A data-driven epidemic model with human mobility and vaccination protection for COVID-19 prediction.

Epidemiological models allow for quantifying the dynamic characteristics of large-scale outbreaks. However, capturing detailed and accurate epidemiological information often requires consideration of multiple kinetic mechanisms and parameters. Due to the uncertainty of pandemic evolution, such as pathogen variation, host immune response and changes in mitigation strategies, the parameter evaluation and state prediction of complex epidemiological models are challenging. Here, we develop a data-driven epidemic model with a generalized SEIR mechanistic structure that includes new compartments, human mobility and vaccination protection. To address the issue of model complexity, we embed the epidemiological model dynamics into physics-informed neural networks (PINN), taking the observed series of time instances as direct input of the network to simultaneously infer unknown parameters and unobserved dynamics of the underlying model. Using actual data during the COVID-19 outbreak in Australia, Israel, and Switzerland, our model framework demonstrates satisfactory performance in multi-step ahead predictions compared to several benchmark models. Moreover, our model infers time-varying parameters such as transmission rates, hospitalization ratios, and effective reproduction numbers, as well as calculates the latent period and asymptomatic infection count, which are typically unreported in public data. Finally, we employ the proposed data-driven model to analyze the impact of different mitigation strategies on COVID-19.

Source-Sink Dynamics in a Two-Patch SI Epidemic Model with Life Stages and No Recovery from Infection.

This study presents a comprehensive analysis of a two-patch, two-life stage SI model without recovery from infection, focusing on the dynamics of disease spread and host population viability in natural populations. The model, inspired by real-world ecological crises like the decline of amphibian populations due to chytridiomycosis and sea star populations due to Sea Star Wasting Disease, aims to understand the conditions under which a sink host population can present ecological rescue from a healthier, source population. Mathematical and numerical analyses reveal the critical roles of the basic reproductive numbers of the source and sink populations, the maturation rate, and the dispersal rate of juveniles in determining population outcomes. The study identifies basic reproduction numbers R 0 for each of the patches, and conditions for the basic reproduction numbers to produce a receiving patch under which its population. These findings provide insights into managing natural populations affected by disease, with implications for conservation strategies, such as the importance of maintaining reproductively viable refuge populations and considering the effects of dispersal and maturation rates on population recovery. The research underscores the complexity of host-pathogen dynamics in spatially structured environments and highlights the need for multi-faceted approaches to biodiversity conservation in the face of emerging diseases.

Infection-induced increases to population size during cycles in a discrete-time epidemic model.

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the 'hydra effect,' i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called 'infection-induced hydra effect.' Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

Global stability and optimal control of an age-structured SVEIR epidemic model with waning immunity and relapses.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number R 0 and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of R 0 is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

Final epidemic size of a two-community SIR model with asymmetric coupling.

Communities are commonly not isolated but interact asymmetrically with each other, allowing the propagation of infectious diseases within the same community and between different communities. To reveal the impact of asymmetrical interactions and contact heterogeneity on disease transmission, we formulate a two-community SIR epidemic model, in which each community has its contact structure while communication between communities occurs through temporary commuters. We derive an explicit formula for the basic reproduction number R 0 , give an implicit equation for the final epidemic size z, and analyze the relationship between them. Unlike the typical positive correlation between R 0 and z in the classic SIR model, we find a negatively correlated relationship between counterparts of our model deviating from homogeneous populations. Moreover, we investigate the impact of asymmetric coupling mechanisms on R 0 . The results suggest that, in scenarios with restricted movement of susceptible individuals within a community, R 0 does not follow a simple monotonous relationship, indicating that an unbending decrease in the movement of susceptible individuals may increase R 0 . We further demonstrate that network contacts within communities have a greater effect on R 0 than casual contacts between communities. Finally, we develop an epidemic model without restriction on the movement of susceptible individuals, and the numerical simulations suggest that the increase in human flow between communities leads to a larger R 0 .

On cognitive epidemic models: spatial segregation versus nonpharmaceutical interventions.

Fick's law and the Fokker-Planck law of diffusion are applied to manifest the cognitive dispersal of individuals in two reaction-diffusion SEIR epidemic models, where the disease transmission is illustrated by nonlocal infection mechanisms in heterogeneous environments. Building upon the well-posedness of solutions, threshold dynamics are discussed in terms of the basic reproduction numbers for the two cognitive epidemic models. The numerical investigation reveals that the Fokker-Planck law can better describe the diffusion of individuals by taking different dispersal strategies of exposed individuals in our cognitive epidemic models, and provides some insights on spatial segregation and nonpharmaceutical interventions: (i) spatial segregation occurs in the random diffusion model when the nonlocal infection radius is small, while it appears in the symmetric diffusion model when the radius is large; (ii) nonpharmaceutical interventions on restricting the dispersal of exposed and infected individuals do not contribute to reducing the infection proportion, but rather eliminate the disease in a region, which expands as the nonlocal infection radius increases. We additionally find that the final infection size in the random diffusion model is significantly smaller than that in the symmetric diffusion model and decreases as the nonlocal infection radius increases.

Dynamical Analysis of an Improved Bidirectional Immunization SIR Model in Complex Network.

In order to investigate the impact of two immunization strategies-vaccination targeting susceptible individuals to reduce their infection rate and clinical medical interventions targeting infected individuals to enhance their recovery rate-on the spread of infectious diseases in complex networks, this study proposes a bilinear SIR infectious disease model that considers bidirectional immunization. By analyzing the conditions for the existence of endemic equilibrium points, we derive the basic reproduction numbers and outbreak thresholds for both homogeneous and heterogeneous networks. The epidemic model is then reconstructed and extensively analyzed using continuous-time Markov chain (CTMC) methods. This analysis includes the investigation of transition probabilities, transition rate matrices, steady-state distributions, and the transition probability matrix based on the embedded chain. In numerical simulations, a notable concordance exists between the outcomes of CTMC and mean-field (MF) simulations, thereby substantiating the efficacy of the CTMC model. Moreover, the CTMC-based model adeptly captures the inherent stochastic fluctuation in the disease transmission, which is consistent with the mathematical properties of Markov chains. We further analyze the relationship between the system's steady-state infection density and the immunization rate through MCS. The results suggest that the infection density decreases with an increase in the immunization rate among susceptible individuals. The current research results will enhance our understanding of infectious disease transmission patterns in real-world scenarios, providing valuable theoretical insights for the development of epidemic prevention and control strategies.

Age-related model for estimating the symptomatic and asymptomatic transmissibility of COVID-19 patients.

Estimation of age-dependent transmissibility of COVID-19 patients is critical for effective policymaking. Although the transmissibility of symptomatic cases has been extensively studied, asymptomatic infection is understudied due to limited data. Using a dataset with reliably distinguished symptomatic and asymptomatic statuses of COVID-19 cases, we propose an ordinary differential equation model that considers age-dependent transmissibility in transmission dynamics. Under a Bayesian framework, multi-source information is synthesized in our model for identifying transmissibility. A shrinkage prior among age groups is also adopted to improve the estimation behavior of transmissibility from age-structured data. The added values of accounting for age-dependent transmissibility are further evaluated through simulation studies. In real-data analysis, we compare our approach with two basic models using the deviance information criterion (DIC) and its extension. We find that the proposed model is more flexible for our epidemic data. Our results also suggest that the transmissibility of asymptomatic infections is significantly lower (on average, 76.45% with a credible interval (27.38%, 88.65%)) than that of symptomatic cases. In both symptomatic and asymptomatic patients, the transmissibility mainly increases with age. Patients older than 30 years are more likely to develop symptoms with higher transmissibility. We also find that the transmission burden of asymptomatic cases is lower than that of symptomatic patients.

Estimating contact network properties by integrating multiple data sources associated with infectious diseases.

To effectively mitigate the spread of communicable diseases, it is necessary to understand the interactions that enable disease transmission among individuals in a population; we refer to the set of these interactions as a contact network. The structure of the contact network can have profound effects on both the spread of infectious diseases and the effectiveness of control programs. Therefore, understanding the contact network permits more efficient use of resources. Measuring the structure of the network, however, is a challenging problem. We present a Bayesian approach to integrate multiple data sources associated with the transmission of infectious diseases to more precisely and accurately estimate important properties of the contact network. An important aspect of the approach is the use of the congruence class models for networks. We conduct simulation studies modeling pathogens resembling SARS-CoV-2 and HIV to assess the method; subsequently, we apply our approach to HIV data from the University of California San Diego Primary Infection Resource Consortium. Based on simulation studies, we demonstrate that the integration of epidemiological and viral genetic data with risk behavior survey data can lead to large decreases in mean squared error (MSE) in contact network estimates compared to estimates based strictly on risk behavior information. This decrease in MSE is present even in settings where the risk behavior surveys contain measurement error. Through these simulations, we also highlight certain settings where the approach does not improve MSE.

Analysis of a multi-type resurgence of Mycobacterium bovis in cattle and badgers in Southwest France, 2007-2019.

Although control measures to tackle bovine tuberculosis (bTB) in cattle have been successful in many parts of Europe, this disease has not been eradicated in areas where Mycobacterium bovis circulates in multi-host systems. Here we analyzed the resurgence of 11 M. bovis genotypes (defined based on spoligotyping and MIRU-VNTR) detected in 141 farms between 2007 and 2019, in an area of Southwestern France where wildlife infection was also detected from 2012 in 65 badgers. We used a spatially-explicit model to reconstruct the simultaneous diffusion of the 11 genotypes in cattle farms and badger populations. Effective reproduction number R was estimated to be 1.34 in 2007-2011 indicating a self-sustained M. bovis transmission by a maintenance community although within-species Rs were both < 1, indicating that neither cattle nor badger populations acted as separate reservoir hosts. From 2012, control measures were implemented, and we observed a decrease of R below 1. Spatial contrasts of the basic reproduction ratio suggested that local field conditions may favor (or penalize) local spread of bTB upon introduction into a new farm. Calculation of generation time distributions showed that the spread of M. bovis has been more rapid from cattle farms (0.5-0.7 year) than from badger groups (1.3-2.4 years). Although eradication of bTB appears possible in the study area (since R < 1), the model suggests it is a long-term prospect, because of the prolonged persistence of infection in badger groups (2.9-5.7 years). Supplementary tools and efforts to better control bTB infection in badgers (including vaccination for instance) appear necessary.

Twice evasions of Omicron variants explain the temporal patterns in six Asian and Oceanic countries.

BACKGROUND: The ongoing coronavirus 2019 (COVID-19) pandemic has emerged and caused multiple pandemic waves in the following six countries: India, Indonesia, Nepal, Malaysia, Bangladesh and Myanmar. Some of the countries have been much less studied in this devastating pandemic. This study aims to assess the impact of the Omicron variant in these six countries and estimate the infection fatality rate (IFR) and the reproduction number [Formula: see text] in these six South Asia, Southeast Asia and Oceania countries. METHODS: We propose a Susceptible-Vaccinated-Exposed-Infectious-Hospitalized-Death-Recovered model with a time-varying transmission rate [Formula: see text] to fit the multiple waves of the COVID-19 pandemic and to estimate the IFR and [Formula: see text] in the aforementioned six countries. The level of immune evasion and the intrinsic transmissibility advantage of the Omicron variant are also considered in this model. RESULTS: We fit our model to the reported deaths well. We estimate the IFR (in the range of 0.016 to 0.136%) and the reproduction number [Formula: see text] (in the range of 0 to 9) in the six countries. Multiple pandemic waves in each country were observed in our simulation results. CONCLUSIONS: The invasion of the Omicron variant caused the new pandemic waves in the six countries. The higher [Formula: see text] suggests the intrinsic transmissibility advantage of the Omicron variant. Our model simulation forecast implies that the Omicron pandemic wave may be mitigated due to the increasing immunized population and vaccine coverage.

Modeling Syphilis and HIV Coinfection: A Case Study in the USA.

Syphilis and HIV infections form a dangerous combination. In this paper, we propose an epidemic model of HIV-syphilis coinfection. The model always has a unique disease-free equilibrium, which is stable when both reproduction numbers of syphilis and HIV are less than 1. If the reproduction number of syphilis (HIV) is greater than 1, there exists a unique boundary equilibrium of syphilis (HIV), which is locally stable if the invasion number of HIV (syphilis) is less than 1. Coexistence equilibrium exists and is stable when all reproduction numbers and invasion numbers are greater than 1. Using data of syphilis cases and HIV cases from the US, we estimated that both reproduction numbers for syphilis and HIV are slightly greater than 1, and the boundary equilibrium of syphilis is stable. In addition, we observed competition between the two diseases. Treatment for primary syphilis is more important in mitigating the transmission of syphilis. However, it might lead to increase of HIV cases. The results derived here could be adapted to other multi-disease scenarios in other regions.

Global stability of latency-age/stage-structured epidemic models with differential infectivity.

In this paper, we first formulate a system of ODEs-PDE to model diseases with latency-age and differential infectivity. Then, based on the ways how latent individuals leave the latent stage, one ODE and two DDE models are derived. We only focus on the global stability of the models. All the models have some similarities in the existence of equilibria. Each model has a threshold dynamics for global stability, which is completely characterized by the basic reproduction number. The approach is the Lyapunov direct method. We propose an idea on constructing Lyapunov functionals for the two DDE and the original ODEs-PDE models. During verifying the negative (semi-)definiteness of derivatives of the Lyapunov functionals along solutions, a novel positive definite function and a new inequality are used. The idea here is also helpful in applying the Lyapunov direct method to prove the global stability of some epidemic models with age structure or delays.

Perceptive movement of susceptible individuals with memory.

The perception of susceptible individuals naturally lowers the transmission probability of an infectious disease but has been often ignored. In this paper, we formulate and analyze a diffusive SIS epidemic model with memory-based perceptive movement, where the perceptive movement describes a strategy for susceptible individuals to escape from infections. We prove the global existence and boundedness of a classical solution in an n-dimensional bounded smooth domain. We show the threshold-type dynamics in terms of the basic reproduction number [Formula: see text]: when [Formula: see text], the unique disease-free equilibrium is globally asymptotically stable; when [Formula: see text], there is a unique constant endemic equilibrium, and the model is uniformly persistent. Numerical analysis exhibits that when [Formula: see text], solutions converge to the endemic equilibrium for slow memory-based movement and they converge to a stable periodic solution when memory-based movement is fast. Our results imply that the memory-based movement cannot determine the extinction or persistence of infectious disease, but it can change the persistence manner.

Threshold behavior and exponential ergodicity of an sir epidemic model: the impact of random jamming and hospital capacity.

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease's persistence or eradication depends on the basic reproduction number [Formula: see text]. If [Formula: see text], the disease is eradicated with a probability of 1, while [Formula: see text] results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if [Formula: see text], the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.

SIRC epidemic model with cross-immunity and multiple time delays.

Multi-strain diseases lead to the development of some degree of cross-immunity among people. In the present paper, we propose a multi-delayed SIRC epidemic model with incubation and immunity time delays. Here we aim to examine and investigate the effects of incubation delay [Formula: see text] and the impact of vaccine which provides partial/cross-immunity with immunity delay parameter ([Formula: see text]) on the disease dynamics. Also, we study the impact of the strength of cross-immunity [Formula: see text] on the disease prevalence. The positivity and boundedness of the solutions of the epidemic model have been established. Two different types of equilibrium points (disease-free and endemic) have been deduced. Expression for basic reproduction number has been derived. The stability conditions and Hopf-bifurcation about both the equilibrium points in the absence and presence of both delays have been discussed. The Lyapunov stability conditions about the endemic equilibrium point have been established. Numerical simulations have been performed to support our analytical results. We quantitatively demonstrate how oscillations and Hopf-bifurcation allow time delays to alter the dynamics of the system. The combined impacts of both the delays on disease prevalence has been studied. Through parameter sensitivity analysis, we observe that the infected population decreases with an increase in vaccination rate and the system starts to stabilize early with the increase in cross-immunity rate. Global sensitivity analysis for the basic reproduction number has been performed using Latin hypercube sampling and partial rank correlation coefficients techniques. The combined effect of vaccination rate with transmission rate and vaccination rate with re-infection probability (i.e. strength of cross-immunity) on [Formula: see text] have been discussed. Our research underlines the need to take cross-immunity and time delays into account in the epidemic model in order to better understand disease dynamics.

Novel spatial profiles of some diffusive SIS epidemic models.

In this paper, we are concerned with two SIS epidemic reaction-diffusion models with mass action infection mechanism of the form SI, and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small. For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval. In sharp contrast, for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals, our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant, and the infected population may concentrate only at some isolated highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat, depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations. Numerical simulations are performed to support and complement our theoretical findings.

Imperfect and Bogdanov-Takens bifurcations in biological models: from harvesting of species to isolation of infectives.

A natural biological system under human interventions may exhibit complex dynamical behaviors which could lead to either the collapse or stabilization of the system. The bifurcation theory plays an important role in understanding this evolution process by modeling and analyzing the biological system. In this paper, we examine two types of biological models that Fred Brauer made pioneer contributions: predator-prey models with stocking/harvesting and epidemic models with importation/isolation. First we consider the predator-prey model with Holling type II functional response whose dynamics and bifurcations are well-understood. By considering human interventions such as constant harvesting or stocking of predators, we show that the system under human interventions undergoes imperfect bifurcation and Bogdanov-Takens bifurcation, which induces much richer dynamical behaviors such as the existence of limit cycles or homoclinic loops. Then we consider an epidemic model with constant importation/isolation of infective individuals and observe similar imperfect and Bogdanov-Takens bifurcations when the constant importation/isolation rate varies.

The impact of household structure on disease-induced herd immunity.

The disease-induced herd immunity level [Formula: see text] is the fraction of the population that must be infected by an epidemic to ensure that a new epidemic among the remaining susceptible population is not supercritical. For a homogeneously mixing population [Formula: see text] equals the classical herd immunity level [Formula: see text], which is the fraction of the population that must be vaccinated in advance of an epidemic so that the epidemic is not supercritical. For most forms of heterogeneous mixing [Formula: see text], sometimes dramatically so. For an SEIR (susceptible [Formula: see text] exposed [Formula: see text] infective [Formula: see text] recovered) model of an epidemic among a population that is partitioned into households, in which individuals mix uniformly within households and, in addition, uniformly at a much lower rate in the population at large, we show that [Formula: see text] unless variability in the household size distribution is sufficiently large. Thus, introducing household structure into a model typically has the opposite effect on disease-induced herd immunity than most other forms of population heterogeneity. We reach this conclusion by considering an approximation [Formula: see text] of [Formula: see text], supported by numerical studies using real-world household size distributions. For [Formula: see text], we prove that [Formula: see text] when all households have size n, and conjecture that this inequality holds for any common household size n. We prove results comparing [Formula: see text] and [Formula: see text] for epidemics which are highly infectious within households, and also for epidemics which are weakly infectious within households.

Effectiveness of feedback control and the trade-off between death by COVID-19 and costs of countermeasures.

We provided a framework of a mathematical epidemic modeling and a countermeasure against the novel coronavirus disease (COVID-19) under no vaccines and specific medicines. The fact that even asymptomatic cases are infectious plays an important role for disease transmission and control. Some patients recover without developing the disease; therefore, the actual number of infected persons is expected to be greater than the number of confirmed cases of infection. Our study distinguished between cases of confirmed infection and infected persons in public places to investigate the effect of isolation. An epidemic model was established by utilizing a modified extended Susceptible-Exposed-Infectious-Recovered model incorporating three types of infectious and isolated compartments, abbreviated as SEIIIHHHR. Assuming that the intensity of behavioral restrictions can be controlled and be divided into multiple levels, we proposed the feedback controller approach to implement behavioral restrictions based on the active number of hospitalized persons. Numerical simulations were conducted using different detection rates and symptomatic ratios of infected persons. We investigated the appropriate timing for changing the degree of behavioral restrictions and confirmed that early initiating behavioral restrictions is a reasonable measure to reduce the burden on the health care system. We also examined the trade-off between reducing the cumulative number of deaths by the COVID-19 and saving the cost to prevent the spread of the virus. We concluded that a bang-bang control of the behavioral restriction can reduce the socio-economic cost, while a control of the restrictions with multiple levels can reduce the cumulative number of deaths by infection.