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 higher-risk individuals who play a greater role in transmission. Therefore, a limited ``first wave'' may leave behind a residual population that cannot support a second wave once interventions are lifted. In this paper, we systematically analyse a number of mean-field models for networks and other structured populations to address issues relevant to the Covid-19 pandemic. In particular, we consider herd-immunity under several scenarios. We confirm that, in networks with high degree heterogeneity, the first wave confers herd-immunity with significantly fewer infections than equivalent models with lower degree heterogeneity. However, if modelling the intervention as a change in the contact network, then this effect might become more subtle. Indeed, modifying the structure can shield highly connected nodes from becoming infected during the first wave and make the second wave more substantial. We confirm this finding by using an age-structured compartmental model parameterised with real data and comparing lockdown periods implemented either as a global scaling of the mixing matrix or age-specific structural changes. We find that results regarding herd immunity levels are strongly dependent on the model, the duration of lockdown and how lockdown is implemented.
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 subst…
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.
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…
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…
Time varying susceptibility of host at individual level due to waning and boosting immunity is known to induce rich long-term behavior of disease transmission dynamics. Meanwhile, the impact of the time varying heterogeneity of host susceptibility on the shot-term behavior of epidemics is not well-studied, even though the large amount of the available epidemiological data are the short-term epidemics. Here we constructed a parsimonious mathematical model describing the short-term transmission dynamics taking into account natural-boosting immunity by reinfection, and obtained the explicit solut…
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…
The epidemic threshold of a spreading process indicates the condition for the occurrence of the wide spreading regime, thus representing a predictor of the network vulnerability to the epidemic. Such threshold depends on the natural history of the disease and on the pattern of contacts of the network with its time variation. Based on the theoretical framework introduced in (Valdano et al. PRX 2015) for a susceptible-infectious-susceptible model, we formulate here an infection propagator approach to compute the epidemic threshold accounting for more realistic effects regarding a varying force o…
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 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…
SIRS epidemic models assume that individual immunity (from infection and vaccination) wanes in one big leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that's become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and where booster vaccines are given to increase immunity. This paper considers an epidemic model allowing for such gradual waning of immunity (either linear or exponential waning) thereby extending SIRS epidemics, and also incorporates vaccination. The two v…
This paper examines the interplay of the effect of cross immunity and antibody-dependent enhancement (ADE) in mutistrain diseases. Motivated by dengue fever, we study a model for the spreading of epidemics in a population with multistrain interactions mediated by both partial temporary cross immunity and ADE. Although ADE models have previously been observed to cause chaotic outbreaks, we show analytically that weak cross immunity has a stabilizing effect on the system. That is, the onset of disease fluctuations requires a larger value of ADE with small cross immunity than without. However, st…
Bovine tuberculosis, a disease that affects cattle and badgers in Ireland, was studied via stochastic epidemic modeling using incidence data from the Four Area Project (Griffin et al., 2005). The Four Area Project was a large scale field trial conducted in four diverse farming regions of Ireland over a five-year period (1997-2002) to evaluate the impact of badger culling on bovine tuberculosis incidence in cattle herds. Based on the comparison of several models, the model with no between-herd transmission and badger-to-herd transmission proportional to the total number of infected badgers cull…
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 introduce a modified SIR model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states susceptible (${\bf S}$), infected (${\bf I}$) or recovered (${\bf R}$). In the state ${\bf R}$ an individual is assumed to stay immune within a finite time interval. In the first part, we introduce a random life time 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 rando…
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…
Mathematical modelling of the spread of epidemics has been an interesting challenge in the field of epidemiology. The SIR Model proposed by Kermack and McKendrick in 1927 is a prototypical model of epidemiology. However, it has its limitations. In this paper, we show two independent ways of generalizing this model, the first one if the vaccine isn't discovered or ready to use and the next one, if the vaccine is discovered and ready to use. In the first part, we have pointed out a major over-simplification, i.e., assumption of variation of the time derivatives of the variables with the linear o…
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…
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 …
Between pandemics, the influenza virus exhibits periods of incremental evolution via a process known as antigenic drift. This process gives rise to a sequence of strains of the pathogen that are continuously replaced by newer strains, preventing a build up of immunity in the host population. In this paper, a parsimonious epidemic model is defined that attempts to capture the dynamics of evolving strains within a host population. The `evolving strains' epidemic model has many properties that lie in-between the Susceptible-Infected-Susceptible and the Susceptible-Infected-Removed epidemic models…
In this paper we conduct a simulation study of the spread of an epidemic like COVID-19 with temporary immunity on finite spatial and non-spatial network models. In particular, we assume that an epidemic spreads stochastically on a scale-free network and that each infected individual in the network gains a temporary immunity after its infectious period is over. After the temporary immunity period is over, the individual becomes susceptible to the virus again. When the underlying contact network is embedded in Euclidean geometry, we model three different intervention strategies that aim to contr…
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…
The Beyond-Linear-Use-of-Equation-Superposition (BLUES) function method is extended to coupled nonlinear ordinary differential equations and applied to the epidemiological SIRS model with vaccination. Accurate analytic approximations are obtained for the time evolution of the susceptible and infected population fractions. The results are compared with those obtained with alternative methods, notably Adomian decomposition, variational iteration and homotopy perturbation. In contrast with these methods, the BLUES iteration converges rapidly, globally, and captures the exact asymptotic behavior f…
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…
An epidemic carries human and fiscal costs. In the case of imported pandemics, the first-best solution is to restrict national borders to identify and isolate infected individuals. However, when that opportunity is not fully seized and there is no preventative intervention available, second-best options must be chosen. In this article we develop a system of differential equations that simulate both the fiscal and human costs associated to different mitigation measurements. After simulating several scenarios, we conclude that herd immunity (or unleashing the pandemic) is the worst policy in ter…
Many pathogens spread primarily via direct contact between infected and susceptible hosts. Thus, the patterns of contacts or contact network of a population fundamentally shapes the course of epidemics. While there is a robust and growing theory for the dynamics of single epidemics in networks, we know little about the impacts of network structure on long term epidemic or endemic transmission. For seasonal diseases like influenza, pathogens repeatedly return to populations with complex and changing patterns of susceptibility and immunity acquired through prior infection. Here, we develop two m…
Seasonal influenza kills hundreds of thousands every year, with multiple constantly-changing strains in circulation at any given time. A high mutation rate enables the influenza virus to evade recognition by the human immune system, including immunity acquired through past infection and vaccination. Here, we capture the genetic similarity of influenza strains and their evolutionary dynamics with genotype networks. We show that the genotype networks of influenza A (H3N2) hemagglutinin are characterized by heavy-tailed distributions of module sizes and connectivity, suggesting critical-like beha…
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…
Genotype networks drive oscillating endemicity and epidemic trajectories in viral evolution
University of Zaragoza（萨拉戈萨大学）；Institute of Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza（萨拉戈萨大学复杂系统生物计算与物理研究所 (BIFI)）；Universitat Rovira i Virgili（罗维拉-伊-维尔吉利大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；University of Kobe（神户大学）
Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, and states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the R0 alternative" of [52, 51], which states that the local stability condition of the disease free equilibrium may be expressed as R0 < 1, where R0 is the famous basic reproduction number, which plays also a major role in the theory of branching processes. The literature suggests that it is impossible to find general laws concerning the endemic points. However, it is quite co…
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…
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）
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 …
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…
Infection can spread easily on networks with heterogeneous degree distribution. Here, we considered targeted immunization on such networks, wherein a fraction of individuals with the highest connectivity are immunized. To quantify the effect of this targeted immunization approach on population immunity, we proposed a method using the type reproduction number. Consequently, we derived a precise and simple formula that can yield the immunization threshold, which, to the best of our knowledge, is the first such result presented in the literature.
