Age-structured Model Research Articles

Effectiveness and efficiency of immunisation strategies to prevent RSV among infants and older adults in Germany: a modelling study

BackgroundRecently, several novel RSV immunisation products that protect infants and older adults against RSV disease have been licensed in Europe. We estimated the effectiveness and efficiency of introducing these RSV immunisation strategies in Germany.MethodsWe used a Bayesian framework to fit a deterministic age-structured dynamic transmission model of RSV to sentinel surveillance and RSV-specific hospitalisation data in Germany from 2015 to 2019. The calibrated model was used to evaluate different RSV intervention strategies over 5 years: long-acting, single-dose monoclonal antibodies (mAbs) in high-risk infants aged 1–5 months; long-acting mAbs in all infants aged 1–5 months; seasonal vaccination of pregnant women and one-time seasonal vaccination of older adults (75 + /65 + /55 + years). We performed sensitivity analysis on vaccine uptake, seasonal vs. year-round maternal vaccination, and the effect of under-ascertainment for older adults.ResultsThe model was able to match the various RSV datasets. Replacing the current short-acting mAB for high-risk infants with long-acting mAbs prevented 1.1% of RSV-specific hospitalisations in infants per year at the same uptake. Expanding the long-acting mAB programme to all infants prevented 39.3% of infant hospitalisations per year. Maternal vaccination required a larger number to be immunised to prevent one additional hospitalisation than a long-acting mAB for the same uptake. Vaccination of adults older than 75 years at an uptake of 40% in addition to Nirsevimab in all infants prevented an additional 4.5% of all RSV hospitalisations over 5 years, with substantial uncertainty in the correction for under-ascertainment of the RSV burden.ConclusionsImmunisation has the potential to reduce the RSV disease burden in Germany.

Marine reserves can buffer against environmental fluctuations for overexploited but not sustainably harvested fisheries.

Globally, decision-makers are seeking management levers that can mitigate the negative effects of climate change on ecosystems that have already been transformed from their natural state by the effects of fishing. An important question is whether marine reserves can provide buffering (i.e., population-level resilience) against climate disturbances to fished populations. Here, we examine one aspect of this question, by asking whether marine reserves can reduce the variability in either overall biomass or in fishery yield, in the face of environmental variability. This could happen because greater reproduction of longer-lived, larger fish inside reserves could supplement recruitment to the fished portion of the population. We addressed this question using age-structured population models, assuming a system where some proportion of the coastline is protected in marine reserves (0%-30%), and the remainder is fished (at a range of possible harvest rates). We modeled populations with sedentary adults and dispersal via a larval pool. Since recent extreme climate events (e.g., marine heatwaves) have reduced juvenile survival for some fish species, we assumed that environmental variability affected the survival of the first age class in our model. We viewed population variability as a question of buffering, measured as the proportion of time a simulated population spent below a target reference point, with the idea that marine reserves could prevent the population from reaching low levels in the face of fishing and environmental variability. We found that fisheries with more area in marine reserves always had less variability in biomass. However, adding marine reserves only reduced variability in fisheries yield when the fished part of the population was being harvested at a rate exceeding the maximum sustainable yield. This new result on reducing variability is in line with previous findings that the "spillover" effects of marine reserve benefits to fishery yields only accrue when the fishery outside reserve boundaries is being overharvested.

Global stability and optimal control of an age-structured quit-smoking model with quit-smoking duration

In this paper, we introduce an age-structured quit-smoking model that incorporates the duration of smoking cessation, a crucial parameter related to smoking cessation counseling and disease treatment. We identify the threshold value [Formula: see text], which determines the existence and stability of the smoking-free steady state. Our findings show that the smoking-free steady state is locally and globally stable if [Formula: see text], whereas it becomes unstable, and the unique steady state with present smokers emerges and is locally and globally stable if [Formula: see text]. We propose an optimal control strategy, and our results show its ability to effectively decrease the number of smokers and promote smoking cessation. To validate our findings and to study the influence of varying parameters on the model’s dynamics, numerical simulations are carried out. A concise conclusion is then shared at the end of the paper.

A convergence analysis for the approximation to the solution of an age-structured population model with infinite lifespan

Considering the numerical approximation of the density distribution for an age-structured population model with unbounded lifespan on a compact interval [0,T], we prove second order of convergence for a discretization that adaptively selects its truncated age-interval according to the exponential rate of decay with age of the solution of the model. It appears that the adaptive capacity of the length in the truncated age-interval of the discretization to the infinity lifespan is a very convenient approach for a long-time integration of the model to establish the asymptotic behavior of its dynamics numerically. The analysis of convergence uses an appropriate weighted maximum norm with exponential weights to cope with the unbounded age lifespan. We report experiments to exhibit numerically the theoretical results and the asymptotic behaviour of the dynamics for an age-structured squirrel population model introduced by Sulsky.

Vaccination subsidy allocation under budget constraints considering the human interpersonal contact pattern and vaccine protection effect in epidemics

Vaccination subsidy is one of the most important measures to increase the willingness of people to be vaccinated for epidemic prevention and control. Given the budget constraint of vaccination subsidy and the different roles of various age groups population in the epidemic transmission process, a novel approach is proposed to optimize the utilization of the limited vaccination subsidy budget. First, an age-structured SIR model that integrates the human interpersonal contact pattern and vaccine protection effect is developed to simulate the epidemic trend under a certain vaccination state. Then, a decision model is constructed to determine the optimal vaccination subsidy allocation scheme for various age groups based on the age-structured SIR model. A case study of Wuhan City is used to verify the decision approach, and a series of sensitivity analyses are conducted. The result shows that the peak time of an epidemic is positively related to the total treatment cost for all the infections, and more subsidies should be allocated to the populations with higher interpersonal contact frequencies for epidemic prevention and control.

On the necessity of accounting for age structure in human malaria transmission modelling

Malaria is one of the most common mosquito-borne diseases widespread in tropical and subtropical regions, causing thousands of deaths every year in the world. In a previous paper, we formulated an age-structured model containing three structural variables: (i) the chronological age of human and mosquito populations, (ii) the time since they are infected, and (iii) humans waning immunity (i.e. the progressive loss of protective antibodies after recovery). In the present paper, we expand the analysis of this age-structured model and focus on the derivation of entomological and epidemiological results commonly used in the literature, following the works of Smith and McKenzie. We generalize their results to the age-structured case. In order to quantify the impact of neglecting structuring variables such as chronological age, we assigned values from the literature to our model parameters. While some parameters values are readily accessible from the literature, at least those about the human population, the parameters concerning mosquitoes are less commonly documented and the values of a number of them (e.g. mosquito survival in the presence or in absence of infection) can be discussed extensively. Our analysis, informed by parameter values from the literature, demonstrates that overlooking those structural variables of human and mosquito populations may result in inaccurate epidemiological predictions and suboptimal control strategies. We highlight the epidemiological implications of these findings and emphasize the necessity of considering age structure in future malaria control programs.

Influenza vaccine allocation in tropical settings under constrained resources.

Influenza virus seasonality, synchronicity, and vaccine supply differ substantially between temperate and tropical settings, and optimal vaccination strategy may differ on this basis. Many national vaccine recommendations focus on high-risk groups, elderly populations, and healthcare workers despite previous analyses demonstrating broad benefits to vaccinating younger high-contact age groups. In this study, we parameterized an age-structured nonseasonal asynchronous epidemiological model of influenza virus transmission for a tropical low-income setting. We evaluated timing and age allocation of vaccines across vaccine supplies ranging from 10 to 90% using decade-based age groups. Year-round vaccination was beneficial when compared with more concentrated annual vaccine distribution. When targeting a single age group for vaccine prioritization, maximum vaccine allocation to the 10-19 high-contact age group minimized annual influenza mortality for all but one vaccine supply. When evaluating across all possible age allocations, optimal strategies always allocated a plurality of vaccines to school-age children (10-19). The converse, however, was not true as not all strategies allocating a plurality to children aged 10-19 minimized mortality. Allocating a high proportion of vaccine supply to the 10-19 age group is necessary but not sufficient to minimize annual mortality as distribution of remaining vaccine doses to other age groups also needs to be optimized. Strategies focusing on indirect benefits (vaccinating children) showed higher variance in mortality outcomes than strategies focusing on direct benefits (vaccinating the elderly). However, the indirect benefit approaches showed a lower mean mortality and a lower minimum mortality than vaccination focused on the elderly.

Assessing vaccine strategies for mpox outbreak in New York City using an age-structure model

BackgroundSince May 7 2022, mpox has been endemic in many countries which has attracted the attention of health authorities in various countries and made control decisions, in which vaccination is the mainstream strategy. However, the shortage of vaccine doses and the reduction of protective efficacy have led to unresolved issues such as vaccine allocation decisions and evaluation of transmission scale.MethodsWe developed an epidemiological model to describe the prevalence of the mpox virus in New York City and calibrated the model to match surveillance data from May 19 to November 3, 2022. Finally, we adjusted the model to simulate and compare several scenarios of non-vaccination and pre-pandemic vaccination.ResultsRelative to the status quo, if vaccination is not carried out, the number of new infections increases to about 385%, and the transmission time will be extended to about 350%, while if vaccinated before the epidemic, the number of new infections decreases to 94.2-96%.ConclusionsThe mpox outbreak in New York City may be linked to the Pride event. However, with current vaccine coverage, there will be no more large-scale outbreaks of mpox, even if there is another similar activity. For areas with limited vaccines, priority is given to high-risk groups in the age group [34–45] years as soon as possible.

Positivity and Boundedness Preserving Numerical Scheme for a Stochastic Multigroup Susceptible-Infected-Recovering Epidemic Model with Age Structure.

Since the stochastic age-structured multigroup susceptible-infected-recovering (SIR) epidemic model is nonlinear, the solution of this model is hard to be explicitly represented. It is necessary to construct effective numerical methods so as to predict the number of infections. In addition, the stochastic age-structured multigroup SIR model has features of positivity and boundedness of the solution. Therefore, in this article, in order to ensure that the numerical and analytical solutions must have the same properties, by modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving EM (PBPEM) method on temporal space for stochastic age-structured multigroup SIR model, which is proved to have a strong convergence to the true solution over finite time intervals. Moreover, by combining the standard finite element method and the PBPEM method, we propose a full-discrete scheme to show the numerical solutions, as well as analyze the error estimations. Finally, the full-discrete scheme is applied to a general stochastic two-group SIR model and the Chlamydia epidemic model, which shows the superiority of the numerical method.

Dynamics of an age-structured SIS epidemic model with local dispersal and general incidence functions

In this paper, an age-structured SIS epidemic model with local dispersal and general incidence functions is formulated. We first analytically derive the well-posedness of solutions, the basic reproduction number R0 and the existence of steady states. Then, we show that the disease-free steady state is global asymptotically stable if R0≤1 and disease in the model is uniformly persistent if R0>1. Moreover, we show global asymptotic stability of the endemic steady state under some cases. Finally, infection age, media coverage and spatial diffusion on the epidemic of infectious diseases are demonstrated by numerical simulations.

A retrospective assessment of forecasting the peak of the SARS-CoV-2 Omicron BA.1 wave in England.

We discuss the invasion of the Omicron BA.1 variant into England as a paradigm for real-time model fitting and projection. Here we use a mixture of simple SIR-type models, analysis of the early data and a more complex age-structure model fit to the outbreak to understand the dynamics. In particular, we highlight that early data shows that the invading Omicron variant had a substantial growth advantage over the resident Delta variant. However, early data does not allow us to reliably infer other key epidemiological parameters-such as generation time and severity-which influence the expected peak hospital numbers. With more complete epidemic data from January 2022 are we able to capture the true scale of the epidemic in terms of both infections and hospital admissions, driven by different infection characteristics of Omicron compared to Delta and a substantial shift in estimated precautionary behaviour during December. This work highlights the challenges of real time forecasting, in a rapidly changing environment with limited information on the variant's epidemiological characteristics.

Dynamic analysis of HCV infection and drug resistance using an age-structured multiscale model

Direct-acting antiviral agents (DAAs) are known to interfere with various intracellular stages of the hepatitis C virus (HCV) life cycle and have demonstrated efficacy in treating HCV infection. However, DAA monotherapy can lead to drug resistance due to mutations. This paper explores the impact of DAA therapy on HCV dynamics using a multiscale age-structured partial differential equation (PDE) model that incorporates intracellular viral RNA replication within infected cells and two strains of viruses representing a drug-sensitive strain and a drug-resistant mutant variant, respectively. We derived an equivalent ordinary differential equation (ODE) model from the PDE model to simplify mathematical analysis and numerical simulations. We studied the dynamics of the two virus strains before treatment and investigated the impact of mutations on the evolution kinetics of drug-sensitive and drug-resistant viruses, as well as the competition between the two strains during treatment. We also explored the role of DAAs in blocking HCV RNA replication and releasing new virus particles from cells. During treatment, mutations do not significantly influence the dynamics of various virus strains; however, they can generate low-level HCV that may be completely inhibited due to their poor fitness. The fitness of the mutant strain compared to the drug-sensitive strain determines which strain dominates the virus population. We also investigated the prevalence and drug resistance evolution of HCV variants during DAA treatment.

Health and economic impacts of Vaccae vaccination incorporating active case finding in India and South Africa: a modelling study

IntroductionTuberculosis (TB) is a major global health issue, particularly, in India and South Africa. We aim to evaluate the potential of the new TB vaccine, Vaccae, to enhance TB control...

A heterogeneous continuous age-structured model of mumps with vaccine

In classical mumps models, individuals are generally assumed to be uniformly mixed (homogeneous), ignoring population heterogeneity (preference, activity, etc.). Age is the key to catching mixed patterns in developing mathematical models for mumps. A continuous heterogeneous age-structured model for mumps with vaccines has been developed in this paper. The stability of age-structured models is a difficult question. An explicit formula of R0 was defined for the various mixing modes (isolation, proportional and heterogeneous mixing) with or without the vaccine. The results show that the endemic steady state is unique and locally stable if R0 > 1 without any additional conditions. A number of numerical examples are given to support the theory.

Dynamic Vaccine Allocation for Control of Human-Transmissible Disease.

During pandemics, such as COVID-19, supplies of vaccines can be insufficient for meeting all needs, particularly when vaccines first become available. Our study develops a dynamic methodology for vaccine allocation, segmented by region, age, and timeframe, using a time-sensitive, age-structured compartmental model. Based on the objective of minimizing a weighted sum of deaths and cases, we used the Sequential Least Squares Quadratic Programming method to search for a locally optimal COVID-19 vaccine allocation for the United States, for the period from 16 December 2020 to 30 June 2021, where regions corresponded to the 50 states in the United States (U.S.). We also compared our solution to actual allocations of vaccines. From our model, we estimate that approximately 1.8 million cases and 9 thousand deaths could have been averted in the U.S. with an improved allocation. When case reduction is prioritized over death reduction, we found that young people (17 and younger) should receive priority over old people due to their potential to expose others. However, if death reduction is prioritized over case reduction, we found that more vaccines should be allocated to older people, due to their propensity for severe disease. While we have applied our methodology to COVID-19, our approach generalizes to other human-transmissible diseases, with potential application to future epidemics.

Variety is the spice of life: nongenetic variation in life histories influences population growth and evolvability.

Individual vital rates are key determinants of lifetime reproductive success, and variability in these rates shapes population dynamics. Previous studies have found that this vital rate hetero- geneity can influence demographic properties including population growth rates, however, the explicit effects of the amount of variation within and the covariance between vital rates that can also vary throughout the lifespan on population growth remains unknown. Here, we explore the analytical consequences of nongenetic heterogeneity on long-term population growth rates and rates of evolution by modifying traditional age-structured population projection matrices to incorporate variation among individual vital rates. The model allows vital rates to be permanent throughout life ("fixed condition") or to change over the lifespan ("dynamic condition"). We reduce the complexity associated with adding individual heterogeneity to age-structured models through a novel application of matrix collapsing ("phenotypic collapsing"), showing how to col- lapse in a manner that preserves the asymptotic and transient dynamics of the original matrix. The main conclusion is that nongenetic individual heterogeneity can strongly impact the long-term growth rate and rates of evolution. The magnitude and sign of this impact depends heavily on how the heterogeneity covaries across the lifespan of an organism. Our results emphasize that nongenetic variation cannot simply be viewed as random noise, but rather that it has consistent, predictable effects on fitness and evolvability.

The impact of household size on measles transmission: A long-term perspective

Households play an important role in the transmission of infectious diseases due to the close contact therein. Previous modeling studies on disease transmission with household-level mixing have explored the relationship between household size distribution and epidemic characteristics such as final epidemic sizes and the basic reproduction number but have not considered the epidemic impact of declining household sizes caused by demographic shifts. Here, we use a disease transmission model that incorporates demographic changes in household sizes to study the long-term transmission dynamics of measles in communities with varying household size distributions. We explore the impact of incorporating both household- and age-structured mixing on the dynamic properties of the transmission model and compare these dynamics across different household size distributions. Our analysis, based on the household- and age-structured model, shows that communities with larger household sizes require higher vaccination thresholds and bear a greater burden of infections. However, simulations show the apparent impact of changing household sizes is the combined result of changing birth rates and household mixing, and that changing birth rates likely play a larger role than changes in household mixing in shaping measles transmission dynamics (n.b, life-long immunity makes replenishment of population susceptibility from births a crucial transmission driver for measles). In addition, simulations of endemic transmission of measles within a hypothetical population formulated using aggregated world demographic data suggest the decline in household size (driven by changing fertility rates of the population), in addition to increasing vaccination coverage, could have had a significant impact on the incidence of measles over time.

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 < 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 < 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

Overcompensation of transient and permanent death rate increases in age-structured models with cannibalistic interactions

There has been renewed interest in understanding the mathematical structure of ecological population models that lead to overcompensation, the process by which a population recovers to a higher level after suffering a permanent increase in predation or harvesting. Here, we apply a recently formulated kinetic population theory to formally construct an age-structured single-species population model that includes a cannibalistic interaction in which older individuals prey on younger ones. Depending on the age-dependent structure of this interaction, our model can exhibit transient or steady-state overcompensation of an increased death rate as well as oscillations of the total population, both phenomena that have been observed in ecological systems. Analytic and numerical analysis of our model reveals sufficient conditions for overcompensation and oscillations. We also show how our structured population partial integrodifferential equation (PIDE) model can be reduced to coupled ODE models representing piecewise constant parameter domains, providing additional mathematical insight into the emergence of overcompensation.

Analysis and optimal control of a hierarchical age-structured population model in a polluted environment

For individual biological populations, examining the impacts of competition within the species and the presence of toxic substances in the environment can effectively facilitate control measures. However research in this area remains limited. This article introduces a toxicant-population model incorporating a hierarchical age-structured and examines the minimum deviation problem within this model. The aim is to explore effective control strategies that strike a balance between improving the environment and population benefits. Through a novel approach, we reformulate the problem as an abstract Cauchy problem on a non-dense domain and then apply integrated semigroup theory to establish the existence, uniqueness, and positivity of solutions to the model. The existence of optimal control is derived using Ekeland's variational principle in functional analysis. The expression for optimal control is given by the Pontryagin's minimum principle. Lastly, we conducted numerical experiments on the model, applied optimality conditions to delineate optimal control, and illustrated the comparison between crucial population parameters under optimal control and those without any control.