Fundamental Approaches towards Predictive Epidemic Modelling

Mathematical models are increasingly adopted for setting disease prevention and control targets. As model-informed policies are implemented, however, the inaccuracies of some forecasts become apparent, for example overprediction of infection burdens and intervention impacts. Here, we attribute these discrepancies to methodological limitations in capturing the heterogeneities of real-world systems. The mechanisms underpinning risk factors of infection and their interactions determine individual propensities to acquire disease. These factors are potentially so numerous and complex that to attain a full mechanistic description is likely unfeasible. To contribute constructively to the development of health policies, model developers either leave factors out (reductionism) or adopt a broader but coarse description (holism). In our view, predictive capacity requires holistic descriptions of heterogeneity which are currently underutilised in infectious disease epidemiology, in comparison to other population disciplines, such as non-communicable disease epidemiology, demography, ecology and evolution.

The dynamics of the COVID-19 pandemic is greatly influenced by vaccine quality, as well as by vaccination rates and the behaviour of infected individuals, both of which reflect public health policies. We develop a model for the dynamics of relevant cohorts within a fixed population, taking extreme care to model the reduced social contact of infected individuals in a rigorous self-consistent manner. The basic reproduction number R₀ is then derived in terms of the parameters of the model. Analysis of R₀ reveals two interesting possibilities, both of which are plausible based on known characteristics of COVID-19. Firstly, if the population in general moderates social contact, while infected individuals who display clinical symptoms tend not to isolate, then increased vaccination can drive the epidemic towards a disease-free equilibrium (DFE). However, if the reverse is true, then increased vaccination can destabilise the DFE and yield an endemic state. This surprising result is due to the fact that the vaccines are leaky, and can lead to an increase in asymptomatic individuals who unknowingly spread the disease. Therefore, this work shows that public policy regarding the monitoring and release of health data should be combined judiciously with modeling-informed vaccination policy to control COVID-19.

This study aims to develop a novel mathematical epidemic compartmental model that includes a compartment or class for individuals who become infected and experience severe illness due to the infection. These individuals require hospitalization and the use of specialized medical equipment, such as ventilators, ICU beds, etc, during an outbreak. This compartment is referred to as the 'hospitalized population compartment' throughout this study. Additionally, the model incorporates a saturated incidence rate for new infection cases and the hospitalization rate for individuals severely affected by the infection, intending to create a more realistic scenario of the dynamics of disease transmission. The model is developed by integrating a compartment for hospitalized individuals into the standard susceptible-infected-recovered compartmental model and is subsequently mathematically analyzed for qualitative behavior. In this model, the saturated hospitalization rate reflects that the number of severely infected individuals who can be hospitalized is limited at any given time due to constraints in sufficient hospital infrastructure availability. The incidence rate of susceptibles becoming infected is modeled using the Holling Type II functional form, which incorporates inhibitory effects observed within the population. The study analyzes the mathematical model for two types of equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). To investigate the stability of both equilibria, the basic reproduction number, ${R_0}$, is calculated using the next-generation matrix method. The findings indicate that when ${R_0} < 1$, the DFE is locally asymptotically stable. Conversely, when ${R_0} > 1$, the DFE becomes unstable, leading to the emergence of a positive EE. Additionally, the study explores the occurrence of forward and backward transcritical bifurcations under specific conditions when ${R_0} = 1$. Furthermore, the study delves into both the local and global stability behaviors of the EE. Numerical simulations of the model are also performed to support the theoretical findings.

We study two meta-population models of infectious diseases in heterogeneous networks. We distinguish between asymptomatic and symptomatic infections and these two go through the different courses of infection and recovery. We consider that asymptomatic infections are described by an SIS model and symptomatic infections by an SIR or SIRS model depending on the immunity upon recovery. By introducing the probability of being infected asymptomatically, we combine an SIS model for asymptomatic infections with an SIR or SIRS model for symptomatic infections to obtain the SIS-SIR and SIS-SIRS models. We use a heterogeneous mean-field theory and Monte Carlo simulations to analyze two models and find that both models undergo nonequilibrium continuous phase transitions from the endemic phase to the disease-free phase at certain critical thresholds as we vary the proportion of asymptomatic infections. It suggests that it may be possible to maintain the population in the disease-free phase by controlling the proportion of asymptomatic infections. The SIS-SIRS model shows that asymptomatic infection drives symptomatic infection and vice versa. In addition, the spreading of infections eventually ceases as the population decreases even at a fixed proportion of asymptomatic infections corresponding to the endemic phase. The results provide a theoretical basis for understanding the epidemiological facts that social distancing and reducing asymptomatic infections are important factors in optimizing quarantine measures to prevent the epidemic outbreaks of infectious diseases.

We simulate spreads of diseases for the susceptible–infected–recovered (SIR) model on contact networks with a particular focus on incorporated protective measures such as face masks. We consider the small-world network model. By using a large-deviation approach, in particular the $1/t$ Wang–Landau algorithm, we obtained the full probability density function (pdf) of the cumulative number C of infected people over the full range of its support. In this way we are able to reach probabilities as small as 10⁻⁵⁰. We obtain distinct characteristics in the pdf such as non-analyticities induced by the onset of the protective measures. Still, the results indicate that the mathematical large-deviation principle also holds for this extended SIR model, meaning that the size-dependence enters in P(C) in a simple fashion and the distribution is determined by the so-called rate function. We observe different phases in the pdf, which we investigate by analyzing the corresponding infection courses, i.e. time series, and and their correlations to the observed values of C.

Optimal protocols of vaccine administration to minimize the effects of infectious diseases depend on a number of variables that admit different degrees of control. Examples include the characteristics of the disease and how it impacts on different groups of individuals as a function of sex, age or socioeconomic status, its transmission mode, or the demographic structure of the affected population. Here we introduce a compartmental model of infection propagation with vaccination and reinfection and analyze the effect that variations on the rates of these two processes have on the progression of the disease and on the number of fatalities. The population is split into two groups to highlight the overall effects on disease caused by different relationships between vaccine administration and various demographic structures. As a practical example, we study COVID-19 dynamics in various countries using real demographic data. The model can be easily applied to any other disease transmitted through direct interaction between infected and susceptible individuals, and any demographic structure, through a suitable estimation of parameter values. Two main conclusions stand out. First, the higher the fraction of reinfected individuals, the higher the likelihood that the disease becomes quasi-endemic. Second, optimal vaccine roll-out depends on demographic structure and disease fatality, so there is no unique vaccination protocol, valid for all countries, that minimizes the effects of a specific disease. Simulations of the general model can be carried out at this interactive webpage Atienza (2021 S²iyrd model simulator).

The existence of an exponential growth phase during early stages of a pandemic is often taken for granted. However, for the 2019 novel coronavirus epidemic, the early exponential phase lasted only for about six days, while the quadratic growth prevailed for forty days until it spread to other countries and continued, again quadratically, but with a shorter time constant. Here we show that this rapid phase is followed by a subsequent slow-down where the coefficient is reduced to almost the original value at the outbreak. This can be explained by the merging of previously disconnected sites that occurred after the disease jumped (nonlocally) to a relatively small number of separated sites. Subsequent variations in the slope with continued growth can qualitatively be explained as a result of reinfections and variations in their rate. We demonstrate that the observed behavior can be described by a standard epidemiological model with spatial extent and reinfections included. Time-dependent changes in the spatial diffusion coefficient can also model corresponding variations in the slope.

We study a discrete susceptible–infected–recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to infinity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution.

A significant proportion of the infections driving the current SARS-CoV-2 pandemic are transmitted asymptomatically. Here we introduce and study a simple epidemic model with separate compartments comprising asymptomatic and symptomatic infected individuals. The linear dynamics determining the outbreak condition of the model is equivalent to a renewal theory approach with exponential waiting time distributions. Exploiting a nontrivial conservation law of the full nonlinear dynamics, we derive analytic bounds on the peak number of infections in the absence and presence of mitigation through isolation and testing. The bounds are compared to numerical solutions of the differential equations.

Global strategies to contain a pandemic, such as social distancing and protective measures, are designed to reduce the overall transmission rate between individuals. Despite such measures, essential institutions, including hospitals, schools, and food producing plants, remain focal points of local outbreaks. Here we develop a model for the stochastic infection dynamics that predicts the statistics of local outbreaks from observables of the underlying global epidemics. Specifically, we predict two key outbreak characteristics: the probability of proliferation from a first infection in the local community, and the establishment size, which is the threshold size of local infection clusters where proliferation becomes likely. We derive these results using a contact network model of communities, and we show how the proliferation probability depends on the contact degree of the first infected individual. Based on this model, we suggest surveillance protocols by which individuals are tested proportionally to their degree in the contact network. We characterize the efficacy of contact-based protocols as a function of the epidemiological and the contact network parameters, and we show numerically that such protocols outperform random testing.

We derive a generalized Hamiltonian formalism for a modified susceptible–infectious–recovered/removed (SIR) epidemic model taking into account the population V of vaccinated persons. The resulting SIRV model is shown to admit three possible functionally independent Hamiltonians and hence three associated Poisson structures. The reduced case of vanishing vaccinated sector shows a complete correspondence with the known Poisson structures of the SIR model. The SIRV model is shown to be expressible as an almost Nambu system, except for a scale factor function breaking the divergenceless property. In the autonomous case with time-independent stationary ratios k and b, the SIRV model is shown to be a maximally super-integrable system. For this case we test the accuracy of numerical schemes that are suited to solve the stiff set of SIRV differential equations.

During the COVID pandemic, periods of exponential growth of the disease have been mitigated by containment measures that in different occasions have resulted in a power-law growth of the number of cases. The first observation of such behaviour has been obtained from 2020 late spring data coming from China by Ziff and Ziff in reference Ziff and Ziff (2020 Fractal kinetics of COVID-19 pandemic MedRxiv). After this important observation the power-law scaling (albeit with different exponents) has also been observed in other countries during periods of containment of the spread. Early interpretations of these results suggest that this phenomenon might be due to spatial effects of the spread. Here we show that temporal modulations of infectivity of individuals due to containment measures can also cause power-law growth of the number of cases over time. To this end we propose a stochastic well-mixed susceptible-infected-removed model of epidemic spreading in presence of containment measures resulting in a time dependent infectivity and we explore the statistical properties of the resulting branching process at criticality. We show that at criticality it is possible to observe power-law growth of the number of cases with exponents ranging between one and two. Our asymptotic analytical results are confirmed by extensive Monte Carlo simulations. Although these results do not exclude that spatial effects might be important in modulating the power-law growth of the number of cases at criticality, this work shows that even well-mixed populations may already feature non trivial power-law exponents at criticality.

We employ individual-based Monte Carlo computer simulations of a stochastic SEIR model variant on a two-dimensional Newman–Watts small-world network to investigate the control of epidemic outbreaks through periodic testing and isolation of infectious individuals, and subsequent quarantine of their immediate contacts. Using disease parameters informed by the COVID-19 pandemic, we investigate the effects of various crucial mitigation features on the epidemic spreading: fraction of the infectious population that is identifiable through the tests; testing frequency; time delay between testing and isolation of positively tested individuals; and the further time delay until quarantining their contacts as well as the quarantine duration. We thus determine the required ranges for these intervention parameters to yield effective control of the disease through both considerable delaying the epidemic peak and massively reducing the total number of sustained infections.

Controlling the COVID-19 pandemic is an urgent global challenge. The rapid geographic spread of SARS-CoV-2 directly reflects the social structure. Before effective vaccines and treatments are widely available, we have to rely on alternative, non-pharmaceutical interventions, including frequent testing, contact tracing, social distancing, mask wearing, and hand-washing, as public health practises to slow down the spread of the disease. However, frequent testing is the key in the absence of any alternative. We propose a network approach to determine the optimal low resources setting oriented pool testing strategies that identifies infected individuals in a small number of tests and few rounds of testing, at low prevalence of the virus. We simulate stochastic infection curves on societies under quarantine. Allowing some social interaction is possible to keep the COVID-19 curve flat. However, similar results can be strategically obtained searching and isolating infected persons to preserve a healthier social structure. Here, we analyze which are the best strategies to contain the virus applying an algorithm that combine samples and testing them in groups [1]. A relevant parameter to keep infection curves flat using this algorithm is the daily frequency of testing at zones where a high infection rate is reported. On the other hand, the algorithm efficiency is low for random search of infected people.

The earlier analytical analysis (part A) of the susceptible–infectious–recovered (SIR) epidemics model for a constant ratio k of infection to recovery rates is extended here to the semi-time case which is particularly appropriate for modeling the temporal evolution of later (than the first) pandemic waves when a greater population fraction from the first wave has been infected. In the semi-time case the SIR model does not describe the quantities in the past; instead they only hold for times later than the initial time t = 0 of the newly occurring wave. Simple exact and approximative expressions are derived for the final and maximum values of the infected, susceptible and recovered/removed population fractions as well the daily rate and cumulative number of new infections. It is demonstrated that two types of temporal evolution of the daily rate of new infections j(τ) occur depending on the values of k and the initial value of the infected fraction I(0) = η: in the decay case for k ⩾ 1 − 2η the daily rate monotonically decreases at all positive times from its initial maximum value j(0) = η(1 − η). Alternatively, in the peak case for k < 1 − 2η the daily rate attains a maximum at a finite positive time. By comparing the approximated analytical solutions for j(τ) and J(τ) with the exact ones obtained by numerical integration, it is shown that the analytical approximations are accurate within at most only 2.5 percent. It is found that the initial fraction of infected persons sensitively influences the late time dependence of the epidemics, the maximum daily rate and its peak time. Such dependencies do not exist in the earlier investigated all-time case.

By characterizing the time evolution of COVID-19 in term of its 'velocity' (log of the new cases per day) and its rate of variation, or 'acceleration', we show that in many countries there has been a deceleration even before lockdowns were issued. This feature, possibly due to the increase of social awareness, can be rationalized by a susceptible-hidden-infected-recovered model introduced by Barnes, in which a hidden (isolated from the virus) compartment H is gradually populated by susceptible people, thus reducing the effectiveness of the virus spreading. By introducing a partial hiding mechanism, for instance due to the impossibility for a fraction of the population to enter the hidden state, we obtain a model that, although still sufficiently simple, faithfully reproduces the different deceleration trends observed in several major countries.

We study the epidemic spreading on spatial networks where the probability that two nodes are connected decays with their distance as a power law. As the exponent of the distance dependence grows, model networks smoothly transition from the random network limit to the regular lattice limit. We show that despite keeping the average number of contacts constant, the increasing exponent hampers the epidemic spreading by making long-distance connections less frequent. The spreading dynamics is influenced by the distance-dependence exponent as well and changes from exponential growth to power-law growth. The observed power-law growth is compatible with recent analyses of empirical data on the spreading of COVID-19 in numerous countries.

We revisit the susceptible-infectious-recovered/removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t = 0, where data is assumed to be available. We address the question of how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be relaxed in part B. Relaxing this assumption allows us to formulate initial conditions incompatible with the original SIR model.

Vaccination is commonly used for reducing the spread of infectious diseases; however, we know that not all vaccinations are completely effective. Thus, it is important to study the impacts of vaccine failure on the spreading dynamics of infectious diseases. In this study, we investigate the dynamics of an epidemic model with three types of imperfect vaccinations on complex networks. More specifically, the present model follows a susceptible-infected-susceptible process with a vaccinated compartment that permit leaky, all-or-nothing, and waning vaccines. A threshold value ${\mathcal{R}}_{v}$ is first presented to ensure the disease-free equilibrium is asymptotically stable. Next, we derive a necessary and sufficient criterion that assures the presence of a backward (or subcritical) bifurcation when ${\mathcal{R}}_{v}=1$. From this criterion, we can observe that the leaky vaccine plays an important role in leading to such a bifurcation, and interestingly, we also find that this condition is independent of the network structure. The disease is also demonstrated to persist whenever ${\mathcal{R}}_{v}{ >}1$. Numerical simulations are conducted to validate the theoretical results. Our results show that imperfect vaccination can cause a backward bifurcation, which usually has serious consequences for disease control. Thus, appropriate infection control policies need to be further developed.