A simple model for complex dynamical transitions in epidemics

Effect of homophily on coupled behavior-disease dynamics near a tipping point

Understanding the interplay between social activities and disease dynamics is crucial for effective public health interventions. Recent studies using coupled behavior-disease models assumed homogeneous populations. However, heterogeneity in population, such as different social groups, cannot be ignored. In this study, we divided the population into social media users and non-users, and investigated the impact of homophily (the tendency for individuals to associate with others similar to themselves) and online events on disease dynamics. Our results reveal that homophily hinders the adoption of vaccinating strategies, hastening the approach to a tipping point after which the population converges to an endemic equilibrium with no vaccine uptake. Furthermore, we find that online events can significantly influence disease dynamics, with early discussions on social media platforms serving as an early warning signal of potential disease outbreaks. Our model provides insights into the mechanisms underlying these phenomena and underscores the importance of considering homophily in disease modeling and public health strategies.

The impact of household size on measles transmission: A long-term perspective

Households play an important role in the transmission of infectious diseases due to the close contact therein. Previous modeling studies on disease transmission with household-level mixing have explored the relationship between household size distribution and epidemic characteristics such as final epidemic sizes and the basic reproduction number but have not considered the epidemic impact of declining household sizes caused by demographic shifts. Here, we use a disease transmission model that incorporates demographic changes in household sizes to study the long-term transmission dynamics of measles in communities with varying household size distributions. We explore the impact of incorporating both household- and age-structured mixing on the dynamic properties of the transmission model and compare these dynamics across different household size distributions. Our analysis, based on the household- and age-structured model, shows that communities with larger household sizes require higher vaccination thresholds and bear a greater burden of infections. However, simulations show the apparent impact of changing household sizes is the combined result of changing birth rates and household mixing, and that changing birth rates likely play a larger role than changes in household mixing in shaping measles transmission dynamics (n.b, life-long immunity makes replenishment of population susceptibility from births a crucial transmission driver for measles). In addition, simulations of endemic transmission of measles within a hypothetical population formulated using aggregated world demographic data suggest the decline in household size (driven by changing fertility rates of the population), in addition to increasing vaccination coverage, could have had a significant impact on the incidence of measles over time.

Control, bi-stability, and preference for chaos in time-dependent vaccination campaign

In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of a constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., >0.95). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and, in some cases, via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.

Unraveling the Role of Viral Interference in Disrupting Biennial RSV Epidemics in Northern Stockholm

Respiratory syncytial virus (RSV) primarily impacts infants and older adults, with seasonal winter outbreaks in temperate countries. Biennial cycles of RSV activity have also been identified in Northern Europe and some states in the United States. Delayed RSV activity was reported worldwide during the 2009 influenza pandemic, and a disrupted biennial pattern of RSV activity was observed in northern Stockholm following the pandemic. Biennial patterns shifted to early/large outbreaks in even-numbered years and late/small outbreaks in odd-numbered years. However, the mechanisms underpinning this change in pattern remain unknown. In this work, we constructed an age-stratified mechanistic model to explicitly test three factors that could lead to the change in RSV transmission dynamics: 1) birth rates, 2) temperatures, and 3) viral interference. By fitting the model to weekly RSV admission data over a 20-year period and comparing different models, we found that viral interference from influenza was the only mechanism that explained the shifted biennial pattern. Our work demonstrates the complex interplay between different respiratory viruses, providing evidence that supports the presence of interactions between the H1N1 pandemic influenza virus and RSV at the population level, with implications for future public health interventions.

Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

Optimal Timing Regularly Outperforms Higher Coverage in Preventative Measles Supplementary Immunization Campaigns

Measles threatens the lives and livelihoods of tens of millions of children and there are countries where routine immunization systems miss enough individuals to create the risk of large outbreaks. To help address this threat, measles supplementary immunization activities are time-limited, coordinated campaigns to immunize en masse a target population. Timing campaigns to be concurrent with building outbreak risk is an important consideration, but current programmatic standards focus on campaigns achieving a high coverage of at least 95%. We show that there is a dramatic trade-off between campaign timeliness and coverage. Optimal timing at coverages as low as 50% for areas with weak routine immunization systems is shown to outperform the current standard, which is delayed by as little as 6 months. Measured coverage alone is revealed as a potentially misleading performance metric.

Chaos theory in the understanding of COVID-19 pandemic dynamics

The chaos theory, a field of study in mathematics and physics, offers a unique lens through which to understand the dynamics of the COVID-19 pandemic. This theory, which deals with complex systems whose behavior is highly sensitive to initial conditions, can provide insights into the unpredictable and seemingly random nature of the pandemic's spread. In this review, we will discuss some literature data with the aim of showing how chaos theory could provide valuable perspectives in understanding the complex and dynamic nature of the COVID-19 pandemic. In particular, we will emphasize how the chaos theory can help in dissecting the unpredictable, non- linear progression of the disease, the importance of initial conditions, and the complex interactions between various factors influencing its spread. These insights are crucial for developing effective strategies to manage and mitigate the impact of the pandemic.

Dispersal synchronizes giant kelp forests

Spatial synchrony is the tendency for population fluctuations to be correlated among different locations. This phenomenon is a ubiquitous feature of population dynamics and is important for ecosystem stability, but several aspects of synchrony remain unresolved. In particular, the extent to which any particular mechanism, such as dispersal, contributes to observed synchrony in natural populations has been difficult to determine. To address this gap, we leveraged recent methodological improvements to determine how dispersal structures synchrony in giant kelp (Macrocystis pyrifera), a global marine foundation species that has served as a useful system for understanding synchrony. We quantified population synchrony and fecundity with satellite imagery across 11 years and 880 km of coastline in southern California, USA, and estimated propagule dispersal probabilities using a high-resolution ocean circulation model. Using matrix regression models that control for the influence of geographic distance, resources (seawater nitrate), and disturbance (destructive waves), we discovered that dispersal was an important driver of synchrony. Our findings were robust to assumptions about propagule mortality during dispersal and consistent between two metrics of dispersal: (1) the individual probability of dispersal and (2) estimates of demographic connectivity that incorporate fecundity (the number of propagules dispersing). We also found that dispersal and environmental conditions resulted in geographic clusters with distinct patterns of synchrony. This study is among the few to statistically associate synchrony with dispersal in a natural population and the first to do so in a marine organism. The synchronizing effects of dispersal and environmental conditions on foundation species, such as giant kelp, likely have cascading effects on the spatial stability of biodiversity and ecosystem function.

How immune dynamics shape multi-season epidemics: a continuous-discrete model in one dimensional antigenic space

We extend a previously published model for the dynamics of a single strain of an influenza-like infection. The model incorporates a waning acquired immunity to infection and punctuated antigenic drift of the virus, employing a set of coupled integral equations within a season and a discrete map between seasons. The long term behaviour of the model is demonstrated by examples where immunity to infection depends on the time since a host was last infected, and where immunity depends on the number of times that a host has been infected. The first scenario leads to complicated dynamics in some regions of parameter space, and to regions of parameter space with more than one attractor. The second scenario leads to a stable fixed point, corresponding to an identical epidemic each season. We also examine the model with both paradigms in combination, almost always but not exclusively observing a stable fixed point or periodic solution. Adding stochastic perturbations to the between season map fails to destroy the model's qualitative dynamics. Our results suggest that if the level of host immunity depends on the elapsed time since the last infection then the epidemiological dynamics may be unpredictable.

Predicting the impact of COVID-19 non-pharmaceutical intervention on short- and medium-term dynamics of enterovirus D68 in the US

Recent outbreaks of enterovirus D68 (EV-D68) infections, and their causal linkage with acute flaccid myelitis (AFM), continue to pose a serious public health concern. During 2020 and 2021, the dynamics of EV-D68 and other pathogens have been significantly perturbed by non-pharmaceutical interventions against COVID-19; this perturbation presents a powerful natural experiment for exploring the dynamics of these endemic infections. In this study, we analyzed publicly available data on EV-D68 infections, originally collected through the New Vaccine Surveillance Network, to predict their short- and long-term dynamics following the COVID-19 interventions. Although long-term predictions are sensitive to our assumptions about underlying dynamics and changes in contact rates during the NPI periods, the likelihood of a large outbreak in 2023 appears to be low. Comprehensive surveillance data are needed to accurately characterize future dynamics of EV-D68. The limited incidence of AFM cases in 2022, despite large EV-D68 outbreaks, poses further questions for the timing of the next AFM outbreaks.

Simulating real-life scenarios to better understand the spread of diseases under different contexts

Current statistical models to simulate pandemics miss the most relevant information about the close atomic interactions between individuals which is the key aspect of virus spread. Thus, they lack a proper visualization of such interactions and their impact on virus spread. In the field of computer graphics, and more specifically in computer animation, there have been many crowd simulation models to populate virtual environments. However, the focus has typically been to simulate reasonable paths between random or semi-random locations in a map, without any possibility of analyzing specific individual behavior. We propose a crowd simulation framework to accurately simulate the interactions in a city environment at the individual level, with the purpose of recording and analyzing the spread of human diseases. By simulating the whereabouts of agents throughout the day by mimicking the actual activities of a population in their daily routines, we can accurately predict the location and duration of interactions between individuals, thus having a model that can reproduce the spread of the virus due to human-to-human contact. Our results show the potential of our framework to closely simulate the virus spread based on real agent-to-agent contacts. We believe that this could become a powerful tool for policymakers to make informed decisions in future pandemics and to better communicate the impact of such decisions to the general public.

The probability of epidemic burnout in the stochastic SIR model with vital dynamics

We present an approach to computing the probability of epidemic "burnout," i.e., the probability that a newly emergent pathogen will go extinct after a major epidemic. Our analysis is based on the standard stochastic formulation of the Susceptible-Infectious-Removed (SIR) epidemic model including host demography (births and deaths) and corresponds to the standard SIR ordinary differential equations (ODEs) in the infinite population limit. Exploiting a boundary layer approximation to the ODEs and a birth-death process approximation to the stochastic dynamics within the boundary layer, we derive convenient, fully analytical approximations for the burnout probability. We demonstrate-by comparing with computationally demanding individual-based stochastic simulations and with semi-analytical approximations derived previously-that our fully analytical approximations are highly accurate for biologically plausible parameters. We show that the probability of burnout always decreases with increased mean infectious period. However, for typical biological parameters, there is a relevant local minimum in the probability of persistence as a function of the basic reproduction number [Formula: see text]. For the shortest infectious periods, persistence is least likely if [Formula: see text]; for longer infectious periods, the minimum point decreases to [Formula: see text]. For typical acute immunizing infections in human populations of realistic size, our analysis of the SIR model shows that burnout is almost certain in a well-mixed population, implying that susceptible recruitment through births is insufficient on its own to explain disease persistence.

Waning immunity can drive repeated waves of infections

In infectious disease models, it is known that mechanisms such as births, seasonality in transmission and pathogen evolution can generate oscillations in infection numbers. We show how waning immunity is also a mechanism that is sufficient on its own to enable sustained oscillations. When previously infected or vaccinated individuals lose full protective immunity, they become partially susceptible to reinfections. This partial immunity subsequently wanes over time, making individuals more susceptible to reinfections and potentially more infectious if infected. Losses of full and partial immunity lead to a surge in infections, which is the precursor of oscillations. We present a discrete-time Susceptible-Infectious-Immune-Waned-Infectious (SIRWY) model that features the waning of fully immune individuals (as a distribution of time at which individuals lose fully immunity) and the gradual loss of partial immunity (as increases in susceptibility and potential infectiousness over time). A special case of SIRWY is the discrete-time SIRS model with geometric distributions for waning and recovery. Its continuous-time analogue is the classic SIRS with exponential distributions, which does not produce sustained oscillations for any choice of parameters. We show that the discrete-time version can produce sustained oscillations and that the oscillatory regime disappears as discrete-time tends to continuous-time. A different special case of SIRWY is one with fixed times for waning and recovery. We show that this simpler model can also produce sustained oscillations. In conclusion, under certain feature and parameter choices relating to how exactly immunity wanes, fluctuations in infection numbers can be sustained without the need for any additional mechanisms.

Malaria, climate variability, and interventions: modelling transmission dynamics

Assessment of the relative impact of climate change on malaria dynamics is a complex problem. Climate is a well-known factor that plays a crucial role in driving malaria outbreaks in epidemic transmission areas. However, its influence in endemic environments with intensive malaria control interventions is not fully understood, mainly due to the scarcity of high-quality, long-term malaria data. The demographic surveillance systems in Africa offer unique platforms for quantifying the relative effects of weather variability on the burden of malaria. Here, using a process-based stochastic transmission model, we show that in the lowlands of malaria endemic western Kenya, variations in climatic factors played a key role in driving malaria incidence during 2008-2019, despite high bed net coverage and use among the population. The model captures some of the main mechanisms of human, parasite, and vector dynamics, and opens the possibility to forecast malaria in endemic regions, taking into account the interaction between future climatic conditions and intervention scenarios.

Colonization and transmission of Staphylococcus aureus in schools: a citizen science project

Aggregation of children in schools has been established to be a key driver of transmission of infectious diseases. Mathematical models of transmission used to predict the impact of control measures, such as vaccination and testing, commonly depend on self-reported contact data. However, the link between self-reported social contacts and pathogen transmission has not been well described. To address this, we used Staphylococcus aureus as a model organism to track transmission within two secondary schools in England and test for associations between self-reported social contacts, test positivity and the bacterial strain collected from the same students. Students filled out a social contact survey and their S. aureus colonization status was ascertained through self-administered swabs from which isolates were sequenced. Isolates from the local community were also sequenced to assess the representativeness of school isolates. A low frequency of genome-linked transmission precluded a formal analysis of links between genomic and social networks, suggesting that S. aureus transmission within schools is too rare to make it a viable tool for this purpose. Whilst we found no evidence that schools are an important route of transmission, increased colonization rates found within schools imply that school-age children may be an important source of community transmission.

Global analysis and prediction scenario of infectious outbreaks by recurrent dynamic model and machine learning models: A case study on COVID-19

It is essential to evaluate patient outcomes at an early stage when dealing with a pandemic to provide optimal clinical care and resource management. Many methods have been proposed to provide a roadmap against different pandemics, including the recent pandemic disease COVID-19. Due to recurrent epidemic waves of COVID-19, which have been observed in many countries, mathematical modeling and forecasting of COVID-19 are still necessary as long as the world continues to battle against the pandemic. Modeling may aid in determining which interventions to try or predict future growth patterns. In this article, we design a combined approach for analyzing any pandemic in two separate parts. In the first part of the paper, we develop a recurrent SEIRS compartmental model to predict recurrent outbreak patterns of diseases. Due to its time-varying parameters, our model is able to reflect the dynamics of infectious diseases, and to measure the effectiveness of the restrictive measures. We discuss the stable solutions of the corresponding autonomous system with frozen parameters. We focus on the regime shifts and tipping points; then we investigate tipping phenomena due to parameter drifts in our time-varying parameters model that exhibits a bifurcation in the frozen-in case. Furthermore, we propose an optimal numerical design for estimating the system’s parameters. In the second part, we introduce machine learning models to strengthen the methodology of our paper in data analysis, particularly for prediction scenarios. We use MLP, RBF, LSTM, ANFIS, and GRNN for training and evaluation of COVID-19. Then, we compare the results with the recurrent dynamical system in the fitting process and prediction scenario. We also confirm results by implementing our methods on the released data on COVID-19 by WHO for Italy, Germany, Iran, and South Africa between 1/22/2020 and 7/24/2021, when people were engaged with different variants including Alpha, Beta, Gamma, and Delta. The results of this article show that the dynamic model is adequate for long-term analysis and data fitting, as well as obtaining parameters affecting the epidemic. However, it is ineffective in providing a long-term forecast. In contrast machine learning methods effectively provide disease prediction, although they do not provide analysis such as dynamic models. Finally, some metrics, including RMSE, R-Squared, and accuracy, are used to evaluate the machine learning models. These metrics confirm that ANFIS and RBF perform better than other methods in training and testing zones.

Waves of infection emerging from coupled social and epidemiological dynamics

The coronavirus (SARS-CoV-2) exhibited waves of infection in 2020 and 2021 in Japan. The number of infected had multiple distinct peaks at intervals of several months. One possible process causing these waves of infection is people switching their activities in response to the prevalence of infection. In this paper, we present a simple model for the coupling of social and epidemiological dynamics. The assumptions are as follows. Each person switches between active and restrained states. Active people move more often to crowded areas, interact with each other, and suffer a higher rate of infection than people in the restrained state. The rate of transition from restrained to active states is enhanced by the fraction of currently active people (conformity), whereas the rate of backward transition is enhanced by the abundance of infected people (risk avoidance). The model may show transient or sustained oscillations, initial-condition dependence, and various bifurcations. The infection is maintained at a low level if the recovery rate is between the maximum and minimum levels of the force of infection. In addition, waves of infection may emerge instead of converging to the stationary abundance of infected people if both conformity and risk avoidance of people are strong.

Pediatric burden and seasonality of human metapneumovirus over 5 years in Managua, Nicaragua

BACKGROUND

Human metapneumovirus (hMPV) is an important cause of pediatric respiratory infection. We leveraged the Nicaraguan Pediatric Influenza Cohort Study (NPICS) to assess the burden and seasonality of symptomatic hMPV infection in children.

METHODS

NPICS is an ongoing prospective study of children in Managua, Nicaragua. We assessed children for hMPV infection via real-time reverse-transcription polymerase chain reaction (RT-PCR). We used classical additive decomposition analysis to assess the temporal trends, and generalized growth models (GGMs) were used to estimate effective reproduction numbers.

RESULTS

From 2011 to 2016, there were 564 hMPV symptomatic infections, yielding an incidence rate of 5.74 cases per 100 person-years (95% CI 5.3, 6.2). Children experienced 3509 acute lower respiratory infections (ALRIs), of which 160 (4.6%) were associated with hMPV infection. Children under the age of one had 55% of all symptomatic hMPV infections (62/112) develop into hMPV-associated ALRIs and were five times as likely as children over one to have an hMPV-associated ALRI (rate ratio 5.5 95% CI 4.1, 7.4 p < 0.001). Additionally, symptomatic reinfection with hMPV was common. In total, 87 (15%) of all observed symptomatic infections were detected reinfections. The seasonality of symptomatic hMPV outbreaks varied considerably. From 2011 to 2016, four epidemic periods were observed, following a biennial seasonal pattern. The mean ascending phase of the epidemic periods were 7.7 weeks, with an overall mean estimated reproductive number of 1.2 (95% CI 1.1, 1.4).

CONCLUSIONS

Symptomatic hMPV infection was associated with substantial burden among children in the first year of life. Timing and frequency of symptomatic hMPV incidence followed biennial patterns.

A mathematical model for simulating the spread of a disease through a country divided into geographical regions with different population densities

The SIR (susceptible-infectious-recovered) model is a well known method for predicting the number of people (or animals) in a population who become infected by and then recover from a disease. Modifications can include categories such people who have been exposed to the disease but are not yet infectious or those who die from the disease. However, the model has nearly always been applied to the entire population of a country or state but there is considerable observational evidence that diseases can spread at different rates in densely populated urban regions and sparsely populated rural areas. This work presents a new approach that applies a SIR type model to a country or state that has been divided into a number of geographical regions, and uses different infection rates in each region which depend on the population density in that region. Further, the model contains a simple matrix based method for simulating the movement of people between different regions. The model is applied to the spread of disease in the United Kingdom and the state of Rio Grande do Sul in Brazil.

Predicting the effects of climate change on the cross-scale epidemiological dynamics of a fungal plant pathogen

The potential for climate change to exacerbate the burden of human infectious diseases is increasingly recognized, but its effects on infectious diseases of plants have received less attention. Understanding the impacts of climate on the epidemiological dynamics of plant pathogens is imperative, as these organisms play central roles in natural ecosystems and also pose a serious threat to agricultural production and food security. We use the fungal ‘flax rust’ pathogen (Melampsora lini) and its subalpine wildflower host Lewis flax (Linum lewisii) to investigate how climate change might affect the dynamics of fungal plant pathogen epidemics using a combination of empirical and modeling approaches. Our results suggest that climate change will initially slow transmission at both the within- and between-host scales. However, moderate resurgences in disease spread are predicted as warming progresses, especially if the rate of greenhouse gas emissions continues to increase at its current pace. These findings represent an important step towards building a holistic understanding of climate effects on plant infectious disease that encompasses demographic, epidemiological, and evolutionary processes. A core result is that neglecting processes at any one scale of plant pathogen transmission may bias projections of climate effects, as climate drivers have variable and cascading impacts on processes underlying transmission that occur at different scales.

Study of the Effect of Urban Densification and Micrometeorology on the Sustainability of a Coronavirus-Type Pandemic

This research examines the persistence of a pandemic in urban environments subjected to intensive densification processes, applying chaotic analysis tools to hourly time series constructed by relating accumulated patients with meteorological and pollutant variables (measured at ground level). To investigate this objective, seven communes of the metropolitan region of Santiago de Chile that present intensive urbanization processes that affect urban micrometeorology, favoring the concentration of pollutants, were considered. Quotients were constructed between the number of hourly patients with SARS-CoV-2 that accumulated in each commune over a period of two years and the hourly variables of urban micrometeorology (temperature, magnitude of wind speed, relative humidity) and pollutant concentration (tropospheric ozone, particulate material of 2.5 and 10 μm) constituting a new family of time series. Chaos theory was applied to these new time series, obtaining the chaotic parameters Lyapunov coefficient, correlation entropy, Lempel–Ziv complexity, Hurst coefficient and the fractal dimension in each measurement commune. The results showed that the accumulated patients (2020–2022), of the order of 400,000, belonged to the five communes (with a built area of approximately 300,000 m2 in recent years) that had the highest urban densification, which affected urban meteorology, favored the concentration of pollutants and made the SARS-Cov-2 pandemic more persistent. The “ideal” density of built housing should balance a pandemic and nullify its expansion.

Population growth of Gyrodactylus kobayashii in goldfish (Carassius auratus) associated with host density

Host density is a key regulatory factor in parasite transmission. The goldfish (Carassius auratus)-Gyrodactylus kobayashii model was used to investigate effects of host density on population growth of gyrodactylids. A donor fish infected by five gravid gyrodactylids was mixed with 11 parasite-free goldfish at five host densities. There was a significant positive correlation between host density and mean abundance of G. kobayashii throughout the 58-day experiment. During early infection (days 15–24), mean abundance in medium high (0.5 fish L−1) and high host density groups (1 and 2 fish L−1) was significantly higher than that in the low host density groups (0.125 and 0.25 fish L−1). At high host density, prevalence increased more rapidly, and the peak prevalence was higher. Fitting of an exponential growth model showed that the population growth rate of the parasite increased with host density. A hypothesis was proposed that higher host density contributed to increased reinfection of detached gyrodactylids. A reinfection experiment was designed to test this hypothesis. Both mean abundance and prevalence at a host density of 1 fish L−1 were significantly higher than those at 0.25 fish L−1 on days 1 and 3, which suggested that more reinfections of G. kobyashii occurred at the higher host density. Density-dependent transmission during the early infection was an important contributor of population growth of G. kobayashii, as well as density-dependent reinfection of the detached gyrodactylids.

Impact of population mixing between vaccinated and unvaccinated subpopulations on infectious disease dynamics: implications for SARS-CoV-2 transmission

Background:

The speed of vaccine development has been a singular achievement during the COVID-19 pandemic, although uptake has not been universal. Vaccine opponents often frame their opposition in terms of the rights of the unvaccinated. We sought to explore the impact of mixing of vaccinated and unvaccinated populations on risk of SARS-CoV-2 infection among vaccinated people.

Methods:

We constructed a simple susceptible–infectious–recovered compartmental model of a respiratory infectious disease with 2 connected subpopulations: people who were vaccinated and those who were unvaccinated. We simulated a spectrum of patterns of mixing between vaccinated and unvaccinated groups that ranged from random mixing to complete like-with-like mixing (complete assortativity), in which people have contact exclusively with others with the same vaccination status. We evaluated the dynamics of an epidemic within each subgroup and in the population as a whole.

Results:

We found that the risk of infection was markedly higher among unvaccinated people than among vaccinated people under all mixing assumptions. The contact-adjusted contribution of unvaccinated people to infection risk was disproportionate, with unvaccinated people contributing to infections among those who were vaccinated at a rate higher than would have been expected based on contact numbers alone. We found that as like-with-like mixing increased, attack rates among vaccinated people decreased from 15% to 10% (and increased from 62% to 79% among unvaccinated people), but the contact-adjusted contribution to risk among vaccinated people derived from contact with unvaccinated people increased.

Interpretation:

Although risk associated with avoiding vaccination during a virulent pandemic accrues chiefly to people who are unvaccinated, their choices affect risk of viral infection among those who are vaccinated in a manner that is disproportionate to the portion of unvaccinated people in the population.

Dissecting recurrent waves of pertussis across the boroughs of London

Pertussis has resurfaced in the UK, with incidence levels not seen since the 1980s. While the fundamental causes of this resurgence remain the subject of much conjecture, the study of historical patterns of pathogen diffusion can be illuminating. Here, we examined time series of pertussis incidence in the boroughs of Greater London from 1982 to 2013 to document the spatial epidemiology of this bacterial infection and to identify the potential drivers of its percolation. The incidence of pertussis over this period is characterized by 3 distinct stages: a period exhibiting declining trends with 4-year inter-epidemic cycles from 1982 to 1994, followed by a deep trough until 2006 and the subsequent resurgence. We observed systematic temporal trends in the age distribution of cases and the fade-out profile of pertussis coincident with increasing national vaccine coverage from 1982 to 1990. To quantify the hierarchy of epidemic phases across the boroughs of London, we used the Hilbert transform. We report a consistent pattern of spatial organization from 1982 to the early 1990s, with some boroughs consistently leading epidemic waves and others routinely lagging. To determine the potential drivers of these geographic patterns, a comprehensive parallel database of borough-specific features was compiled, comprising of demographic, movement and socio-economic factors that were used in statistical analyses to predict epidemic phase relationships among boroughs. Specifically, we used a combination of a feed-forward neural network (FFNN), and SHapley Additive exPlanations (SHAP) values to quantify the contribution of each covariate to model predictions. Our analyses identified a number of predictors of a borough's historical epidemic phase, specifically the age composition of households, the number of agricultural and skilled manual workers, latitude, the population of public transport commuters and high-occupancy households. Univariate regression analysis of the 2012 epidemic identified the ratio of cumulative unvaccinated children to the total population and population of Pakistan-born population to have moderate positive and negative association, respectively, with the timing of epidemic. In addition to providing a comprehensive overview of contemporary pertussis transmission in a large metropolitan population, this study has identified the characteristics that determine the spatial spread of this bacterium across the boroughs of London.

Epidemic Spreading in Trajectory Networks

Epidemics of infectious diseases, such as the one caused by the rapid spread of the coronavirus disease 2019 (COVID-19), have tested the world's more advanced health systems and have caused an enormous societal and economic damage. The mechanism of contagion is well understood. As people move around, over time, they regularly engage in social interactions. The spatiotemporal network representing these interactions constitutes the backbone on which an epidemic spreads, causing outbreaks. At the same time, advanced technological responses have claimed some success in controlling the epidemic based on digital contact tracing technologies. Motivated by these observations, we design, develop and evaluate a stochastic agent-basedSEIRmodel of epidemic spreading in spatiotemporal networks informed by mobility data of individuals (trajectories). The model focuses on individual variation in mobility patterns that affects the degree of exposure to the disease. Understanding the role that individual nodes play in the process of disease spreading through network effects is fundamental as it allows to (i) assess the risk of infection of individuals, (ii) assess the size of a disease outbreak due to specific individuals, and (iii) assess targeted intervention strategies that aim to control the epidemic spreading. We perform a comprehensive analysis of the model employing COVID-19 as a use case. The results indicate that simple individual-based intervention strategies that exhibit significant network effects can effectively control the spread of an epidemic. We have also demonstrated that targeted interventions can outperform generic intervention strategies. Overall, our work provides an evidence-based data-driven model to support decision making and inform public policy regarding intervention strategies for containing or mitigating the epidemic spread.

A behavioural modelling approach to assess the impact of COVID-19 vaccine hesitancy

We introduce a compartmental epidemic model to describe the spread of COVID-19 within a population, assuming that a vaccine is available, but vaccination is not mandatory. The model takes into account vaccine hesitancy and the refusal of vaccination by individuals, which take their decision on vaccination based on both the present and past information about the spread of the disease. Theoretical analysis and simulations show that voluntary vaccination can certainly reduce the impact of the disease but is unable to eliminate it. We also demonstrate how the information-related parameters affect the dynamics of the disease. In particular, vaccine hesitancy and refusal are better contained in case of widespread information coverage and short-term memory. Finally, the possible impact of seasonality on the spread of the disease is investigated.

Infectious disease in an era of global change

The twenty-first century has witnessed a wave of severe infectious disease outbreaks, not least the COVID-19 pandemic, which has had a devastating impact on lives and livelihoods around the globe. The 2003 severe acute respiratory syndrome coronavirus outbreak, the 2009 swine flu pandemic, the 2012 Middle East respiratory syndrome coronavirus outbreak, the 2013-2016 Ebola virus disease epidemic in West Africa and the 2015 Zika virus disease epidemic all resulted in substantial morbidity and mortality while spreading across borders to infect people in multiple countries. At the same time, the past few decades have ushered in an unprecedented era of technological, demographic and climatic change: airline flights have doubled since 2000, since 2007 more people live in urban areas than rural areas, population numbers continue to climb and climate change presents an escalating threat to society. In this Review, we consider the extent to which these recent global changes have increased the risk of infectious disease outbreaks, even as improved sanitation and access to health care have resulted in considerable progress worldwide.

Complexity of COVID-19 Dynamics

With population explosion and globalization, the spread of infectious diseases has been a major concern. In 2019, a newly identified type of Coronavirus caused an outbreak of respiratory illness, popularly known as COVID-19, and became a pandemic. Although enormous efforts have been made to understand the spread of COVID-19, our knowledge of the COVID-19 dynamics still remains limited. The present study employs the concepts of chaos theory to examine the temporal dynamic complexity of COVID-19 around the world. The false nearest neighbor (FNN) method is applied to determine the dimensionality and, hence, the complexity of the COVID-19 dynamics. The methodology involves: (1) reconstruction of a single-variable COVID-19 time series in a multi-dimensional phase space to represent the underlying dynamics; and (2) identification of “false” neighbors in the reconstructed phase space and estimation of the dimension of the COVID-19 series. For implementation, COVID-19 data from 40 countries/regions around the world are studied. Two types of COVID-19 data are analyzed: (1) daily COVID-19 cases; and (2) daily COVID-19 deaths. The results for the 40 countries/regions indicate that: (1) the dynamics of COVID-19 cases exhibit low- to medium-level complexity, with dimensionality in the range 3 to 7; and (2) the dynamics of COVID-19 deaths exhibit complexity anywhere from low to high, with dimensionality ranging from 3 to 13. The results also suggest that the complexity of the dynamics of COVID-19 deaths is greater than or at least equal to that of the dynamics of COVID-19 cases for most (three-fourths) of the countries/regions. These results have important implications for modeling and predicting the spread of COVID-19 (and other infectious diseases), especially in the identification of the appropriate complexity of models.

Forecast of the COVID-19 trend in India: A simple modelling approach

By February 2021, the overall impact of the COVID-19 pandemic in India had been relatively mild in terms of total reported cases and deaths. Surprisingly, the second wave in early April becomes devastating and attracts worldwide attention. Multiple factors (e.g., Delta variants with increased transmissibility) could have driven the rapid growth of the epidemic in India and led to a large number of deaths within a short period. We aim to reconstruct the transmission rate, estimate the infection fatality rate and forecast the epidemic size. We download the reported COVID-19 mortality data in India and formulate a simple mathematical model with a flexible transmission rate. We use iterated filtering to fit our model to deaths data. We forecast the infection attack rate in a month ahead. Our model simulation matched the reported deaths well and is reasonably close to the results of the serological study. We forecast that the infection attack rate (IAR) could have reached 43% by July 24, 2021, under the current trend. Our estimated infection fatality rate is about 0.07%. Under the current trend, the IAR will likely reach a level of 43% by July 24, 2021. Our estimated infection fatality rate appears unusually low, which could be due to a low case to infection ratio reported in previous study. Our approach is readily applicable in other countries and with other types of data (e.g., excess deaths).

A stochastic model explains the periodicity phenomenon of influenza on network

Influenza is an infectious disease with obvious periodic changes over time. It is of great practical significance to explore the non-environment-related factors that cause this regularity for influenza control and individual protection. In this paper, based on the randomness of population number and the heterogeneity of population contact, we have established a stochastic infectious disease model about influenza based on the degree of the network, and obtained the power spectral density function by using the van Kampen expansion method of the master equation. The relevant parameters are obtained by fitting the influenza data of sentinel hospitals. The results of the numerical analysis show that: (1) for the infected, the infection period of patients who go to the sentinel hospitals is particularly different from the others who do not; (2) for all the infected, there is an obvious nonlinear relationship between their infection period and the visiting rate of the influenza sentinel hospitals, the infection rate and the degree. Among them, only the infection period of patients who do not go to the sentinel hospitals decreased monotonously with the infection rate (increased monotonously with the visiting rate), while the rest had a non-monotonic relationship.

A generalized differential equation compartmental model of infectious disease transmission

For decades, mathematical models of disease transmission have provided researchers and public health officials with critical insights into the progression, control, and prevention of disease spread. Of these models, one of the most fundamental is the SIR differential equation model. However, this ubiquitous model has one significant and rarely acknowledged shortcoming: it is unable to account for a disease's true infectious period distribution. As the misspecification of such a biological characteristic is known to significantly affect model behavior, there is a need to develop new modeling approaches that capture such information. Therefore, we illustrate an innovative take on compartmental models, derived from their general formulation as systems of nonlinear Volterra integral equations, to capture a broader range of infectious period distributions, yet maintain the desirable formulation as systems of differential equations. Our work illustrates a compartmental model that captures any Erlang distributed duration of infection with only 3 differential equations, instead of the typical inflated model sizes required by traditional differential equation compartmental models, and a compartmental model that captures any mean, standard deviation, skewness, and kurtosis of an infectious period distribution with 4 differential equations. The significance of our work is that it opens up a new class of easy-to-use compartmental models to predict disease outbreaks that do not require a complete overhaul of existing theory, and thus provides a starting point for multiple research avenues of investigation under the contexts of mathematics, public health, and evolutionary biology.

The effect of population size for pathogen transmission on prediction of COVID-19 spread

Extreme public health interventions play a critical role in mitigating the local and global prevalence and pandemic potential. Here, we use population size for pathogen transmission to measure the intensity of public health interventions, which is a key characteristic variable for nowcasting and forecasting of COVID-19. By formulating a hidden Markov dynamic system and using nonlinear filtering theory, we have developed a stochastic epidemic dynamic model under public health interventions. The model parameters and states are estimated in time from internationally available public data by combining an unscented filter and an interacting multiple model filter. Moreover, we consider the computability of the population size and provide its selection criterion. With applications to COVID-19, we estimate the mean of the effective reproductive number of China and the rest of the globe except China (GEC) to be 2.4626 (95% CI: 2.4142-2.5111) and 3.0979 (95% CI: 3.0968-3.0990), respectively. The prediction results show the effectiveness of the stochastic epidemic dynamic model with nonlinear filtering. The hidden Markov dynamic system with nonlinear filtering can be used to make analysis, nowcasting and forecasting for other contagious diseases in the future since it helps to understand the mechanism of disease transmission and to estimate the population size for pathogen transmission and the number of hidden infections, which is a valid tool for decision-making by policy makers for epidemic control.

Transient disease dynamics across ecological scales

Analyses of transient dynamics are critical to understanding infectious disease transmission and persistence. Identifying and predicting transients across scales, from within-host to community-level patterns, plays an important role in combating ongoing epidemics and mitigating the risk of future outbreaks. Moreover, greater emphases on non-asymptotic processes will enable timely evaluations of wildlife and human diseases and lead to improved surveillance efforts, preventive responses, and intervention strategies. Here, we explore the contributions of transient analyses in recent models spanning the fields of epidemiology, movement ecology, and parasitology. In addition to their roles in predicting epidemic patterns and endemic outbreaks, we explore transients in the contexts of pathogen transmission, resistance, and avoidance at various scales of the ecological hierarchy. Examples illustrate how (i) transient movement dynamics at the individual host level can modify opportunities for transmission events over time; (ii) within-host energetic processes often lead to transient dynamics in immunity, pathogen load, and transmission potential; (iii) transient connectivity between discrete populations in response to environmental factors and outbreak dynamics can affect disease spread across spatial networks; and (iv) increasing species richness in a community can provide transient protection to individuals against infection. Ultimately, we suggest that transient analyses offer deeper insights and raise new, interdisciplinary questions for disease research, consequently broadening the applications of dynamical models for outbreak preparedness and management.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s12080-021-00514-w.

Epidemiological dynamics of enterovirus D68 in the United States and implications for acute flaccid myelitis

Acute flaccid myelitis (AFM) recently emerged in the United States as a rare but serious neurological condition since 2012. Enterovirus D68 (EV-D68) is thought to be a main causative agent, but limited surveillance of EV-D68 in the United States has hampered the ability to assess their causal relationship. Using surveillance data from the BioFire Syndromic Trends epidemiology network in the United States from January 2014 to September 2019, we characterized the epidemiological dynamics of EV-D68 and found latitudinal gradient in the mean timing of EV-D68 cases, which are likely climate driven. We also demonstrated a strong spatiotemporal association of EV-D68 with AFM. Mathematical modeling suggested that the recent dominant biennial cycles of EV-D68 dynamics may not be stable. Nonetheless, we predicted that a major EV-D68 outbreak, and hence an AFM outbreak, would have still been possible in 2020 under normal epidemiological conditions. Nonpharmaceutical intervention efforts due to the ongoing COVID-19 pandemic are likely to have reduced the sizes of EV-D68 and AFM outbreaks in 2020, illustrating the broader epidemiological impact of the pandemic.

Climate Change-Driven Regime Shifts in a Planktonic Food Web

AbstractPredicting how food webs will respond to global environmental change is difficult because of the complex interplay between the abiotic forcing and biotic interactions. Mechanistic models of species interactions in seasonal environments can help understand the effects of global change in different ecosystems. Seasonally ice-covered lakes are warming faster than many other ecosystems and undergoing pronounced food web changes, making the need to forecast those changes especially urgent. Using a seasonally forced food web model with a generalist zooplankton grazer and competing cold-adapted winter and warm-adapted summer phytoplankton, we show that with declining ice cover, the food web moves through different dynamic regimes, from annual to biennial cycles, with decreasing and then disappearing winter phytoplankton blooms and a shift of maximum biomass to summer season. Interestingly, when predator-prey interactions were not included, a declining ice cover did not cause regime shifts, suggesting that both are needed for regime transitions. A cluster analysis of long-term data from Lake Baikal, Siberia, supports the model results, revealing a change from regularly occurring winter blooms of endemic diatoms to less frequent winter bloom years with decreasing ice cover. Together, the results show that even gradual environmental change, such as declining ice cover duration, may cause discontinuous or abrupt transitions between dynamic regimes in food webs.

Epidemiological dynamics of enterovirus D68 in the US: implications for acute flaccid myelitis

The lack of active surveillance for enterovirus D68 (EV-D68) in the US has hampered the ability to assess the relationship with predominantly biennial epidemics of acute flaccid myelitis (AFM), a rare but serious neurological condition. Using novel surveillance data from the BioFire® Syndromic Trends (Trend) epidemiology network, we characterize the epidemiological dynamics of EV-D68 and demonstrate strong spatiotemporal association with AFM. Although the recent dominant biennial cycles of EV-D68 dynamics may not be stable, we show that a major EV-D68 epidemic, and hence an AFM outbreak, would still be possible in 2020 under normal epidemiological conditions. Significant social distancing due to the ongoing COVID-19 pandemic could reduce the size of an EV-D68 epidemic in 2020, illustrating the potential broader epidemiological impact of the pandemic.

Chaos in the peroxidase-oxidase oscillator

The peroxidase-oxidase (PO) reaction involves the oxidation of reduced nicotinamide adenine dinucleotide by molecular oxygen. When both reactants are supplied continuously to a reaction mixture containing the enzyme and a phenolic compound, the reaction will exhibit oscillatory behavior. In fact, the reaction exhibits a zoo of dynamical behaviors ranging from simple periodic oscillations to period-doubled and mixed mode oscillations to quasiperiodicity and chaos. The routes to chaos involve period-doubling, period-adding, and torus bifurcations. The dynamic behaviors in the experimental system can be simulated by detailed semiquantitative models. Previous models of the reaction have omitted the phenolic compound from the reaction scheme. In the current paper, we present new experimental results with the oscillating PO reaction that add to our understanding of its rich dynamics, and we describe a new variant of a previous model, which includes the chemistry of the phenol in the reaction mechanism. This new model can simulate most of the experimental behaviors of the experimental system including the new observations presented here. For example, the model reproduces the two main routes to chaos observed in experiments: (i) a period-doubling scenario, which takes place at low pH, and a period-adding scenario involving mixed mode oscillations (MMOs), which occurs at high pH. Our simulations suggest alternative explanations for the pH-sensitivity of the dynamics. We show that the MMO domains are separated by narrow parameter regions of chaotic behavior or quasiperiodicity. These regions start as tongues of secondary quasiperiodicity and develop into strange attractors through torus breakdown.

Development and dissemination of infectious disease dynamic transmission models during the COVID-19 pandemic: what can we learn from other pathogens and how can we move forward?

The current COVID-19 pandemic has resulted in the unprecedented development and integration of infectious disease dynamic transmission models into policy making and public health practice. Models offer a systematic way to investigate transmission dynamics and produce short-term and long-term predictions that explicitly integrate assumptions about biological, behavioural, and epidemiological processes that affect disease transmission, burden, and surveillance. Models have been valuable tools during the COVID-19 pandemic and other infectious disease outbreaks, able to generate possible trajectories of disease burden, evaluate the effectiveness of intervention strategies, and estimate key transmission variables. Particularly given the rapid pace of model development, evaluation, and integration with decision making in emergency situations, it is necessary to understand the benefits and pitfalls of transmission models. We review and highlight key aspects of the history of infectious disease dynamic models, the role of rigorous testing and evaluation, the integration with data, and the successful application of models to guide public health. Rather than being an expansive history of infectious disease models, this Review focuses on how the integration of modelling can continue to be advanced through policy and practice in appropriate and conscientious ways to support the current pandemic response.

How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors

Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed upon a monotonic decay needs a different description from that of the standard chaotic dynamics. The concept of snapshot attractors and their natural distribution has been adopted from the field of the latest climate change research. This shows the importance of the finite-time chaotic effect and ensemble interpretation while investigating the spread of a disease. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics-even for those values of contact parameters where originally regular behaviour is expected. We argue that anomalous outbreaks of the infectious class cannot die out until transient chaos is presented in the system. Nevertheless, this fact becomes apparent by using an ensemble approach rather than a single trajectory representation. These findings are applicable generally in explicitly time-dependent epidemic systems regardless of parameter values and time scales.

Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate

In this paper, Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate and the effect of periodic and bounded noise on the chaotic motion of SIR model possessing homoclinic orbits are detailed investigated. Based on homoclinic bifurcation, necessary conditions for possible chaotic motion as well as sufficient condition are derived by the random Melnikov theorem, and to establish the threshold of bounded noise amplitude for the onset of chaos.

Patterns of smallpox mortality in London, England, over three centuries

Smallpox is unique among infectious diseases in the degree to which it devastated human populations, its long history of control interventions, and the fact that it has been successfully eradicated. Mortality from smallpox in London, England was carefully documented, weekly, for nearly 300 years, providing a rare and valuable source for the study of ecology and evolution of infectious disease. We describe and analyze smallpox mortality in London from 1664 to 1930. We digitized the weekly records published in the London Bills of Mortality (LBoM) and the Registrar General's Weekly Returns (RGWRs). We annotated the resulting time series with a sequence of historical events that might have influenced smallpox dynamics in London. We present a spectral analysis that reveals how periodicities in reported smallpox mortality changed over decades and centuries; many of these changes in epidemic patterns are correlated with changes in control interventions and public health policies. We also examine how the seasonality of reported smallpox mortality changed from the 17th to 20th centuries in London.

Epidemiological Characteristics, Seasonal Dynamic Patterns, and Associations with Meteorological Factors of Rubella in Shaanxi Province, China, 2005-2018

Rubella occurs worldwide, causing approximately 100,000 cases annually of congenital rubella syndrome, leading to severe birth defects. Better targeting of public health interventions is needed to achieve rubella elimination goals. To that end, we measured the epidemiological characteristics and seasonal dynamic patterns of rubella and determined its association with meteorological factors in Shaanxi Province, China. Data on rubella cases in Shaanxi Province from 2005 to 2018 were obtained from the Chinese National Notifiable Disease Reporting System. The Morlet wavelet analysis was used to estimate temporal periodicity of rubella incidence. Mixed generalized additive models were used to measure associations between meteorological variables (temperature and relative humidity) and rubella incidence. A total of 17,185 rubella cases were reported in Shaanxi during the study period, for an annual incidence of 3.27 cases per 100,000 population. Interannual oscillations in rubella incidence of 0.8-1.4 years, 3.8-4.8 years, and 0.5 years were detected. Both temperature and relative humidity exhibited nonlinear associations with the incidence of rubella. The accumulative relative risk of transmission for the overall pooled estimates was maximized at a temperature of 0.23°C and relative humidity of 41.6%. This study found that seasonality and meteorological factors have impact on the transmission of rubella; public health interventions to eliminate rubella must consider periodic and seasonal fluctuations as well as meteorological factors.

Estimating epidemic coupling between populations from the time to invasion

Identifying the mechanisms by which diseases spread among populations is important for understanding and forecasting patterns of epidemics and pandemics. Estimating transmission coupling among populations is challenging because transmission events are difficult to observe in practice, and connectivity among populations is often obscured by local disease dynamics. We consider the common situation in which an epidemic is seeded in one population and later spreads to a second population. We present a method for estimating transmission coupling between the two populations, assuming they can be modelled as susceptible-infected-removed (SIR) systems. We show that the strength of coupling between the two populations can be estimated from the time taken for the disease to invade the second population. Confidence in the estimate is low if only a single invasion event has been observed, but is substantially improved if numerous independent invasion events are observed. Our analysis of this simplest, idealized scenario represents a first step toward developing and verifying methods for estimating epidemic coupling among populations in an ever-more-connected global human population.

Transient indicators of tipping points in infectious diseases

The majority of known early warning indicators of critical transitions rely on asymptotic resilience and critical slowing down. In continuous systems, critical slowing down is mathematically described by a decrease in magnitude of the dominant eigenvalue of the Jacobian matrix on the approach to a critical transition. Here, we show that measures of transient dynamics, specifically, reactivity and the maximum of the amplification envelope, also change systematically as a bifurcation is approached in an important class of models for epidemics of infectious diseases. Furthermore, we introduce indicators designed to detect trends in these measures and find that they reliably classify time series of case notifications simulated from stochastic models according to levels of vaccine uptake. Greater attention should be focused on the potential for systems to exhibit transient amplification of perturbations as a critical threshold is approached, and should be considered when searching for generic leading indicators of tipping points. Awareness of this phenomenon will enrich understanding of the dynamics of complex systems on the verge of a critical transition.

Fast estimation of time-varying infectious disease transmission rates

Compartmental epidemic models have been used extensively to study the historical spread of infectious diseases and to inform strategies for future control. A critical parameter of any such model is the transmission rate. Temporal variation in the transmission rate has a profound influence on disease spread. For this reason, estimation of time-varying transmission rates is an important step in identifying mechanisms that underlie patterns in observed disease incidence and mortality. Here, we present and test fast methods for reconstructing transmission rates from time series of reported incidence. Using simulated data, we quantify the sensitivity of these methods to parameters of the data-generating process and to mis-specification of input parameters by the user. We show that sensitivity to the user's estimate of the initial number of susceptible individuals-considered to be a major limitation of similar methods-can be eliminated by an efficient, "peak-to-peak" iterative technique, which we propose. The method of transmission rate estimation that we advocate is extremely fast, for even the longest infectious disease time series that exist. It can be used independently or as a fast way to obtain better starting conditions for computationally expensive methods, such as iterated filtering and generalized profiling.

Optimal public health intervention in a behavioural vaccination model: the interplay between seasonality, behaviour and latency period

Hesitancy and refusal of vaccines preventing childhood diseases are spreading due to 'pseudo-rational' behaviours: parents overweigh real and imaginary side effects of vaccines. Nonetheless, the 'Public Health System' (PHS) may enact public campaigns to favour vaccine uptake. To determine the optimal time profiles for such campaigns, we apply the optimal control theory to an extension of the susceptible-infectious-removed (SIR)-based behavioural vaccination model by d'Onofrio et al. (2012, PLoS ONE, 7, e45653). The new model is of susceptible-exposed-infectious-removed (SEIR) type under seasonal fluctuations of the transmission rate. Our objective is to minimize the total costs of the disease: the disease burden, the vaccination costs and a less usual cost: the economic burden to enact the PHS campaigns. We apply the Pontryagin minimum principle and numerically explore the impact of seasonality, human behaviour and latency rate on the control and spread of the target disease. We focus on two noteworthy case studies: the low (resp. intermediate) relative perceived risk of vaccine side effects and relatively low (resp. very low) speed of imitation. One general result is that seasonality may produce a remarkable impact on PHS campaigns aimed at controlling, via an increase of the vaccination uptake, the spread of a target infectious disease. In particular, a higher amplitude of the seasonal variation produces a higher effort and this, in turn, beneficially impacts the induced vaccine uptake since the larger is the strength of seasonality, the longer the vaccine propensity remains large. However, such increased effort is not able to fully compensate the action of seasonality on the prevalence.

Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate

A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate kIS 1 + β I + α I 2 ( β > - 2 α such that 1 + β I + α I 2 > 0 for all I ≥ 0 ) is considered in this paper. It is shown that the basic reproduction number R 0 does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold value R ∗ ( < 1 ) such that: (i) if R 0 < R ∗ , then the disease-free equilibrium is globally asymptotically stable; (ii) if R 0 = R ∗ , then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if R ∗ < R 0 < 1 , then there are two endemic equilibria, one is a weak focus of multiplicity at least three, the other is a saddle; (iv) if R 0 ≥ 1 , then there is again a unique endemic equilibrium which is a weak focus of multiplicity at least three. As parameters vary, the model undergoes saddle-node bifurcation, backward bifurcation, Bogdanov-Takens bifurcation of codimension three, Hopf bifurcation, and degenerate Hopf bifurcation of codimension three. Moreover, it is shown that there exists a critical value α 0 for the psychological effect α , a critical value k 0 for the infection rate k, and two critical values β 0 , β 1 ( β 1 < β 0 ) for β that will determine whether the disease dies out or persists in the form of positive periodic coexistent oscillations or coexistent steady states under different initial populations. Numerical simulations are given to demonstrate the existence of one, two or three limit cycles.

Predicting re-emergence times of dengue epidemics at low reproductive numbers: DENV1 in Rio de Janeiro, 1986-1990

Predicting arbovirus re-emergence remains challenging in regions with limited off-season transmission and intermittent epidemics. Current mathematical models treat the depletion and replenishment of susceptible (non-immune) hosts as the principal drivers of re-emergence, based on established understanding of highly transmissible childhood diseases with frequent epidemics. We extend an analytical approach to determine the number of ‘skip’ years preceding re-emergence for diseases with continuous seasonal transmission, population growth and under-reporting. Re-emergence times are shown to be highly sensitive to small changes in low R₀ (secondary cases produced from a primary infection in a fully susceptible population). We then fit a stochastic Susceptible–Infected–Recovered (SIR) model to observed case data for the emergence of dengue serotype DENV1 in Rio de Janeiro. This aggregated city-level model substantially over-estimates observed re-emergence times either in terms of skips or outbreak probability under forward simulation. The inability of susceptible depletion and replenishment to explain re-emergence under ‘well-mixed’ conditions at a city-wide scale demonstrates a key limitation of SIR aggregated models, including those applied to other arboviruses. The predictive uncertainty and high skip sensitivity to epidemiological parameters suggest a need to investigate the relevant spatial scales of susceptible depletion and the scaling of microscale transmission dynamics to formulate simpler models that apply at coarse resolutions.

A global-scale ecological niche model to predict SARS-CoV-2 coronavirus infection rate

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A Maximum-Entropy Ecological Niche Model is used to estimate a global-scale probability distribution of COVID-19 high infection rate.

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Environmental parameters (surface air temperature, precipitation, and elevation) and humanrelated parameters (CO₂ emission and population density) are used in the model.

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The model is trained only with data of Italian provinces with high infection rate, but predicts known actual infection focuses, e.g. the Hubei province in China.

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A risk index is proposed, which correctly classifies most World countries, which have reported high COVID-19 spread rate, as zones with high-risk of infection rate increase.

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The methodology follows an Open-science approach where the model is published as a standardized Web service that maximises re-usability on new data and new diseases, and guarantees the transparency of the approach and the results.

COVID-19 pandemic is a global threat to human health and economy that requires urgent prevention and monitoring strategies. Several models are under study to control the disease spread and infection rate and to detect possible factors that might favour them, with a focus on understanding the correlation between the disease and specific geophysical parameters. However, the pandemic does not present evident environmental hindrances in the infected countries. Nevertheless, a lower rate of infections has been observed in some countries, which might be related to particular population and climatic conditions.

In this paper, infection rate of COVID-19 is modelled globally at a 0.5^∘ resolution, using a Maximum Entropy-based Ecological Niche Model that identifies geographical areas potentially subject to a high infection rate. The model identifies locations that could favour infection rate due to their particular geophysical (surface air temperature, precipitation, and elevation) and human-related characteristics (CO² and population density). It was trained by facilitating data from Italian provinces that have reported a high infection rate and subsequently tested using datasets from World countries’ reports. Based on this model, a risk index was calculated to identify the potential World countries and regions that have a high risk of disease increment.

The distribution outputs foresee a high infection rate in many locations where real-world disease outbreaks have occurred, e.g. the Hubei province in China, and reports a high risk of disease increment in most World countries which have reported significant outbreaks (e.g. Western U.S.A.). Overall, the results suggest that a complex combination of the selected parameters might be of integral importance to understand the propagation of COVID-19 among human populations, particularly in Europe. The model and the data were distributed through Open-science Web services to maximise opportunities for re-usability regarding new data and new diseases, and also to enhance the transparency of the approach and results.