Modelling the effect of a booster vaccination on disease epidemiology

Scenarios of future mpox outbreaks among men who have sex with men: a modelling study based on cross-sectional seroprevalence data from the Netherlands, 2022

BackgroundFollowing the 2022-2023 mpox outbreak, crucial knowledge gaps exist regarding orthopoxvirus-specific immunity in risk groups and its impact on future outbreaks.AimWe combined cross-sectional seroprevalence studies in two cities in the Netherlands with mathematical modelling to evaluate scenarios of future mpox outbreaks among men who have sex with men (MSM).MethodsSerum samples were obtained from 1,065 MSM attending Centres for Sexual Health (CSH) in Rotterdam or Amsterdam following the peak of the Dutch mpox outbreak and the introduction of vaccination. For MSM visiting the Rotterdam CSH, sera were linked to epidemiological and vaccination data. An in-house developed ELISA was used to detect vaccinia virus (VACV)-specific IgG. These observations were combined with published data on serial interval and vaccine effectiveness to inform a stochastic transmission model that estimates the risk of future mpox outbreaks.ResultsThe seroprevalence of VACV-specific antibodies was 45.4% and 47.1% in Rotterdam and Amsterdam, respectively. Transmission modelling showed that the impact of risk group vaccination on the original outbreak was likely small. However, assuming different scenarios, the number of mpox cases in a future outbreak would be markedly reduced because of vaccination. Simultaneously, the current level of immunity alone may not prevent future outbreaks. Maintaining a short time-to-diagnosis is a key component of any strategy to prevent new outbreaks.ConclusionOur findings indicate a reduced likelihood of large future mpox outbreaks among MSM in the Netherlands under current conditions, but emphasise the importance of maintaining population immunity, diagnostic capacities and disease awareness.

Mathematical analysis of pulse vaccination in controlling the dynamics of measles transmission

Although the incidence of measles has been significantly reduced through vaccination, it remains an important public health problem. In this paper, a measles model with pulse vaccination is formulated to investigate the influential pulse vaccination on the period of time for the extinction of the disease. The threshold value of the formulated model, called the control reproduction number and denoted by R ∗ , is derived. It is found that the disease-free periodic solution of the model exists and is globally attractivity whenever R ∗ < 1 in the sense that measles is eliminated. If R ∗ > 1 , the positive solution of the model exists and is permanent which indicates the disease persists in the community. Theoretical conditions for disease eradication under various constraints are given. The effect of pulse vaccination is explored using data from Thailand. The results obtained can guide policymakers in deciding on the optimal scheduling in order to achieve the strategic plan of measles elimination by vaccination.

Dynamic behavior analysis of an SVIR epidemic model with two time delays associated with the COVID-19 booster vaccination time

Since the outbreak of COVID-19, there has been widespread concern in the community, especially on the recent heated debate about when to get the booster vaccination. In order to explore the optimal time for receiving booster shots, here we construct an SVIR model with two time delays based on temporary immunity. Second, we theoretically analyze the existence and stability of equilibrium and further study the dynamic properties of Hopf bifurcation. Then, the statistical analysis is conducted to obtain two groups of parameters based on the official data, and numerical simulations are carried out to verify the theoretical analysis. As a result, we find that the equilibrium is locally asymptotically stable when the booster vaccination time is within the critical value. Moreover, the results of the simulations also exhibit globally stable properties, which might be more beneficial for controlling the outbreak. Finally, we propose the optimal time of booster vaccination and predict when the outbreak can be effectively controlled.

An SVEIRE Model of Tuberculosis to Assess the Effect of an Imperfect Vaccine and Other Exogenous Factors

This study extends a deterministic mathematical model for the dynamics of tuberculosis transmission to examine the impact of an imperfect vaccine and other exogenous factors, such as re-infection among treated individuals and exogenous re-infection. The qualitative behaviors of the model are investigated, covering many distinct aspects of the transmission of the disease. The proposed model is observed to show a backward bifurcation, even when Rv<1. As such, we assume that diminishing Rv to less than unity is not effective for the elimination of tuberculosis. Furthermore, the results reveal that an imperfect tuberculosis vaccine is always effective at reducing the spread of infectious diseases within the population, though the general effect increases with the increase in effectiveness and coverage. In particular, it is shown that a limited portion of people being vaccinated at steady-state and vaccine efficacy assume a equivalent role in decreasing disease burden. From the numerical simulation, it is shown that using an imperfect vaccine lead to effective control of tuberculosis in a population, provided that the efficacy of the vaccine and its coverage are reasonably high.

Measuring Infection Transmission in a Stochastic SIV Model with Infection Reintroduction and Imperfect Vaccine

An additional compartment of vaccinated individuals is considered in a SIS stochastic epidemic model with infection reintroduction. The quantification of the spread of the disease is modeled by a continuous time Markov chain. A well-known measure of the initial transmission potential is the basic reproduction number [Formula: see text], which determines the herd immunity threshold or the critical proportion of immune individuals required to stop the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the typical infective and previously infected individuals, [Formula: see text] overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to [Formula: see text] are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic.

Delay in booster schedule as a control parameter in vaccination dynamics

The use of multiple vaccine doses has proven to be essential in providing high levels of protection against a number of vaccine-preventable diseases at the individual level. However, the effectiveness of vaccination at the population level depends on several key factors, including the dose-dependent protection efficacy of vaccine, coverage of primary and booster doses, and in particular, the timing of a booster dose. For vaccines that provide transient protection, the optimal scheduling of a booster dose remains an important component of immunization programs and could significantly affect the long-term disease dynamics. In this study, we developed a vaccination model as a system of delay differential equations to investigate the effect of booster schedule using a control parameter represented by a fixed time-delay. By exploring the stability analysis of the model based on its reproduction number, we show the disease persistence in scenarios where the booster dose is sub-optimally scheduled. The findings indicate that, depending on the protection efficacy of primary vaccine series and the coverage of booster vaccination, the time-delay in a booster schedule can be a determining factor in disease persistence or elimination. We present model results with simulations for a vaccine-preventable bacterial disease, Heamophilus influenzae serotype b, using parameter estimates from the previous literature. Our study highlights the importance of timelines for multiple-dose vaccination in order to enhance the population-wide benefits of herd immunity.

Mathematical analysis of a tuberculosis model with imperfect vaccine

Since 1921, the Bacille Calmette–Guerin (BCG) vaccine continues to be the most widely used vaccine for the prevention of Tuberculosis (TB). However, the immunity induced by BCG wanes out after some time making the vaccinated individual susceptible to TB infection. In this work, we formulate a mathematical model that incorporates the vaccination of newly born children and older susceptible individuals in the transmission dynamics of TB in a population, with a vaccine that can confer protection on older susceptible individuals. In the absence of disease-induced deaths, the model is shown to undergo the phenomenon of backward bifurcation where a stable disease-free equilibrium (DFE) co-exists with a stable positive (endemic) equilibrium when the associated reproduction number is less than unity. It is shown that this phenomenon does not exist in the absence of imperfect vaccine, exogenous reinfection, and reinfection of previously treated individuals. It is further shown that a special case of the model has a unique endemic equilibrium point (EEP), which is globally asymptotically stable when the associated reproduction number exceeds unity. Uncertainty and sensitivity analysis are carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population using the total population of latently infected individuals, total number of actively infected individuals, disease incidence, and the effective reproduction number as output responses. The analysis shows that the top five parameters of the model that have the greatest influence on the effective reproduction number of the model are the transmission rate, the fraction of fast disease progression, modification parameter which accounts for reduced likelihood to infection by vaccinated individuals due to imperfect vaccine, rate of progression from latent to active TB, and the treatment rate of actively infected individuals, with other key parameters influencing the outcomes of the other output responses. Numerical simulations suggest that with higher vaccination rate of older susceptible individuals, fewer new born children need to be vaccinated, in order to achieve disease eradication.

Logistical constraints lead to an intermediate optimum in outbreak response vaccination

Dynamic models in disease ecology have historically evaluated vaccination strategies under the assumption that they are implemented homogeneously in space and time. However, this approach fails to formally account for operational and logistical constraints inherent in the distribution of vaccination to the population at risk. Thus, feedback between the dynamic processes of vaccine distribution and transmission might be overlooked. Here, we present a spatially explicit, stochastic Susceptible-Infected-Recovered-Vaccinated model that highlights the density-dependence and spatial constraints of various diffusive strategies of vaccination during an outbreak. The model integrates an agent-based process of disease spread with a partial differential process of vaccination deployment. We characterize the vaccination response in terms of a diffusion rate that describes the distribution of vaccination to the population at risk from a central location. This generates an explicit trade-off between slow diffusion, which concentrates effort near the central location, and fast diffusion, which spreads a fixed vaccination effort thinly over a large area. We use stochastic simulation to identify the optimum vaccination diffusion rate as a function of population density, interaction scale, transmissibility, and vaccine intensity. Our results show that, conditional on a timely response, the optimal strategy for minimizing outbreak size is to distribute vaccination resource at an intermediate rate: fast enough to outpace the epidemic, but slow enough to achieve local herd immunity. If the response is delayed, however, the optimal strategy for minimizing outbreak size changes to a rapidly diffusive distribution of vaccination effort. The latter may also result in significantly larger outbreaks, thus suggesting a benefit of allocating resources to timely outbreak detection and response.

It has long been recognized that an epidemic of infectious disease can be prevented if a sufficient proportion of the susceptible population is vaccinated in advance. This logic also holds for vaccine-based outbreak response to stop an outbreak of a novel, or re-emerging pathogen, but with an important caveat. If vaccination is used in response to an outbreak, then it will necessarily take time to achieve the required level of vaccination coverage, during which time the outbreak may continue to spread. Thus, one must consider the logistical and operational constraints of vaccine distribution to assess the ability of outbreak response vaccination to slow or stop an advancing epidemic. We develop a simple mathematical framework for representing vaccine distribution in response to an epidemic and solve for the optimal distribution strategy under realistic constraints of total vaccination effort. Focused deployment near the outbreak epicenter concentrates resources in the area most in need, but may allow the outbreak to spread outside of the response zone. Broad deployment over the whole population may spread vaccination resources too thin, creating shortages and delays at the local scale that fail to prevent the advancing epidemic. Thus we found that, in general, the best strategy is an intermediate optimum that deploys vaccine neither too slow to prevent escape from the outbreak epicenter, nor too fast to spread resources too thin. The specific optimum rate for any given outbreak depends on the infectiousness of the pathogen, the population density, the range of contacts amongst individuals, the timeliness of the response, and the vaccine intensity. This insight only emerges from linking an epidemic model with a realistic model of outbreak response and highlights the need for further work to merge operations research with epidemic models to develop operationally relevant response strategies.

MODELING VACCINE PREVENTABLE VECTOR-BORNE INFECTIONS: YELLOW FEVER AS A CASE STUDY

In this paper, we propose and simulate a deterministic model for a vector-borne disease in the presence of a vaccine. The model allows the assessment of the impact of an imperfect vaccine with various characteristics, which include waning protective immunity, incomplete vaccine-induced protection and adverse events. We find three threshold parameters which govern the existence and stability of the equilibrium points. Our stability analysis suggests that the reduction in the mosquito fertility theoretically is the most effective factor of reducing disease prevalence in both low and high transmission areas. To illustrate the theoretical results, the model is simulated by the example of yellow fever. We also perform sensitivity analyses to determine the importance of both vaccine-induced mortality rate and disease-induced mortality rate for determining a control strategy. We found that there is an optimum vaccination rate, above which people die by the vaccination and below which people die by the disease.

Evolution and Use of Dynamic Transmission Models for Measles and Rubella Risk and Policy Analysis

The devastation caused by periodic measles outbreaks motivated efforts over more than a century to mathematically model measles disease and transmission. Following the identification of rubella, which similarly presents with fever and rash and causes congenital rubella syndrome (CRS) in infants born to women first infected with rubella early in pregnancy, modelers also began to characterize rubella disease and transmission. Despite the relatively large literature, no comprehensive review to date provides an overview of dynamic transmission models for measles and rubella developed to support risk and policy analysis. This systematic review of the literature identifies quantitative measles and/or rubella dynamic transmission models and characterizes key insights relevant for prospective modeling efforts. Overall, measles and rubella represent some of the relatively simplest viruses to model due to their ability to impact only humans and the apparent life-long immunity that follows survival of infection and/or protection by vaccination, although complexities arise due to maternal antibodies and heterogeneity in mixing and some models considered potential waning immunity and reinfection. This review finds significant underreporting of measles and rubella infections and widespread recognition of the importance of achieving and maintaining high population immunity to stop and prevent measles and rubella transmission. The significantly lower transmissibility of rubella compared to measles implies that all countries could eliminate rubella and CRS by using combination of measles- and rubella-containing vaccines (MRCVs) as they strive to meet regional measles elimination goals, which leads to the recommendation of changing the formulation of national measles-containing vaccines from measles only to MRCV as the standard of care.

A two-phase within-host model for immune response and its application to serological profiles of pertussis

We present a simple phenomenological within-host model describing both the interaction between a pathogen and the immune system and the waning of immunity after clearing of the pathogen. We implement the model into a Bayesian hierarchical framework to estimate its parameters for pertussis using Markov chain Monte Carlo methods. We show that the model captures some essential features of the kinetics of titers of IgG against pertussis toxin. We identify a threshold antibody level that separates a large increase in antibody level upon infection from a small increase and accordingly might be interpreted as a threshold separating clinical from subclinical infections. We contrast predictions of the model with observations reported in the literature and based on independent data and find a remarkable correspondence.

Dynamics of infectious diseases

Modern infectious disease epidemiology has a strong history of using mathematics both for prediction and to gain a deeper understanding. However the study of infectious diseases is a highly interdisciplinary subject requiring insights from multiple disciplines, in particular a biological knowledge of the pathogen, a statistical description of the available data and a mathematical framework for prediction. Here we begin with the basic building blocks of infectious disease epidemiology--the SIS and SIR type models--before considering the progress that has been made over the recent decades and the challenges that lie ahead. Throughout we focus on the understanding that can be developed from relatively simple models, although accurate prediction will inevitably require far greater complexity beyond the scope of this review. In particular, we focus on three critical aspects of infectious disease models that we feel fundamentally shape their dynamics: heterogeneously structured populations, stochasticity and spatial structure. Throughout we relate the mathematical models and their results to a variety of real-world problems.

Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems

We analyse a periodically driven SIR epidemic model for childhood related diseases, where the contact rate and vaccination rate parameters are considered periodic. The aim is to define optimal vaccination strategies for control of childhood related infections. Stability analysis of the uninfected solution is the tool for setting up the control function. The optimal solutions are sought within a set of susceptible population profiles. Our analysis reveals that periodic vaccination strategy hardly contributes to the stability of the uninfected solution if the human residence time (life span) is much larger than the contact rate period. However, if the human residence time and the contact rate periods match, we observe some positive effect of periodic vaccination. Such a vaccination strategy would be useful in the developing world, where human life spans are shorter, or basically in the case of vaccination of livestock or small animals whose life-spans are relatively shorter.

Impact of discontinuous treatments on disease dynamics in an SIR epidemic model

We consider an SIR epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we are able to determine the basic reproduction number R₀, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that in the case that an equilibrium is asymptotically stable, the convergence to the equilibrium can actually be achieved in finite time, and we can estimate this time in terms of the model parameters, initial sub-populations and the initial treatment strength. This suggests that from the view point of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we use to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on non-smooth analysis.

A SEASONALLY FORCED EPIDEMIC MODEL WITH TARGETED ANTIVIRAL PROPHYLAXIS

In this paper, we propose a mathematical model to describe the transmission dynamics of infectious diseases with targeted antiviral prophylaxis strategy. Our model incorporates seasonal driving force since seasonal force has a great effect on the spread of infectious diseases. Based on the local stability of disease free equilibrium we derive the control reproduction number [Formula: see text]. Sufficient conditions for the global stability of the disease free equilibrium are obtained. Using the persistence theory for discrete dynamical system, we prove that the infectious disease will remain endemic if [Formula: see text]. Simulation results are also provided to study the effect of targeted antiviral prophylaxis on transmission dynamics of infectious disease and investigate the influence of seasonality on the efficiency of targeted antiviral prophylaxis strategy.

Estimating the duration of pertussis immunity using epidemiological signatures

Case notifications of pertussis have shown an increase in a number of countries with high rates of routine pediatric immunization. This has led to significant public health concerns over a possible pertussis re-emergence. A leading proposed explanation for the observed increase in incidence is the loss of immunity to pertussis, which is known to occur after both natural infection and vaccination. Little is known, however, about the typical duration of immunity and its epidemiological implications. Here, we analyze a simple mathematical model, exploring specifically the inter-epidemic period and fade-out frequency. These predictions are then contrasted with detailed incidence data for England and Wales. We find model output to be most sensitive to assumptions concerning naturally acquired immunity, which allows us to estimate the average duration of immunity. Our results support a period of natural immunity that is, on average, long-lasting (at least 30 years) but inherently variable.

The eradication of vaccine-preventable infectious diseases remains an important public health priority. To achieve this goal, the level of immunity afforded needs to be high and long-lasting. For pertussis, one of the leading causes of mortality in infants, immunity has been shown to wane in some individuals. The epidemiological impacts of this observation depend critically on the duration of protective immunity in the entire population, which remains notoriously difficult to estimate. We approach this problem by exploring the agreement between model dynamics and case notification data from England & Wales. Our estimates suggest the average duration of immunity is much longer than is currently thought (at least 30 years), but that some individuals would lose immunity quite rapidly.

Gaining insights into human viral diseases through mathematics

Mathematical models have been recognized as powerful tools for providing new insights into the understanding of viral dynamics of human diseases at both the population and cellular levels. This article briefly reviews the role of mathematical models and␣their historical precedents for creating new knowledge of the mechanisms of disease pathogenesis, transmission, and control of some human viral infections. Future research in the modelling of infectious diseases will need to rely upon incorporation of the fundamental principles that govern viral dynamics in vivo as well as in the population.