Standard SIR Model Research Articles

Anticipating epidemic transitions in metapopulations with multivariate spectral similarity

Prediction and control of emerging pathogens is a fundamental challenge for public health. To meet this challenge, new analytic tools are needed to characterize the underlying dynamics of the geographical spread of pathogens, identify predictable changes in their dynamics, and support strategic planning for disease elimination and control. Nonparametric and model-independent tools are particularly needed. Here, we propose a multivariate method that uses similarity in cross-spectral density between measured spatial time series of disease prevalence as a feature measuring the proximity of a tipping point, i.e., emergence or elimination. In particular, we show that the increase in the average value of spectral similarity in measured epidemiological time series contains crucial information about the underlying dynamics and proximity to critical points in infectious disease systems. Theoretical analysis of a standard metapopulation SIR model and empirical analysis of case reports of pertussis in the continental USA demonstrate that this increase is observed when the disease approaches elimination. Therefore, this nonparametric indicator provides insight into the fundamental underlying state of the epidemiological system, which is key in developing appropriate strategies to more quickly achieve elimination goals.

A pair formation model with recovery: Application to mpox

The current global outbreaks of mpox is a unique infectious disease in the way it seems to be transmitting: it has been observed to be highly concentrated in communities of men who have sex with men (MSM) through pair formation, and also provides long lasting immunity. This framework of mostly close, prolonged contact spreading a disease that admits immunity after infection is unlike similar infections which either offer little to no immunity post-infection or are lifelong infections. This creates the need for a new model framework that incorporates pair formation structure with recovery. While seemingly a straight forward model, we show how new dynamics arise from the combination of pair formation and recovery that are not present in a standard model with recovery and also not present in a pair formation model without recovery. We see that the combination of these two properties allows for waves of infection that are not seen in a standard SIR model. These dynamics suggest that outbreaks of mpox around the world may require special attention from public health. We also derive a reproduction number for this model and estimate the reproduction number of human mpox to be ≈2.3 using global and Canadian data. The expression derived for R0 can help estimate key parameters for diseases transmission and public health interventions and compare to equivalent models without pair formation.

LSS: A locality-based structure system to evaluate the spreader’s importance in social complex networks

Assessing the importance of spreaders in networks is crucial for investigating the survival and robustness of networks. There are numerous potential applications, such as preventing outbreaks, spreading viruses on computer networks, viral marketing, and sickness spreading. These problems are usually unable to be solved by many heuristics with low time complexity. A number of tentative heuristics have been proposed for different application scenarios. However, we still lack an efficient heuristic to solve this type of problem effectively, for instance, low rating accuracy or high time complexity. To deal with this issue, this paper proposes a new heuristic called Locality-based Structure System (LSS), which is based on local information rather than global information of nodes in a network to determine the importance of spreaders. The proposed LSS takes into account the k-shell, degree, and number of triangles in a network. First, connectivity factors are computed based on the properties of nodes connected to them. Then, each node’s contribution to the importance of lines is computed. Finally, the degree and k-shell of nodes, as well as their contribution to the importance of lines, are taken into account. The proposed LSS is validated on a set of real and synthetic complex networks, where the simulation results under the standard SIR model and Kendall correlation coefficient indicate that LSS can efficiently identify influential spreaders in numerous types of networks without requiring any advanced parameter settings.

Scale-free dynamics of COVID-19 in a Brazilian city

A common basis to address the dynamics of directly transmitted infectious diseases, such as COVID-19, are compartmental (or SIR) models. SIR models typically assume homogenous population mixing, a simplification that is convenient but unrealistic. Here we validate an existing model of a scale-free fractal infection process using high-resolution data on COVID-19 spread in São Caetano, Brazil. We find that transmission can be described by a network in which each infectious individual has a small number of susceptible contacts, of the order of 2–5. This model parameter correlated tightly with physical distancing measured by mobile phone data, such that in periods of greater distancing the model recovered a lower average number of contacts, and vice versa. We show that the SIR model is a special case of our scale-free fractal process model in which the parameter that reflects population structure is set at unity, indicating homogeneous mixing. Our more general framework better explained the dynamics of COVID-19 in São Caetano, used fewer parameters than a standard SIR model and accounted for geographically localized clusters of disease. Our model requires further validation in other locations and with other directly transmitted infectious agents.

Impact of contact rate on epidemic spreading in complex networks.

Contact reduction is an effective strategy to mitigate the spreading of epidemic. However, the existing reaction-diffusion equations for infectious disease are unable to characterize this effect. Thus, we here propose an extended susceptible-infected-recovered model by incorporating contact rate into the standard SIR model, and concentrate on investigating its impact on epidemic transmission. We analytically derive the epidemic thresholds on homogeneous and heterogeneous networks, respectively. The effects of contact rate on spreading speed, scale and outbreak threshold are explored on ER and SF networks. Simulations results show that epidemic dissemination is significantly mitigated when contact rate is reduced. Importantly, epidemic spreads faster on heterogeneous networks while broader on homogeneous networks, and the outbreak thresholds of the former are smaller.

Analysis, predicting, and controlling the COVID-19 pandemic in Iraq through SIR model

Using the standard SIR model with three unknown biological parameters, the COVID-19 pandemic in Iraq has been studied. The least squares method and real data on confirmed infections, deaths, and recoveries over a long time (455 days) were used to estimate these parameters. In this regards, first, we find the basic reproductive number R0 is 0.9422661124 which indicates and predicts that the COVID-19 pandemic in Iraq will gradually subside until it is eradicated permanently with time. Additionally, we develop an optimal vaccination strategy with the goal of reducing COVID-19 infections and preventing their spread in Iraq, thereby putting a clear picture of control this pandemic.

A class of anomalous diffusion epidemic models based on CTRW and distributed delay

In recent years, the epidemic model with anomalous diffusion has gained popularity in the literature. However, when introducing anomalous diffusion into epidemic models, they frequently lack physical explanation, in contrast to the traditional reaction–diffusion epidemic models. The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable. Specifically, based on the continuous-time random walk (CTRW), starting from two stochastic processes of the waiting time and the step length, time-fractional space-fractional diffusion, time-fractional reaction–diffusion and fractional-order diffusion can all be naturally introduced into the SIR (S: susceptible, I: infectious and R: recovered) epidemic models, respectively. The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays. Distributed time delay systems can also be reduced to existing models, such as the standard SIR model, the fractional infectivity model and others, within the proper bounds. Meanwhile, as an application of the above stochastic modeling method, the physical meaning of anomalous diffusion is also considered by taking the SEIR (E: exposed) epidemic model as an example. Similar methods can be used to build other types of epidemic models, including SIVRS (V: vaccine), SIQRS (Q: quarantined) and others. Finally, this paper describes the transmission of infectious disease in space using the real data of COVID-19.

Modifying impact of environmental factors on the course of an epidemic process

Introduction. It is necessary to establish peculiarities and regularities of COVID-19 infection; this task requires further research on how to formalize and build spatial-temporal models of the infection spread. This article focuses on determining non-infectious factors that can modify the epidemic process caused by the COVID-19 coronavirus for further substantiation of integrated solutions that are necessary to ensure sanitary-epidemiological welfare of the RF population. Materials and methods. Our study involved analyzing regularities of regional differentiation in parameters introduced into mathematical models. These models described how the epidemic process developed in RF regions depending on modifying non-infectious factors identified by modelling the dynamics of spread of SARS-CoV-2 delta strain. These modifying factors included anti-epidemic activities; sanitary-epidemiological, sociodemographic, and economic conditions in a region; weather and climate; public healthcare systems and people’s lifestyles in RF regions over 2020-2021. The dynamics of the epidemic process was modelled by using the conventional SIR-model. Relationships between parameters introduced into the model of the epidemic process and modifying regional conditions were examined by using correlation-regression analysis. Results. The modelling made it possible to identify priority risk factors that modified COVID-19 spread authentically (p<0.05) and explained regional differences in intensity of contagion, recovery and lethality. We established that population coverage with vaccination, especially among people aged 31-40 years, had the greatest authentic positive influence on the decline of reproduction index (R0) of the virus (r=-0.37). An increase in monthly average temperatures in autumn and winter as well as over a year made for people moving faster from the susceptible to infected category (r=0.21-0.22). Growing sun insolation over a year, especially in summer, resulted in slower movement of susceptible people into the infected category (r=-0.02-(-0.23)). Next, several sanitary-epidemiological indicators authentically made the infection spread faster; they were improper working conditions (not conforming to the safety standards as per physical indicators) and ambient air quality in settlement not corresponding to the hygienic standards as per chemical indicators and noise (r=0.29-0.24). Recovery took longer in regions where alcohol consumption was comparatively higher (r=-0.32). Limitations. The limitations of the study include modelling the epidemic process using the standard SIR model; limited set of indicators and period of analysis. Conclusions. The existing regional differentiation in development of specific stages in the epidemic process related to the COVID-19 delta strain occurs due to complex interactions and influence exerted by modifying factors that create a certain multi-level and multi-component system. This system is able to transform the epidemic process either potentiating it or slowing it down.

Physics‐informed machine learning for the COVID‐19 pandemic: Adherence to social distancing and short‐term predictions for eight countries

BackgroundThe analysis of COVID‐19 infection data through the eye of Physics‐inspired Artificial Intelligence leads to a clearer understanding of the infection dynamics and assists in predicting future evolution. The spreading of the pandemic during the first half of 2020 was curtailed to a larger or lesser extent through social distancing measures imposed by most countries. In the context of the standard Susceptible‐Infected‐Recovered (SIR) model, changes in social distancing enter through time‐dependent infection rates.MethodsIn this work we use machine learning and the infection dynamical equations of SIR to extract from the infection data the degree of social distancing and, through it, assess the effectiveness of the imposed measures.ResultsQuantitative machine learning analysis is applied to eight countries with infection data from the first viral wave. We find as two extremes Greece and USA where the measures were successful and unsuccessful, respectively, in limiting spreading. This physics‐based neural network approach is employed to the second wave of the infection, and by training the network with the new data, we extract the time‐dependent infection rate and make short‐term predictions with a week‐long or even longer horizon. This algorithmic approach is applied to all eight countries with good short‐term results. The data for Greece is analyzed in more detail from August to December 2020.ConclusionsThe model captures the essential spreading dynamics and gives useful projections for the spreading, both in the short‐term but also for a more intermediate horizon, based on specific social distancing measures that are extracted directly from the data.

Modelling epidemic spread in cities using public transportation as a proxy for generalized mobility trends

We study how public transportation data can inform the modeling of the spread of infectious diseases based on SIR dynamics. We present a model where public transportation data is used as an indicator of broader mobility patterns within a city, including the use of private transportation, walking etc. The mobility parameter derived from this data is used to model the infection rate. As a test case, we study the impact of the usage of the New York City subway on the spread of COVID-19 within the city during 2020. We show that utilizing subway transport data as an indicator of the general mobility trends within the city, and therefore as an indicator of the effective infection rate, improves the quality of forecasting COVID-19 spread in New York City. Our model predicts the two peaks in the spread of COVID-19 cases in NYC in 2020, unlike a standard SIR model that misses the second peak entirely.

Simple model of epidemic dynamics with memory effects

We introduce a compartment model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states S=susceptible, I=infected, or R=recovered (SIR model). In state R an individual is assumed to stay immune within a finite-time interval. In the first part, we introduce a random lifetime or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and not captured by the classical standard SIR model.

Back to the Roots: A Discrete Kermack\u2013McKendrick Model Adapted to Covid-19

A widely used tool for analysing the Covid-19 pandemic is the standard SIR model. It seems often to be used as a black box, not taking into account that this model was derived as a special case of the seminal Kermack–McKendrick theory from 1927. This is our starting point. We explain the setup of the Kermack–McKendrick theory (passing to a discrete approach) and use medical information for specializing to a model called by us an adapted K-McK-model. It includes effects of vaccination, mass testing and mutants. We demonstrate the use of the model by applying it to the development in Germany and show, among others things, that a comparatively mild intervention reducing the time until quarantine by one day would lead to a drastic improvement.

Quantifying the role of airborne transmission in the spread of COVID-19.

There is an ongoing debate on the different transmission modes of SARS-CoV-2 and their relative contributions to the pandemic. In this paper, we employ a simple mathematical model, which incorporates both the human-to-human and environment-to-human transmission routes, to study the transmission dynamics of COVID-19. We focus our attention on the role of airborne transmission in the spread of the disease in a university campus setting. We conduct both mathematical analysis and numerical simulation, and incorporate published experimental data for the viral concentration in the air to fit model parameters. Meanwhile, we compare the outcome to that of the standard SIR model, utilizing a perturbation analysis in the presence of multiple time scales. Our data fitting and numerical simulation results show that the risk of airborne transmission for SARS-CoV-2 strongly depends on how long the virus can remain viable in the air. If the time for this viability is short, the airborne transmission route would be inconsequential in shaping the overall transmission risk and the total infection size. On the other hand, if the infectious virus can persist in aerosols beyond a few hours, then airborne transmission could play a much more significant role in the spread of COVID-19.

Effect of time-dependent infectiousness on epidemic dynamics.

In contrast to the common assumption in epidemic models that the rate of infection between individuals is constant, in reality, an individual's viral load determines their infectiousness. We compare the average and individual reproductive numbers and epidemic dynamics for a model incorporating time-dependent infectiousness and a standard SIR (susceptible-infected-recovered) model for both fully mixed and category-mixed populations. We find that the reproductive number depends only on the total infectious exposure and the largest eigenvalue of the mixing matrix and that these two effects are independent of each other. When we compare our time-dependent mean-field model to the SIR model with equivalent rates, the epidemic peak is advanced, and modifying the infection rate function has a strong effect on the time dynamics of the epidemic. We also observe behavior akin to a traveling wave as individuals transition through infectious states.

Modelling parasite-produced marine diseases: The case of the mass mortality event of Pinna nobilis

The state of the art of epidemic modelling in terrestrial ecosystems is the compartmental SIR model and its extensions from the now classical work of Kermack–Mackendrick. In contrast, epidemic modelling of marine ecosystems is a bit behind, and compartmental models have been introduced only recently. One of the reasons is that many epidemic processes in terrestrial ecosystems can be described through a contact process, while modelling marine epidemics is more subtle in many cases. Here we present a model describing disease outbreaks caused by parasites in bivalve populations. The SIRP model is a multicompartmental model with four compartments, three of which describe the different states of the host, susceptible (i.e. healthy), S, infected, I, and removed (dead), R, and one compartment for the parasite in the marine medium, P, written as a 4-dimensional dynamical system. Even if this is the simplest model one can write to describe this system, it is still too complicated for both direct analytical manipulation and direct comparison with experimental observations, as it depends on four parameters to be fitted. We show that it is possible to simplify the model, including a reduction to the standard SIR model if the parameters fulfil certain conditions. The model is validated with available data for the recent Mass Mortality Event of the noble pen shell Pinna nobilis, a disease caused by the parasite Haplosporidium pinnae, showing that the reduced SIR model is able to fit the data. So, we show that a model in which the species that suffers the epidemics (host) cannot move, and contagion occurs through parasites, can be reduced to the standard SIR model that represents epidemic transmission between mobile hosts. The fit indicates that the assumptions made to simplify the model are reasonable in practice, although it leads to an indeterminacy in three of the original parameters. This opens the possibility of performing direct experiments to be able to solve this question.

Assessing the impact of data-driven limitations on tracing and forecasting the outbreak dynamics of COVID-19

The availability of the epidemiological data strongly affects the reliability of several mathematical models in tracing and forecasting COVID-19 pandemic, hampering a fair assessment of their relative performance. The marked difference between the lethality of the virus when comparing the first and second waves is an evident sign of the poor reliability of the data, also related to the variability over time in the number of performed swabs. During the early epidemic stage, swabs were made only to patients with severe symptoms taken to hospital or intensive care unit. Thus, asymptomatic people, not seeking medical assistance, remained undetected. Conversely, during the second wave of infection, total infectives included also a percentage of detected asymptomatic infectives, being tested due to close contacts with swab positives and thus registered by the health system. Here, we compared the outcomes of two SIR-type models (the standard SIR model and the A-SIR model that explicitly considers asymptomatic infectives) in reproducing the COVID-19 epidemic dynamic in Italy, Spain, Germany, and France during the first two infection waves, simulated separately. We found that the A-SIR model overcame the SIR model in simulating the first wave, whereas these discrepancies are reduced in simulating the second wave, when the accuracy of the epidemiological data is considerably higher. These results indicate that increasing the complexity of the model is useless and unnecessarily wasteful if not supported by an increased quality of the available data.

Dynamics of epidemic spreading on connected graphs.

We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connections between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a collection of numerical simulations for a selection of connected graphs.

A Viral Disease That Damages the Immunity Conferred by Different Viral Diseases or Vaccination

In this paper we modify a standard SIR model used to study the spread of some diseases by incorporating a disease that destroys the immunity that is conferred by having one of the other diseases or being vaccinated against the disease. A specific biological example of this occurs with measles. Studies of recent measles’ patients has shown that many patients have lost some (or all) of their immunity to other diseases from which they were previously protected. In the future, models like those developed here might be helpful in understanding how viruses that affect multiple organ systems can impact the effect the disease has on at risk populations.

Los Angeles County SARS-CoV-2 Epidemic: Critical Role of Multi-generational Intra-household Transmission

We observed wide variation in the incidence of confirmed COVID-19 cases in 300 communities making up Los Angeles County, the largest county by population in the United States. The surge in incidence from October 19, 2020 to January 10, 2021, accounting for two-thirds of all confirmed cases since the start of the epidemic, was concentrated in communities with a high prevalence of multi-generational households. This indicator of household structure was a more important predictor of the surge in incidence than the prevalence of households with low income or with at least one high-risk worker. Based upon a spatial adaptation of the standard SIR model, the cumulative incidence of COVID-19, adjusted for underascertainment of both asymptomatic and symptomatic cases, ranged from under 10% in low multi-generational communities to over 30% in high multi-generational communities.

The social dynamics of COVID-19

We add a simple dynamic process for adaptive “social distancing” measures to a standard SIR model of the COVID pandemic. With a limited attention span and in the absence of a consistent long-term strategy against the pandemic, this process leads to a sweeping of an instability, i.e. fluctuations in the effective reproduction number around its bifurcation value of Reff=1. While mitigating the pandemic in the short-run, this process remains intrinsically fragile and does not constitute a sustainable strategy that societies could follow for an extended period of time.