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.

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.

Prevention and control of Ebola virus transmission: mathematical modelling and data fitting.

The Ebola virus disease (EVD) has been endemic since 1976, and the case fatality rate is extremely high. EVD is spread by infected animals, symptomatic individuals, dead bodies, and contaminated environment. In this paper, we formulate an EVD model with four transmission modes and a time delay describing the incubation period. Through dynamical analysis, we verify the importance of blocking the infection source of infected animals. We get the basic reproduction number without considering the infection source of infected animals. And, it is proven that the model has a globally attractive disease-free equilibrium when the basic reproduction number is less than unity; the disease eventually becomes endemic when the basic reproduction number is greater than unity. Taking the EVD epidemic in Sierra Leone in 2014-2016 as an example, we complete the data fitting by combining the effect of the media to obtain the unknown parameters, the basic reproduction number and its time-varying reproduction number. It is shown by parameter sensitivity analysis that the contact rate and the removal rate of infected group have the greatest influence on the prevalence of the disease. And, the disease-controlling thresholds of these two parameters are obtained. In addition, according to the existing vaccination strategy, only the inoculation ratio in high-risk areas is greater than 0.4, the effective reproduction number can be less than unity. And, the earlier the vaccination time, the greater the inoculation ratio, and the faster the disease can be controlled.

Analysis of the impact of COVID-19 variants and vaccination on the time-varying reproduction number: statistical methods.

Introduction: The COVID-19 pandemic has profoundly impacted global health systems, requiring the monitoring of infection waves and strategies to control transmission. Estimating the time-varying reproduction number is crucial for understanding the epidemic and guiding interventions. Methods: Probability distributions of serial interval are estimated for Pre-Delta and Delta periods. We conducted a comparative analysis of time-varying reproduction numbers, taking into account population immunity and variant differences. We incorporated the regional heterogeneity and age distribution of the population, as well as the evolving variants and vaccination rates over time. COVID-19 transmission dynamics were analyzed with variants and vaccination. Results: The reproduction number is computed with and without considering variant-based immunity. In addition, values of reproduction number significantly differed by variants, emphasizing immunity's importance. Enhanced vaccination efforts and stringent control measures were effective in reducing the transmission of the Delta variant. Conversely, Pre-Delta variant appeared less influenced by immunity levels, due to lower vaccination rates. Furthermore, during the Pre-Delta period, there was a significant difference between the region-specific and the non-region-specific reproduction numbers, with particularly distinct pattern differences observed in Gangwon, Gyeongbuk, and Jeju in Korea. Discussion: This research elucidates the dynamics of COVID-19 transmission concerning the dominance of the Delta variant, the efficacy of vaccinations, and the influence of immunity levels. It highlights the necessity for targeted interventions and extensive vaccination coverage. This study makes a significant contribution to the understanding of disease transmission mechanisms and informs public health strategies.

A hybrid Lagrangian-Eulerian model for vector-borne diseases.

In this paper, a multi-patch and multi-group vector-borne disease model is proposed to study the effects of host commuting (Lagrangian approach) and/or vector migration (Eulerian approach) on disease spread. We first define the basic reproduction number of the model, R 0 , which completely determines the global dynamics of the model system. Namely, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, and if R 0 > 1 , then there exists a unique endemic equilibrium which is globally asymptotically stable. Then, we show that the basic reproduction number has lower and upper bounds which are independent of the host residence times matrix and the vector migration matrix. In particular, nonhomogeneous mixing of hosts and vectors in a homogeneous environment generally increases disease persistence and the basic reproduction number of the model attains its minimum when the distributions of hosts and vectors are proportional. Moreover, R 0 can also be estimated by the basic reproduction numbers of disconnected patches if the environment is homogeneous. The optimal vector control strategy is obtained for a special scenario. In the two-patch and two-group case, we numerically analyze the dependence of the basic reproduction number and the total number of infected people on the host residence times matrix and illustrate the optimal vector control strategy in homogeneous and heterogeneous environments.

Analysis of the COVID-19 model with self-protection and isolation measures affected by the environment.

Since the global outbreak of COVID-19, the virus has continuously mutated and can survive in the air for long periods of time. This paper establishes and analyzes a model of COVID-19 with self-protection and quarantine measures affected by viruses in the environment to investigate the influence of viruses in the environment on the spread of the outbreak, as well as to develop a rational prevention and control measure to control the spread of the outbreak. The basic reproduction number was calculated and Lyapunov functions were constructed to discuss the stability of the model equilibrium points. The disease-free equilibrium point was proven to be globally asymptotically stable when $ R_0 < 1 $, and the endemic equilibrium point was globally asymptotically stable when $ R_0 > 1 $. The model was fitted using data from COVID-19 cases in Chongqing between November 1 to November 25, 2022. Based on the numerical analysis, the following conclusion was obtained: clearing the virus in the environment and strengthening the isolation measures for infected people can control the epidemic to a certain extent, but enhancing the self-protection of individuals can be more effective in reducing the risk of being infected and controlling the transmission of the epidemic, which is more conducive to the practical application.

A deterministic transmission model for analytics-driven optimization of COVID-19 post-pandemic vaccination and quarantine strategies.

This study developed a deterministic transmission model for the coronavirus disease of 2019 (COVID-19), considering various factors such as vaccination, awareness, quarantine, and treatment resource limitations for infected individuals in quarantine facilities. The proposed model comprised five compartments: susceptible, vaccinated, quarantined, infected, and recovery. It also considered awareness and limited resources by using a saturated function. Dynamic analyses, including equilibrium points, control reproduction numbers, and bifurcation analyses, were conducted in this research, employing analytics to derive insights. Our results indicated the possibility of an endemic equilibrium even if the reproduction number for control was less than one. Using incidence data from West Java, Indonesia, we estimated our model parameter values to calibrate them with the real situation in the field. Elasticity analysis highlighted the crucial role of contact restrictions in reducing the spread of COVID-19, especially when combined with community awareness. This emphasized the analytics-driven nature of our approach. We transformed our model into an optimal control framework due to budget constraints. Leveraging Pontriagin's maximum principle, we meticulously formulated and solved our optimal control problem using the forward-backward sweep method. Our experiments underscored the pivotal role of vaccination in infection containment. Vaccination effectively reduces the risk of infection among vaccinated individuals, leading to a lower overall infection rate. However, combining vaccination and quarantine measures yields even more promising results than vaccination alone. A second crucial finding emphasized the need for early intervention during outbreaks rather than delayed responses. Early interventions significantly reduce the number of preventable infections, underscoring their importance.

Modeling different infectious phases of hepatitis B with generalized saturated incidence: An analysis and control.

Hepatitis B is one of the global health issues caused by the hepatitis B virus (HBV), producing 1.1 million deaths yearly. The acute and chronic phases of HBV are significant because worldwide, approximately 250 million people are infected by chronic hepatitis B. The chronic stage is a long-term, persistent infection that can cause liver damage and increase the risk of liver cancer. In the case of multiple phases of infection, a generalized saturated incidence rate model is more reasonable than a simply saturated incidence because it captures the complex dynamics of the different infection phases. In contrast, a simple saturated incidence rate model assumes a fixed shape for the incidence rate curve, which may not accurately reflect the dynamics of multiple infection phases. Considering HBV and its various phases, we constructed a model to present the dynamics and control strategies using the generalized saturated incidence. First, we proved that the model is well-posed. We then found the reproduction quantity and model equilibria to discuss the time dynamics of the model and investigate the conditions for stabilities. We also examined a control mechanism by introducing various controls to the model with the aim to increase the population of those recovered and minimize the infected people. We performed numerical experiments to check the biological significance and control implementation.

Approximating reproduction numbers: a general numerical method for age-structured models.

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

An SIS epidemic model with individual variation.

We study an extension of the stochastic SIS (Susceptible-Infectious-Susceptible) model in continuous time that accounts for variation amongst individuals. By examining its limiting behaviour as the population size grows we are able to exhibit conditions for the infection to become endemic.

A novel within-host model of HIV and nutrition.

In this paper we develop a four compartment within-host model of nutrition and HIV. We show that the model has two equilibria: an infection-free equilibrium and infection equilibrium. The infection free equilibrium is locally asymptotically stable when the basic reproduction number $ \mathcal{R}_0 < 1 $, and unstable when $ \mathcal{R}_0 > 1 $. The infection equilibrium is locally asymptotically stable if $ \mathcal{R}_0 > 1 $ and an additional condition holds. We show that the within-host model of HIV and nutrition is structured to reveal its parameters from the observations of viral load, CD4 cell count and total protein data. We then estimate the model parameters for these 3 data sets. We have also studied the practical identifiability of the model parameters by performing Monte Carlo simulations, and found that the rate of clearance of the virus by immunoglobulins is practically unidentifiable, and that the rest of the model parameters are only weakly identifiable given the experimental data. Furthermore, we have studied how the data frequency impacts the practical identifiability of model parameters.

Threshold dynamics of a switching diffusion SIR model with logistic growth and healthcare resources.

In this article, we have constructed a stochastic SIR model with healthcare resources and logistic growth, aiming to explore the effect of random environment and healthcare resources on disease transmission dynamics. We have showed that under mild extra conditions, there exists a critical parameter, i.e., the basic reproduction number $ R_0/ $, which completely determines the dynamics of disease: when $ R_0/ < 1 $, the disease is eradicated; while when $ R_0/ > 1 $, the disease is persistent. To validate our theoretical findings, we conducted some numerical simulations using actual parameter values of COVID-19. Both our theoretical and simulation results indicated that (1) the white noise can significantly affect the dynamics of a disease, and importantly, it can shift the stability of the disease-free equilibrium; (2) infectious disease resurgence may be caused by random switching of the environment; and (3) it is vital to maintain adequate healthcare resources to control the spread of disease.

The impact of temperature, humidity and closing school on the mumps epidemic: a case study in the mainland of China.

BACKGROUND: To control resurging infectious diseases like mumps, it is necessary to resort to effective control and preventive measures. These measures include increasing vaccine coverage, providing the community with advice on how to reduce exposure, and closing schools. To justify such intervention, it is important to understand how well each of these measures helps to limit transmission. METHODS: In this paper, we propose a simple SEILR (susceptible-exposed-symptomatically infectious-asymptomatically infectious-recovered) model by using a novel transmission rate function to incorporate temperature, humidity, and closing school factors. This new transmission rate function allows us to verify the impact of each factor either separately or combined. Using reported mumps cases from 2004 to 2018 in the mainland of China, we perform data fitting and parameter estimation to evaluate the basic reproduction number  R 0 . As a wide range of one-dose measles, mumps, and rubella (MMR) vaccine programs in China started only in 2008, we use different vaccination proportions for the first Stage I period (from 2004 to 2008) and the second Stage II period (from 2009 to 2018). This allows us to verify the importance of higher vaccine coverage with a possible second dose of MMR vaccine. RESULTS: We find that the basic reproduction number  R 0  is generally between 1 and 3. We then use the Akaike Information Criteria to assess the extent to which each of the three factors contributed to the spread of mumps. The findings suggest that the impact of all three factors is substantial, with temperature having the most significant impact, followed by school opening and closing, and finally humidity. CONCLUSION: We conclude that the strategy of increasing vaccine coverage, changing micro-climate (temperature and humidity), and closing schools can greatly reduce mumps transmission.

Modeling the impact of non-human host predation on the transmission of Chagas disease.

In addition to the traditional transmission route via the biting-and-defecating process, non-human host predation of triatomines is recognized as another significant avenue for Chagas disease transmission. In this paper, we develop an eco-epidemiological model to investigate the impact of predation on the disease's spread. Two critical thresholds, Rvp (the basic reproduction number of triatomines) and R0p (the basic reproduction number of the Chagas parasite), are derived to delineate the model's dynamics. Through the construction of appropriate Lyapunov functions and the application of the Bendixson-Dulac theorem, the global asymptotic stabilities of the equilibria are fully established. The vector-free equilibrium E0 is globally stable when Rvp<1. E1, the disease-free equilibrium, is globally stable when Rvp>1 and R0p<1, while the endemic equilibrium E∗ is globally stable when both Rvp>1 and R0p>1. Numerical simulations highlight that the degree of host predation on triatomines, influenced by non-human hosts activities, can variably increase or decrease the Chagas disease transmission risk. Specifically, low or high levels of host predation can reduce R0p to below unity, while intermediate levels may increase the infected host populations, albeit with a reduction in R0p. These findings highlight the role played by non-human hosts and offer crucial insights for the prevention and control of Chagas disease.

An SEIR network epidemic model with manual and digital contact tracing allowing delays.

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered such that each infected network contact is reported, tested and isolated with some probability and after a random delay. Additionally, digital tracing (based on a tracing app) is triggered if the diagnosed individual is an app-user, and then all of its app-using infectees are immediately notified and isolated. The early phase of the epidemic with manual and/or digital tracing is approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of both contact tracing mechanisms is numerically quantified through the reduction of the reproduction number. This shows that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The relative effectiveness of manual tracing compared to digital tracing increases if: more of the transmission occurs on the network, when the tracing delay is shortened, and when the network degree distribution is heavy-tailed. For realistic values, the combined tracing case can reduce R0 by 20%-30%, so other preventive measures are needed to reduce the reproduction number down to 1.2-1.4 for contact tracing to make it successful in avoiding big outbreaks.

A simulation-based approach for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data.

Tracking pathogen transmissibility during infectious disease outbreaks is essential for assessing the effectiveness of public health measures and planning future control strategies. A key measure of transmissibility is the time-dependent reproduction number, which has been estimated in real-time during outbreaks of a range of pathogens from disease incidence time series data. While commonly used approaches for estimating the time-dependent reproduction number can be reliable when disease incidence is recorded frequently, such incidence data are often aggregated temporally (for example, numbers of cases may be reported weekly rather than daily). As we show, commonly used methods for estimating transmissibility can be unreliable when the timescale of transmission is shorter than the timescale of data recording. To address this, here we develop a simulation-based approach involving Approximate Bayesian Computation for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data. We first use a simulated dataset representative of a situation in which daily disease incidence data are unavailable and only weekly summary values are reported, demonstrating that our method provides accurate estimates of the time-dependent reproduction number under such circumstances. We then apply our method to two outbreak datasets consisting of weekly influenza case numbers in 2019-20 and 2022-23 in Wales (in the United Kingdom). Our simple-to-use approach will allow accurate estimates of time-dependent reproduction numbers to be obtained from temporally aggregated data during future infectious disease outbreaks.

Modelling the impact of precaution on disease dynamics and its evolution.

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number R 0 < 1 . For R 0 > 1 , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics.

A mathematical model for HIV dynamics with multiple infections: implications for immune escape.

Multiple infections enable the recombination of different strains, which may contribute to viral diversity. How multiple infections affect the competition dynamics between the two types of strains, the wild and the immune escape mutant, remains poorly understood. This study develops a novel mathematical model that includes the two strains, two modes of viral infection, and multiple infections. For the representative double-infection case, the reproductive numbers are derived and global stabilities of equilibria are obtained via the Lyapunov direct method and theory of limiting systems. Numerical simulations indicate similar viral dynamics regardless of multiplicities of infections though the competition between the two strains would be the fiercest in the case of quadruple infections. Through sensitivity analysis, we evaluate the effect of parameters on the set-point viral loads in the presence and absence of multiple infections. The model with multiple infections predict that there exists a threshold for cytotoxic T lymphocytes (CTLs) to minimize the overall viral load. Weak or strong CTLs immune response can result in high overall viral load. If the strength of CTLs maintains at an intermediate level, the fitness cost of the mutant is likely to have a significant impact on the evolutionary dynamics of mutant viruses. We further investigate how multiple infections alter the viral dynamics during the combination antiretroviral therapy (cART). The results show that viral loads may be underestimated during cART if multiple-infection is not taken into account.

Effects of fish-human transmission and different life stages of fish on Clonorchiasis: A novel mathematical model.

Clonorchiasis is a zoonotic disease mainly caused by eating raw fish and shrimp, and there is no vaccine to prevent it. More than 30 million people are infected worldwide, of which China alone accounts for about half, and is one of the countries most seriously affected by Clonorchiasis. In this work, we formulate a novel Ordinary Differential Equation (ODE) model to discuss the biological attributes of fish within authentic ecosystems and the complex lifecycle of Clonorchis sinensis. This model includes larval fish, adult fish, infected fish, humans, and cercariae. We derive the basic reproduction number and perform a rigorous stability analysis of the proposed model. Numerically, we use data from 2016 to 2021 in Guangxi, China, to discuss outbreaks of Clonorchiasis and obtain the basic reproduction number R0=1.4764. The fitted curve appropriately reflects the overall trend and replicates a low peak in the case number of Clonorchiasis. By reducing the release rate of cercariae in 2018, the fitted values of Clonorchiasis cases dropped rapidly and almost disappeared. If we decrease the transmission rate from infected fish to humans, Clonorchiasis can be controlled. Our studies also suggest that strengthening publicity education and cleaning water quality can effectively control the transmission of Clonorchiasis in Guangxi, China.

An Efficient Approach to Nowcasting the Time-varying Reproduction Number.

Estimating the instantaneous reproduction number ( ) in near real time is crucial for monitoring and responding to epidemic outbreaks on a daily basis. However, such estimates often suffer from bias due to reporting delays inherent in surveillance systems. We propose a fast and flexible Bayesian methodology to overcome this challenge by estimating while taking into account reporting delays. Furthermore, the method naturally takes into account the uncertainty associated with the nowcasting of cases to get a valid uncertainty estimation of the nowcasted reproduction number. We evaluate the proposed methodology through a simulation study and apply it to COVID-19 incidence data in Belgium.