rate to develop a Universal Differential Equations (UDE) model. Simultaneously, we pre-set a specific form for the recruitment rate and develop a mechanistic model. Data from a COVID Omicron variant outbreak in Shanghai are used to train the UDE model using deep learning methods and to calibrate the mechanistic model using MCMC methods. Subsequently, we project the attack rate and peak of new infections for the first Omicron wave in China after the adjustment of the dynamic zero-COVID policy. Our projections indicate an attack rate and a peak of new infections of 80.06% and 3.17% of the population, respectively, compared with the homogeneous model's projections of 99.97% and 32.78%, thus providing an 18.6% improvement in the prediction accuracy based on the actual data. Our simulations demonstrate that heterogeneity in the susceptibles decreases herd immunity for ~37.36% of the population and prolongs the outbreak period from ~30 days to ~70 days, also aligning with the real case. We consider that this study lays the groundwork for the development of a new class of models and new insights for modelling heterogeneity. Author summary: This study developed a novel modeling framework that explicitly integrates spatial heterogeneity into traditionally homogeneous compartmental models for precise and efficient prediction of epidemics of infectious diseases. We employed the deep learning method to learn the recruitment rate of susceptible individuals from data given its inherent complexity and limited information. Moreover, we predefined the mechanistic form of the recruitment rate to facilitate interpretation and scenario analysis, where we fitted the recruitment rate of mechanistic model to those obtained from deep learning-based approach. Compared to the homogeneous model, our mechanistic model predicts significantly lower peak and final sizes of the epidemic, as well as reduced levels of herd immunity. Using real data as a benchmark, our model's predictive accuracy has markedly improved. The framework of our mechanistic model employs a straightforward approach to integrate heterogeneity into compartment models, thereby facilitating the extension of the modelling framework to encompass various aspects of infectious disease transmission with spatio-temporal heterogeneity among the susceptible population. [ABSTRACT FROM AUTHOR]

Aedes mosquitoes, known as vectors of mosquito-borne diseases, pose significant risks to public health and safety. Modeling the population dynamics of Aedes mosquitoes requires comprehensive approaches due to the complex interplay between biological mechanisms and environmental factors. This study developed a model that couples differential equations with a neural network to simulate the dynamics of mosquito population, and explore the relationships between oviposition rate, temperature, and precipitation. Data from nine cities in Guangdong Province spanning four years were used for model training and parameter estimation, while data from the remaining three cities were reserved for model validation. The trained model successfully simulated the mosquito population dynamics across all twelve cities using the same set of parameters. Correlation coefficients between simulated results and observed data exceeded 0.7 across all cities, with some cities surpassing 0.85, demonstrating high model performance. The coupled neural network in the model effectively revealed the relationships among oviposition rate, temperature, and precipitation, aligning with biological patterns. Furthermore, symbolic regression was used to identify the optimal functional expression for these relationships. By integrating the traditional dynamic model with machine learning, our model can adhere to specific biological mechanisms while extracting patterns from data, thus enhancing its interpretability in biology. Our approach provides both accurate modeling and an avenue for uncovering potential unknown biological mechanisms. Our conclusions can provide valuable insights into designing strategies for controlling mosquito-borne diseases and developing related prediction and early warning systems. Author summary: When simulating population dynamics, scholars often rely on predefined functions to describe processes such as growth, development, and mortality. However, these functions may not be able to accurately capture the underlying biological mechanisms, posing challenges for related research. In this study, we adopted a data-driven approach integrated with mechanistic principles to simulate mosquito dynamics. Given the unknown functional relationships between oviposition rate, temperature, and precipitation, we employed a neural network to model these relationships, resulting in a model that combines the neural network with traditional mechanistic approaches. This method can successfully simulate the changing trends of mosquitoes, infer unknown parameters, train the neural network, and capture the relationships between oviposition rate, temperature, and precipitation. Data from nine cities in Guangdong Province were used for model training, while data from the remaining three cities were used for model validation. Our model successfully fitted data from all cities using the same set of parameters, demonstrating satisfactory performance. On this basis, we successfully identified the functional form between oviposition rate, temperature, and precipitation through the symbolic regression algorithm, which was consistent with the results obtained from the neural network. These findings show that our method can accurately simulate population dynamics while exploring unknown biological mechanisms. [ABSTRACT FROM AUTHOR]

The bifurcation analysis of a predator-prey system with Holling-II functional response function and constant-yield predator releasing rate is carried out in this paper, which is a fundamental model for studying pulse control strategies or switching strategies based on the threshold level of the pest population. We show that the system can have at most two positive equilibria, and can undergo a sequence of bifurcations, including transcritical bifurcation, pitchfork bifurcation, saddle-node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation of codimension 2, and Bogdanov-Takens bifurcation of codimensions 2 and 3. Complex dynamics, such as multi-steady states, the existence of a semi-stable limit cycle, a homoclinic loop, and multiple coexistent periodic orbits, are revealed. These results show that the constant-yield predator releasing rate can be used to organize the complex dynamics and bifurcations of the model. The proposed model with constant-yield predator releasing rate is not only necessary for characterizing many practical problems including integrated pest management, but also can generate more rich bifurcation phenomena and dynamic behaviors. [ABSTRACT FROM AUTHOR]

Due to resource limitation, nonlinear impulsive control tactics related to integrated pest management have been proposed in a generalized pest-natural enemy model, which allows us to address the effects of nonlinear pulse control on the dynamics and successful pest control. The threshold conditions for the existence and global stability of pest-free periodic solution are provided by Floquet theorem and analytic methods. The existence of a nontrivial periodic solution is confirmed by showing the existence of nontrivial fixed point of the stroboscopic mapping determined by time snapshot, which equals to the common impulsive period. In order to address the applications of generalized results and to reveal how the nonlinear impulses affect the successful pest control, as an example the model with Holling II functional response function is investigated carefully. The main results reveal that the pest free periodic solution and a stable interior positive periodic solution can coexist for a wide range of parameters, which indicates that the local stability does not imply the global stability of the pest free periodic solution when nonlinear impulsive control is considered, and consequently the resource limitation (i.e. nonlinear control) may result in difficulties for successful pest control.