Most countries are suffering severely from the ongoing covid-19 pandemic despite various levels of preventive measures. A common question is if and when a country or region will reach herd immunity $h$. The classical herd immunity level $h_C$ is defined as $h_C=1-1/R_0$, where $R_0$ is the basic reproduction number, for covid-19 estimated to lie somewhere in the range 2.2-3.5 depending on country and region. It is shown here that the disease-induced herd immunity level $h_D$, after an outbreak has taken place in a country/region with a set of preventive measures put in place, is actually substantially smaller than $h_C$. As an illustration we show that if $R_0=2.5$ in an age-structured community with mixing rates fitted to social activity studies, and also categorizing individuals into three categories: low active, average active and high active, and where preventive measures affect all mixing rates proportionally, then the disease-induced herd immunity level is $h_D=43\%$ rather than $h_C=1-1/2.5=60\%$. Consequently, a lower fraction infected is required for herd immunity to appear. The underlying reason is that when immunity is induced by disease spreading, the proportion infected in groups with high contact rates is greater than that in groups with low contact rates. Consequently, disease-induced immunity is stronger than when immunity is uniformly distributed in the community as in the classical herd immunity level.
Here we show that herd immunity is not a measure we can implement, trust, control or depend on, and that waiting for herd immunity to play a role is tantamount to hope for the best. Herd immunity is not some protection, as it has been referred to, but a reduction in the chance of infection that a population can achieve only if more infections occur, which is paradoxical. The discussion on when there will be enough herd immunity for the COVID-19 epidemics to halt, delivers the public and policy makers the wrong message that we can wait for something to happen to relax containment measures. Here…
We study a SEIR model considered by Gomes et al. \cite{Gomes2020} and Aguas et al. \cite{Aguas2020} where different individuals are assumed to have different levels of susceptibility or exposure to infection. Under this heterogeneity assumption, epidemic growth is effectively suppressed when the percentage of population having acquired immunity surpasses a critical level - the herd immunity threshold - that is lower than in homogeneous populations. We find explicit formulas to calculate herd immunity thresholds and stable configuration, and explore extensions of the model.
A quantitative COVID-19 model that incorporates hidden asymptomatic patients is developed, and an analytic solution in parametric form is given. The model incorporates the impact of lockdown and resulting spatial migration of population due to announcement of lockdown. A method is presented for estimating the model parameters from real-world data. It is shown that increase of infections slows down and herd immunity is achieved when symptomatic patients are 4-6\% of the population for the European countries we studied, when the total infected fraction is between 50-56 \%. Finally, a method for …
We performed a thorough sensitivity analysis of the herd immunity threshold for discrete-time SIR compartmental models with a static network structure. We find unexpectedly that these models violate classical intuition which holds that the herd immunity threshold should monotonically increase with the transmission parameter. We find the existence of bifurcations in the herd immunity threshold in the high transmission probability regime. The extent of these bifurcations is modulated by the graph heterogeneity, the recovery parameter, and the network size. In the limit of large, well-mixed netwo…
We model and calculate the fraction of infected population necessary to reach herd immunity, taking into account the heterogeneity in infectiousness and susceptibility, as well as the correlation between those two parameters. We show that these cause the effective reproduction number to decrease more rapidly, and consequently have a drastic effect on the estimate of the necessary percentage of the population that has to contract the disease for herd immunity to be reached. We quantify the difference between the size of the infected population when the effective reproduction number decreases be…
In this note we estimate herd immunity levels for the Covid-19 epidemic using a standard SEIR system that models the disease dynamics in homogeneous populations. The results obtained are indicative of values between 80% and 90% for unprotected, fully susceptible populations. Basic protective measures such as hand hygiene and mask wearing may be effective to lower herd immunity levels down to values between 50% and 60% of the total population.
The contact structure of a population plays an important role in transmission of infection. Many ``structured models'' capture aspects of the contact structure through an underlying network or a mixing matrix. An important observation in such models, is that once a fraction $1-1/\mathcal{R}_0$ has been infected, the residual susceptible population can no longer sustain an epidemic. A recent observation of some structured models is that this threshold can be crossed with a smaller fraction of infected individuals, because the disease acts like a targeted vaccine, preferentially immunizing highe…
In constant parameter compartmental models an early onset of herd immunity is at odds with estimates of R values from early stage growth. This paper utilizes a result from the theory of interest rate modeling, namely a bond pricing formula of Vasicek, and an approach inspired by a foundational result in statistics, de Finetti's Theorem, to show how the modeling discrepancy can be explained. Moreover the difference between predictions of classic constant parameter epidemiological models and those with variation and stochastic evolution can be reduced to simple "convexity" formulas. A novel feat…
We study the influence of the rate of the attainment of herd immunity (HI), in the absence of an approved vaccine, on the vulnerable population. We essentially ask the question: how hard the evolution towards the desired herd immunity could be on the life of the vulnerables? We employ mathematical modelling (chemical network theory) and cellular automata based computer simulations to study the human cost of an epidemic spread and an effective strategy to introduce HI. Implementation of different strategies to counter the spread of the disease requires a certain degree of quantitative understan…
COVID-19 will be a continuous threat to human population despite having a few vaccines at hand until we reach the endemic state through natural herd immunity and total immunization through universal vaccination. However, the vaccine acts as a practical tool for reducing the massive public health problem and the emerging economic consequences that the continuing COVID -19 epidemic is causing worldwide, while the vaccine efficacy wanes. In this work, we propose and analyze an epidemic model of Susceptible-Exposed-Infected-Recovered-Vaccinated (SEIRV) population taking into account the rate of va…
We analyse an extended SIRS epidemic model in which immunity at the individual level wanes gradually at exponential rate, but where the waning rate may differ between individuals, for instance as an effect of differences in immune systems. The model also includes vaccination schemes aimed to reach and maintain herd immunity. We consider both the informed situation where the individual waning parameters are known, thus allowing selection of vaccinees being based on both time since last vaccination as well as on the individual waning rate, and the more likely uninformed situation where individua…
Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, super-spreading events lead to burstiness and overdispersion, while long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through parameterization. We deriv…
Countries highly exposed to incoming traffic from China were expected to be at the highest risk of COVID-19 spread. However, COVID-19 case numbers (infection levels) are negatively correlated with incoming traffic-level. Moreover, infection levels are positively correlated with population-size, while the latter should only affect infection-level once herd immunity is reached. These could be explained if a low-virulence strain (LVS) began spreading a few months earlier from China, providing immunity from the later emerging known SARS-CoV-2 high-virulence strain (HVS). We find that the dynamics …
COVID-19 pandemic has been raging all around the world for almost a year now, as of November 1, 2020. In this paper, we try to analyze the variation of the COVID-19 pandemic in different countries in the light of some modifications to the susceptible-infected-recovered (SIR) model. The SIR model was modified by taking time-dependent rate parameters. From this modified SIR model, the basic reproduction number, effective reproduction number, herd immunity, and herd immunity threshold are redefined. The re-outbreak of the COVID-19 is a real threat to various countries. We have used the above-ment…
A population can be immune to epidemics even if not all of its individual members are immune to the disease, so long as sufficiently many are immune - this is the traditional notion of herd immunity. In the smartphone era a population can be immune to epidemics even if not a single one of its members is immune to the disease - a notion we call "digital herd immunity", which is similarly an emergent characteristic of the population. This immunity arises because contact-tracing protocols based on smartphone capabilities can lead to highly efficient quarantining of infected population members and…
Seasonal influenza is a significant public health concern in the United States and globally. While influenza vaccines are the single most effective intervention to reduce influenza morbidity and mortality, there is considerable debate surrounding the merits and consequences of repeated seasonal vaccination. Here, we describe a two-season influenza epidemic contact network model and use it to demonstrate that increasing the level of continuity in vaccination across seasons reduces the burden on public health. We show that revaccination reduces the influenza attack rate not only because it reduc…
The outbreak of Covid-19 led the world to an unprecedent health and economical crisis. In an attempt to responde to this emergency researchers worldwide are intensively studying the Covid-19 pandemic dynamics. In this work, a SIRSi compartmental model is proposed, which is a modification of the known classical SIR model. The proposed SIRSi model considers differences in the immunization within a population, and the possibility of unreported or asymptomatic cases. The model is adjusted to three major cities of São Paulo State, in Brazil, namely, São Paulo, Santos and Campinas, providing estimat…
The COVID-19 pandemic has hit different parts of the world differently: some regions are still in the rise of the first wave, other regions are now facing a decline after a first wave, and yet other regions have started to see a second wave. The current immunity level $\hat i$ in a region is closely related to the cumulative fraction infected, which primarily depends on two factors: a) the initial potential for COVID-19 in the region (often quantified by the basic reproduction number $R_0$), and b) the timing, amount and effectiveness of preventive measures put in place. By means of a mathemat…
We consider the simple epidemiological SIS model for a general heterogeneous population introduced by Lajmanovich and Yorke (1976) in finite dimension, and its infinite dimensional generalization we introduced in previous works. In this model the basic reproducing number $R_0$ is given by the spectral radius of an integral operator. If $R_0>1$, then there exists a maximal endemic equilibrium. In this very general heterogeneous SIS model, we prove that vaccinating according to the profile of this maximal endemic equilibrium ensures herd immunity. Moreover, this vaccination strategy is critical:…
We consider the $SEIRS$ epidemiology model with such features of the COVID-19 outbreak as: abundance of unidentified infected individuals, limited time of immunity and a possibility of vaccination. Within a compartmental realization of this model, we found the disease-free and the endemic stationary states. They exist in their respective restricted regions of the le via linear stability analysis. The expression for the basic reproductive number is obtained as well. The positions and heights of a first peak for the fractions of infected individuals are obtained numerically and are fitted to sim…
Until a vaccine or therapy is found against the SARS-CoV-2 coronavirus, reaching herd immunity appears to be the only mid-term option. However, if the number of infected individuals decreases and eventually fades only beyond this threshold, a significant proportion of susceptible may still be infected until the epidemic is over. A containment strategy is likely the best policy in the worst case where no vaccine or therapy is found. In order to keep the number of newly infected persons to a minimum, a possible strategy is to apply strict containment measures, so that the number of susceptible i…
Using a simple economic model in which social-distancing reduces contagion, we study the implications of waning immunity for the epidemiological dynamics and social activity. If immunity wanes, we find that COVID-19 likely becomes endemic and that social-distancing is here to stay until the discovery of a vaccine or cure. But waning immunity does not necessarily change optimal actions on the onset of the pandemic. Decentralized equilibria are virtually independent of waning immunity until close to peak infections. For centralized equilibria, the relevance of waning immunity decreases in the pr…
The current COVID-19 pandemic and subsequent lockdowns have highlighted the close and delicate relationship between a country's public health and economic health. Macroeconomic models that use preexisting epidemic models to calculate the impacts of a disease outbreak are therefore extremely useful for policymakers seeking to evaluate the best course of action in such a crisis. We develop an SIR model of the COVID-19 pandemic that explicitly considers herd immunity, behavior-dependent transmission rates, remote workers, and indirect externalities of lockdown. This model is presented as an exit …
This paper introduces a new multi-strain epidemic model with reinfection and cross-immunity to provide insights into the resurgence of the COVID-19 epidemic in an area with reportedly high seroprevalence due to a largely unmitigated outbreak: the state of Amazonas, Brazil. Although high seroprevalence could have been expected to trigger herd immunity and prevent further waves in the state, we have observed persistent levels of infection after the first wave and eventually the emergence of a second viral strain just before an augmented second wave. Our experiments suggest that the persistent le…
Bareilly College（巴雷利学院）；CSIR-Central Electronics Engineering Research Institute（CSIR-中央电子工程研究所）；Jawaharlal Nehru University（贾瓦哈拉尔·尼赫鲁大学）
To combat the coronavirus disease 2019 (COVID-19) pandemic, the world has vaccination, plasma therapy, herd immunity, and epidemiological interventions as few possible options. The COVID-19 vaccine development is underway and it may take a significant amount of time to develop the vaccine and after development, it will take time to vaccinate the entire population, and plasma therapy has some limitations. Herd immunity can be a plausible option to fight COVID-19 for small countries. But for a country with huge population like India, herd immunity is not a plausible option, because to acquire he…
I employ a simple mathematical model of an epidemic process to evaluate how four basic quantities: the reproduction number (R), the numbers of sensitive (S) and infectious individuals(I), and total community size (N) affect strategies to control COVID-19. Numerical simulations show that strict suppression measures at the beginning of an epidemic can create low infectious numbers, which thereafter can be managed by mitigation measures over longer periods to flatten the epidemic curve. The stronger the suppression measure, the faster it achieves the low numbers of infections that are conducive t…
The COVID-19 pandemic has placed unprecedented demands on local public health units in Ontario, Canada, one of which was the need for in-house epidemiological-modelling capabilities. To address this need, Ontario Tech University and the Durham Region Health Department developed a native Windows desktop app that performs epidemiological modelling of infectious diseases. The app is an implementation of a multi-stratified compartmental epidemiological model that can accommodate multiple virus variants and levels of vaccination, as well as public health measures such as physical distancing, contac…
We consider the SEIRS compartment epidemiology model suitable for predicting the evolution of the COVID-19 pandemy in the extreme limiting case of no acquired immunity. The disease-free and endemic fixed points are found and their stability is analysed. The expression for the basic reproduction ratio is obtained and discussed, emphasizing on its dependence on the model parameters. The threshold contact ratio is found which determines the possibility for a stable disease-free fixed point existence. Numeric solution for the pandemy evolution is also undertaken together with the approximate analy…
In more than two years since the COVID-19 virus was first detected in China, hundreds of millions of individuals have been infected, and millions have died. Aside from the immediate need for medical solutions (such as vaccines and medications) to treat the epidemic, the Corona pandemic has strengthened the demand for mathematical models that can predict the spread of the pandemic in an ever-changing reality. Here, we present a novel, dynamic particle model based on the basic principles of statistical physics that enables the prediction of the spreading of Covid-19 in the presence of effective …
A mathematical model of COVID-19 with minimal compartments is developed. The model is simple enough to fit data on confirmed cases, estimate the hidden infection figure and incorporate the effect of vaccination. With the effect of the new variant being considered, the model fits well with Philippine data of confirmed cases. With the fitted parameters, results show that a second peak, strikingly similar in magnitude to the first, is to be expected if the current transmission rate is increased by about 5\% and already a complete disaster if it will go up to 20\%. The model is then used to estima…
We consider the dynamics of a virus spreading through a population that produces a mutant strain with the ability to infect individuals that were infected with the established strain. Temporary cross-immunity is included using a time delay, but is found to be a harmless delay. We provide some sufficient conditions that guarantee local and global asymptotic stability of the disease-free equilibrium and the two boundary equilibria when the two strains outcompete one another. It is shown that, due to the immune evasion of the emerging strain, the reproduction number of the emerging strain must be…
The interplay between the virus, infected cells and the immune responses to SARS-CoV-2 is still under debate. Extending the basic model of viral dynamics we propose here a formal approach to describe the neutralizing versus weakly (or non-)neutralizing scenarios and compare with the possible effects of antibody-dependent enhancement (ADE). The theoretical model is consistent with data available from the literature; we show that weakly neutralizing antibodies or ADE can both give rise to either final virus clearance or disease progression, but the immuno-dynamic is different in each case. Given…
Ever since the outbreak of the COVID-19 epidemic, various public health control strategies have been proposed and tested against the coronavirus SARS-CoV-2. We study three specific COVID-19 epidemic control models: the susceptible, exposed, infectious, recovered (SEIR) model with vaccination control; the SEIR model with shield immunity control; and the susceptible, un-quarantined infected, quarantined infected, confirmed infected (SUQC) model with quarantine control. We express the control requirement in metric temporal logic (MTL) formulas (a type of formal specification languages) which can …
In the framework of homogeneous susceptible-infected-recovered (SIR) models, we use a control theory approach to identify optimal pandemic mitigation strategies. We derive rather general conditions for reaching herd immunity while minimizing the costs incurred by the introduction of societal control measures (such as closing schools, social distancing, lockdowns, etc.), under the constraint that the infected fraction of the population does never exceed a certain maximum corresponding to public health system capacity. Optimality is derived and verified by variational and numerical methods for a…
The Susceptible-Infected-Recovered (SIR) model is the cornerstone of epidemiological models. However, this specification depends on two parameters only, which implies a lack of flexibility and the difficulty to replicate the volatile reproduction numbers observed in practice. We extend the classic SIR model by introducing nonlinear stochastic transmission, to get a stochastic SIR model. We derive its exact solution and discuss the condition for herd immunity. The stochastic SIR model corresponds to a population of infinite size. When the population size is finite, there is also sampling uncert…
The global SARS-CoV-2 pandemic suggests a novel type of disease spread dynamics. WHO states that there is currently no evidence that people who have recovered from COVID-19 and have antibodies are immune from a second infection [WHO]. Conventional mathematical models consider cases for which a recovered individual either becomes susceptible again or develops an immunity. Here, we study the case where infected agents recover and only develop immunity if they are continuously infected for some time. Otherwise, they become susceptible again. We show that field theory bounds the peak of the infect…
Background: There is a lack of population-based studies on the effectiveness and durability of the SARS-CoV-2-induced immune protection during the Omicron wave. Methods: This retrospective study included 470 159 cases aged 60 years or older, who tested positive for SARS-CoV-2 between January 1 and June 30, 2022 in Bavaria, Germany. We examined time to death, measured from the earliest date of reporting/symptom onset until day 60. Cox models were used to estimate adjusted hazard ratios (HR) for sex, age, earliest calendar date of documented infection, and level of vaccine-induced or hybrid immu…
The potential waning of the vaccination immunity to COVID-19 could pose threats to public health, as it is tenable that the timing of such waning would synchronize with the near-complete restoration of normalcy. Should also testing be relaxed, we might witness a resurgent COVID-19 wave in winter 2021/2022. In response to this risk, an additional vaccine dose, the booster shot, is being administered worldwide. In a projected study with an outlook of six months, we explore the interplay between the rate at which boosters are distributed and the extent to which testing practices are implemented, …
We generalize the Susceptible-Infected-Removed model for epidemics to take into account generic effects of heterogeneity in the degree of susceptibility to infection in the population. We introduce a single new parameter corresponding to a power-law exponent of the susceptibility distribution that characterizes the population heterogeneity. We show that our generalized model is as simple as the original model which is contained as a limiting case. Because of this simplicity, numerical solutions can be generated easily and key properties of the epidemic wave can still be obtained exactly. In pa…
This technical report describes a dynamic causal model of the spread of coronavirus through a population. The model is based upon ensemble or population dynamics that generate outcomes, like new cases and deaths over time. The purpose of this model is to quantify the uncertainty that attends predictions of relevant outcomes. By assuming suitable conditional dependencies, one can model the effects of interventions (e.g., social distancing) and differences among populations (e.g., herd immunity) to predict what might happen in different circumstances. Technically, this model leverages state-of-t…
We study variants of the SEIR model for interpreting some qualitative features of the statistics of the Covid-19 epidemic in France. Standard SEIR models distinguish essentially two regimes: either the disease is controlled and the number of infected people rapidly decreases, or the disease spreads and contaminates a significant fraction of the population until herd immunity is achieved. After lockdown, at first sight it seems that social distancing is not enough to control the outbreak. We discuss here a possible explanation, namely that the lockdown is creating social heterogeneity: even if …
We propose a Bayesian Heterogeneity Learning approach for Susceptible-Infected-Removal-Susceptible (SIRS) model that allows underlying clustering patterns for transmission rate, recovery rate, and loss of immunity rate for the latest coronavirus (COVID-19) among different regions. Our proposed method provides simultaneously inference on parameter estimation and clustering information which contains both number of clusters and cluster configurations. Specifically, our key idea is to formulates the SIRS model into a hierarchical form and assign the Mixture of Finite mixtures priors for heterogen…
The development and authorization of COVID-19 vaccines has provided the clearest path forward to eliminate community spread hence end the ongoing SARS-CoV-2 pandemic. However, the limited pace at which the vaccine can be administered motivates the question, to what extent must we continue to adhere to social intervention measures such as mask wearing and social distancing? To address this question, we develop a mathematical model of COVID-19 spread incorporating both vaccine dynamics and socio-epidemiological parameters. We use this model to study two important measures of disease control and …
The waning of immunity after recovery or vaccination is a major factor accounting for the severity and prolonged duration of an array of epidemics, ranging from COVID-19 to diphtheria and pertussis. To study the effectiveness of different immunity level-based vaccination schemes in mitigating the impact of waning immunity, we construct epidemiological models that mimic the latter's effect. The total susceptible population is divided into an arbitrarily large number of discrete compartments with varying levels of disease immunity. We then vaccinate various compartments within this framework, co…
The COVID-19 pandemic has severely impacted health systems and economies worldwide. Significant global efforts are therefore ongoing to improve vaccine efficacies, optimize vaccine deployment, and develop new antiviral therapies to combat the pandemic. Mechanistic viral dynamics and quantitative systems pharmacology models of SARS-CoV-2 infection, vaccines, immunomodulatory agents, and antiviral therapeutics have played a key role in advancing our understanding of SARS-CoV-2 pathogenesis and transmission, the interplay between innate and adaptive immunity to influence the outcomes of infection…
Many countries have passed their first COVID-19 epidemic peak. Traditional epidemiological models describe this as a result of non-pharmaceutical interventions that pushed the growth rate below the recovery rate. In this new phase of the pandemic many countries show an almost linear growth of confirmed cases for extended time-periods. This new containment regime is hard to explain by traditional models where infection numbers either grow explosively until herd immunity is reached, or the epidemic is completely suppressed (zero new cases). Here we offer an explanation of this puzzling observati…
We recently described a dynamic causal model of a COVID-19 outbreak within a single region. Here, we combine several of these (epidemic) models to create a (pandemic) model of viral spread among regions. Our focus is on a second wave of new cases that may result from loss of immunity--and the exchange of people between regions--and how mortality rates can be ameliorated under different strategic responses. In particular, we consider hard or soft social distancing strategies predicated on national (Federal) or regional (State) estimates of the prevalence of infection in the population. The mode…
We consider the interplay of vaccination and migration rates on disease persistence in epidemiological systems. We show that short-term and long-term migration can inhibit disease persistence. As a result, we show how migration changes how vaccination rates should be chosen to maintain herd immunity. In a system of coupled SIR models, we analyze how disease eradication depends explicitly on vaccine distribution and migration connectivity. The analysis suggests potentially novel vaccination policies that underscore the importance of optimal placement of finite resources.
Modelling SARS-CoV-2 epidemics via compartmental and cellular automaton SEIRS model with temporal immunity and vaccination
Yukhnovskii Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine（乌克兰国家科学院尤赫诺夫斯基凝聚态物理研究所）；Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University（利沃夫国立理工大学应用数学与基础科学研究所）
We introduce a novel compartmental model accounting for the effects of vaccine efficacy, deployment rates and timing of initiation of deployment. We simulate different scenarios and initial conditions, and we find that higher abundancy and rate of deployment of low efficacy vaccines lowers the cumulative number of deaths in comparison to slower deployment of high efficacy vaccines. We also forecast that, at the same daily deployment rate, the earlier introduction of vaccination schemes with lower efficacy would also lower the number of deaths with respect to a delayed introduction of high effi…
An epidemiological model for COVID-19 was developed and implemented in MATLAB/GNU Octave for use by public health practitioners, policy makers and the general public. The model distinguishes four stages in the disease: infected, sick, seriously sick, and better. The model was preliminarily parameterized based on observations of the spread of the disease. The model is consistent with a mortality rate of 1.5 %. Preliminary simulations with the model indicate that concepts such as "herd immunity" and "flattening the curve" are highly misleading in the context of this virus. Public policies based …
We use daily data from Lombardy, the Italian region most affected by the COVID-19 outbreak, to calibrate a SIR model individually on each municipality. These are all covered by the same health system and, in the post-lockdown phase we focus on, all subject to the same social distancing regulations. We find that municipalities with a higher number of cases at the beginning of the period analyzed have a lower rate of diffusion, which cannot be imputed to herd immunity. In particular, there is a robust and strongly significant negative correlation between the estimated basic reproduction number (…
