Seasonal variation in environmental variables, and in rates of contact among individuals, are fundamental drivers of infectious disease dynamics. Unlike most periodically-forced physical systems, for which the precise pattern of forcing is typically known, underlying patterns of seasonal variation in transmission rates can be estimated approximately at best, and only the period of forcing is accurately known. Yet solutions of epidemic models depend strongly on the forcing function, so dynamical predictions---such as changes in epidemic patterns that can be induced by demographic transitions or mass vaccination---are always subject to the objection that the underlying patterns of seasonality are poorly specified. Here, we demonstrate that the key bifurcations of the standard epidemic model are invariant to the shape of seasonal forcing if the amplitude of forcing is appropriately adjusted. Consequently, analyses applicable to real disease dynamics can be conducted with a smooth, idealized sinusoidal forcing function, and qualitative changes in epidemic patterns can be predicted without precise knowledge of the underlying forcing pattern. We find similar invariance in a seasonally forced predator-prey model, and conjecture that this phenomenon---and the associated robustness of predictions---might be a feature of many other periodically forced dynamical systems.
Deterministic models are developed for the spatial spread of epidemic diseases in geographical settings. The models are focused on outbreaks that arise from a small number of infected hosts imported into sub-regions of the geographical settings. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The models consist of systems of partial differential equations with diffusion terms describing the spatial spread of the underlying microbial infectious agents. The model is compared with real data from seasonal influen…
Common respiratory viruses cause seasonal epidemics in sequential patterns. The underlying mechanisms for this pattern have been debated for some time. For influenza, contenders include temperature, humidity, and vitamin D levels. While such seasonal drivers may be sufficient to explain midwinter peaking it is unclear if other respiratory viruses peaking in spring, summer, or fall have individual environmental drivers. Here we present a dynamic model of interacting diseases, in an effort to explain observed seasonal patterns without requiring multiple seasonal drivers. Our model extends the cl…
Background: Influenza A/H3N2 has been circulating in humans since 1968, causing considerable morbidity and mortality. Although H3N2 incidence is highly seasonal, how such seasonality contributes to global phylogeographic migration dynamics has not yet been established. Results: Incorporating seasonally varying migration rates improves the modeling of migration. In our global model, windows of increased immigration map to the seasonal timing of epidemic spread, while windows of increased emigration map to epidemic decline. Seasonal patterns also correlate with the probability that local lineage…
We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious …
The study of epidemiological systems has generated deep interest in exploring the dynamical complexity of common infectious diseases driven by seasonally varying contact rates. Mathematical modeling and field observations have shown that, under seasonal variation, the incidence rates of some endemic infectious diseases fluctuate dramatically and the dynamics is often characterized by chaotic oscillations in the absence of specific vaccination programs. In fact, the existence of chaotic behavior has been precisely stated in the literature as a noticeable feature in the dynamics of the classical…
The interplay of seasonality, the system's nonlinearities and intrinsic stochasticity is studied for a seasonally forced susceptible-exposed-infective-recovered stochastic model. The model is explored in the parameter region that corresponds to childhood infectious diseases such as measles. The power spectrum of the stochastic fluctuations around the attractors of the deterministic system that describes the model in the thermodynamic limit is computed analytically and validated by stochastic simulations for large system sizes. Size effects are studied through additional simulations. Other effe…
We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle…
The impact of the outbreak of COVID-19 on health has been widely concerned. Disease risk assessment, prediction, and early warning have become a significant research field. Previous research suggests that there is a relationship between air quality and the disease. This paper investigated the impact of the atmospheric environment on the basic reproduction number (R$_0$) in Australia, South Korea, and Italy by using atmospheric environment data, confirmed case data, and the distributed lag non-linear model (DLNM) model based on Quasi-Poisson regression. The results show that the air temperature…
We consider a SIR model with temporary immunity and time dependent transmission rate. We assume time dependent vaccination which confers the same immunity as natural infection. We study two types of vaccination strategies: i) optimal vaccination, in the sense that it minimizes the effort of vaccination in the set of vaccination strategies for which, for any sufficiently small perturbation of the disease free state, the number of infectious individuals is monotonically decreasing; ii) Nash-equilibria strategies where all individuals simultaneously minimize the joint risk of vaccination versus t…
Epidemiological data on seasonal influenza show that the growth rate of the number of infected individuals can increase passing from one exponential growth rate to another one with a larger exponent. Such behavior is not described by conventional epidemiological models. In this work an immuno-epidemiological model is proposed in order to describe this two-stage growth. It takes into account that the growth in the number of infected individuals increases the initial viral load and provides a passage from the first stage of epidemic where only people with weak immune response are infected to the…
Seasonal influenza epidemics cause consistent, considerable, widespread loss annually in terms of economic burden, morbidity, and mortality. With access to accurate and reliable forecasts of a current or upcoming influenza epidemic's behavior, policy makers can design and implement more effective countermeasures. We developed a framework for in-season forecasts of epidemics using a semiparametric Empirical Bayes framework, and applied it to predict the weekly percentage of outpatient doctors visits for influenza-like illness, as well as the season onset, duration, peak time, and peak height, w…
On 11 June the World Health Organization officially raised the phase of pandemic alert (with regard to the new H1N1 influenza strain) to level 6. We use a global structured metapopulation model integrating mobility and transportation data worldwide in order to estimate the transmission potential and the relevant model parameters we used the data on the chronology of the 2009 novel influenza A(H1N1). The method is based on the maximum likelihood analysis of the arrival time distribution generated by the model in 12 countries seeded by Mexico by using 1M computationally simulated epidemics. An e…
In Fall 2020, several European countries reported rapid increases in COVID-19 cases along with growing estimates of the effective reproduction rates. Such an acceleration in epidemic spread is usually attributed to time-dependent effects, e.g. human travel, seasonal behavioral changes, mutations of the pathogen etc. In this case however the acceleration occurred when counter measures such as testing and contact tracing exceeded their capacity limit. Considering Austria as an example, here we show that this dynamics can be captured by a time-independent, i.e. autonomous, compartmental model tha…
The history of infections and epidemics holds famous examples where understanding, containing and ultimately treating an outbreak began with understanding its mode of spread. Influenza, HIV and most computer viruses, spread person to person, device to device, through contact networks; Cholera, Cancer, and seasonal allergies, on the other hand, do not. In this paper we study two fundamental questions of detection: first, given a snapshot view of a (perhaps vanishingly small) fraction of those infected, under what conditions is an epidemic spreading via contact (e.g., Influenza), distinguishable…
The seasonality of respiratory diseases (common cold, influenza, etc.) is a well-known phenomenon studied from ancient times. The development of predictive models is still not only an actual unsolved problem of mathematical epidemiology but also is very important for the safety of public health. Here we show that SIRS (Susceptible-Infected-Recovered-Susceptible) model accurately enough reproduces real curves of flu activity. It contains variable reaction rate, which is a function of mean daily temperature. The proposed alternation of variables represents SIRS equations as the second-order ODE …
Ross-Macdonald models are the building blocks of most vector-borne disease models. Even for the same disease, different authors use different model formulations, but a study of the dynamical consequences of assuming different hypotheses is missing. In this work we present different formulations of the basic Ross-Macdonald model together with a careful discussion of the assumptions behind each model. The most general model presented is an agent based model for which arbitrary distributions for latency and infectious periods for both, host and vectors, is considered. At population level we also …
Many vector-borne disease epidemic models neglect the fact that in modern human civilization, social awareness as well as self-defence system are overwhelming against advanced propagation of the disease. News are becoming more effortlessly accessible through social media and mobile apps, while apparatuses for disease prevention are inclined to be more abundant and affordable. Here we study a simple host-vector model in which media-triggered social awareness and seasonality in vector breeding are taken into account. There appears a certain threshold indicating the alarming outbreak; the number …
Background: Seasonal influenza causes a substantial burden on healthcare services over the winter period when these systems are already under pressure. Policies during the COVID-19 pandemic supressed the transmission of season influenza, making the timing and magnitude of a potential resurgence difficult to predict. Methods: We developed a hierarchical generalised additive model (GAM) for the short-term forecasting of hospital admissions with a positive test for the influenza virus sub-regionally across England. The model incorporates a multi-level structure of spatio-temporal splines, weekly …
Metapopulation (multipatch) models are widely used to study the patterns of spatial spread of epidemics. In this paper we study the impact of inter-patch connection weights on the predictions of these models. We contrast arbitrary, uniform link weights with link weights predicted using a gravity model based on patch populations and distance. In a synthetic system with one large driver city and many small follower cities, we show that under uniform link weights, epidemics in the follower regions are perfectly synchronized. In contrast, gravity-based links allow a more realistic, less synchroniz…
It is well known that behavioral changes in contact patterns may significantly affect the spread of an epidemic outbreak. Here we focus on simple endemic models for recurrent epidemics, by modelling the social contact rate as a function of the available information on the present and past disease prevalence. We show that social behaviour change alone may trigger sustained oscillations. This indicates that human behavior might be a critical explaining factor of oscillations in time-series of endemic diseases. Finally, we briefly show how the inclusion of seasonal variations in contacts may impl…
Influenza Hospitalisations in England during the 2022/23 Season: do different data sources drive divergence in modelled waves? A comparison of surveillance and administrative data
The spread of tick-borne pathogens represents an important threat to human and animal health in many parts of Eurasia. Here, we analysed a 9-year time series of Ixodes ricinus ticks feeding on Apodemus flavicollis mice (main reservoir-competent host for tick-borne encephalitis, TBE) sampled in Trentino (Northern Italy). The tail of the distribution of the number of ticks per host was fitted by three theoretical distributions: Negative Binomial (NB), Poisson-LogNormal (PoiLN), and Power-Law (PL). The fit with theoretical distributions indicated that the tail of the tick infestation pattern on m…
COVID-19 pandemic is one of the major disasters that humanity has ever faced. In this paper, we try to model the effect of vaccination in controlling the pandemic, particularly in context to the third wave which is predicted to hit globally. Here we have modified the SEIRD model by introducing a vaccination term. One of our main assumptions is that the infection rate (\b{eta}(t)) is oscillatory. This oscillatory nature has been discussed earlier in literature with reference to the seasonality of epidemics. However, in our case we invoke this nature of the infection rate (\b{eta}(t)) to model t…
It is possible to model vector-borne infection using the classical Ross-Macdonald model. This attempt, however fails in several respects. First, using measured (or estimated) parameters, the model predicts a much greater number of cases than what is usually observed. Second, the model predicts a single huge outbreaks that is followed after decades of much smaller outbreaks. This is not what is observed. Usually towns or cities report a number of cases that recur for many years, even when environmental changes cannot explain the disappearance of the infection in-between the peaks. In this paper…
