Optimal mix of screening and contact tracing for endemic diseases

Two common means of controlling infectious diseases are screening and contact tracing. Which should be used, and when? We consider the problem of determining the cheapest mix of screening and contact tracing necessary to achieve a desired endemic prevalence of a disease or to identify a specified number of cases. We perform a partial equilibrium analysis of small-scale interventions, assuming that prevalence is unaffected by the intervention; we develop a full equilibrium analysis where we compare the long-term cost of various combinations of screening and contact tracing needed to achieve a given equilibrium prevalence; and we solve the problem of minimizing the total costs of identifying and treating disease cases plus the cost of untreated disease cases. Our analysis provides several insights. First, contact tracing is only cost effective when prevalence is below a threshold value. This threshold depends on the relative cost per case found by screening versus contact tracing. Second, for a given contact tracing policy, the screening rate needed to achieve a given prevalence or identify a specified number of cases is a decreasing function of disease prevalence. As prevalence increases above the threshold (and contact tracing is discontinued), the screening rate jumps discontinuously to a higher level. Third, these qualitative results hold when we consider unchanged or changed prevalence, and short-term or long-term costs.     

Effectiveness of realistic vaccination strategies for contact networks of various degree distributions

A "contact network" that models infection transmission comprises nodes (or individuals) that are linked when they are in contact and can potentially transmit an infection. Through analysis and simulation, we studied the influence of the distribution of the number of contacts per node, defined as degree, on infection spreading and its control by vaccination. Three random contact networks of various degree distributions were examined. In a scale-free network, the frequency of high-degree nodes decreases as the power of the degree (the case of the third power is studied here); the decrease is exponential in an exponential network, whereas all nodes have the same degree in a constant network. Aiming for containment at a very early stage of an epidemic, we measured the sustainability of a specific network under a vaccination strategy by employing the critical transmissibility larger than which the epidemic would occur. We examined three vaccination strategies: mass, ring, and acquaintance. Irrespective of the networks, mass preventive vaccination increased the critical transmissibility inversely proportional to the unvaccinated rate of the population. Ring post-outbreak vaccination increased the critical transmissibility inversely proportional to the unvaccinated rate, which is the rate confined to the targeted ring comprising the neighbors of an infected node; however, the total number of vaccinated nodes could mostly be fewer than 100 nodes at the critical transmissibility. In combination, mass and ring vaccinations decreased the pathogen's "effective" transmissibility each by the factor of the unvaccinated rate. The amount of vaccination used in acquaintance preventive vaccination was lesser than the mass vaccination, particularly under a highly heterogeneous degree distribution; however, it was not as less as that used in ring vaccination. Consequently, our results yielded a quantitative assessment of the amount of vaccination necessary for infection containment, which is universally applicable to contact networks of various degree distributions.     

Epidemics and vaccination on weighted graphs

A Reed-Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge (u, v) in the graph is equipped with two weights W_(u,v) and W_(v,u) that represent the (subjective) strength of the connection and determine the probability that u infects v in case u is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d. weights and for weights that are functions of the degrees. For i.i.d. weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.     

Emergent heterogeneity in declining tuberculosis epidemics

Tuberculosis is a disease of global importance: over 2 million deaths are attributed to this infectious disease each year. Even in areas where tuberculosis is in decline, there are sporadic outbreaks which are often attributed either to increased host susceptibility or increased strain transmissibility and virulence. Using two mathematical models, we explore the role of the contact structure of the population, and find that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneity even in the absence of host or strain variability. We discuss the implications of this finding for tuberculosis control in low incidence settings.     

SIR dynamics in random networks with heterogeneous connectivity

Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE’s. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.      

Household Epidemics: Modelling Effects of Early Stage Vaccination

A Markovian susceptible → infectious → removed (SIR) epidemic model is considered in a community partitioned into households. A vaccination strategy, which is implemented during the early stages of the disease following the detection of infected individuals is proposed. In this strategy, the detection occurs while an individual is infectious and other susceptible household members are vaccinated without further delay. Expressions are derived for the influence on the reproduction numbers of this vaccination strategy for equal and unequal household sizes. We fit previously estimated parameters from influenza and use household distributions for Sweden and Tanzania census data. The results show that the reproduction number is much higher in Tanzania (6 compared with 2) due to larger households, and that infected individuals have to be detected (and household members vaccinated) after on average 5 days in Sweden and after 3.3 days in Tanzania, a much smaller difference.     

Exploring the relationship between incidence and the average age of infection during seasonal epidemics

The inverse relationship between the incidence and the average age of first infection for immunizing agents has become a basic tenet in the theory underlying the mathematical modeling of infectious diseases. However, this relationship assumes that the infection has reached an endemic equilibrium. In reality, most infectious diseases exhibit seasonal and/or long-term oscillations in incidence. We use a seasonally forced age-structured SIR model to explore the relationship between the number of cases and the average age of first infection over a single epidemic cycle. Contrary to the relationship for the equilibrium dynamics, we find that the average age of first infection is greatest at or near the peak of the epidemic when mixing is homogeneous. We explore the sensitivity of our findings to assumptions about the natural history of infection, population mixing behavior, the mechanism of seasonality, and of the timing of case reporting in relation to the infectious period. We conclude that seasonal variation in the average age of first infection tends to be greatest for acute infections, and the relationship between the number of cases and the average age of first infection can vary depending on the nature of population mixing and the natural history of infection.     

A geographical spread of vaccine-resistance in avian influenza epidemics

Vaccination can be a useful tool for control of avian influenza outbreaks in poultry, but its use is reconsidered in most of the countries worldwide because of its negative effects on the disease control. One of the most important negative effects is the potential for emergence of vaccine-resistant viruses. Actually, in the vaccination program in China and Mexico, several vaccine-resistant strains were confirmed. Vaccine-resistant strains usually cause a loss of the protection effectiveness of vaccination. Therefore, a vaccination program that engenders the emergence of the resistant strain might promote the spread of the resistant strain and undermine the control of the infectious disease, even if the vaccination protects against the transmission of a vaccine-sensitive strain. We designed and analyzed a deterministic patch-structured model in heterogeneous areas (with or without vaccination) illustrating transmission of vaccine-sensitive and vaccine-resistant strains during a vaccination program. We found that the vaccination program can eradicate the vaccine-sensitive strain but lead to a prevalence of vaccine-resistant strain. Further, interestingly, the replacement of viral strain could occur in another area without vaccination through a migration of non-infectious individuals due to an illegal trade of poultry. It is also a novel result that only a complete eradication of both strains in vaccination area can achieve the complete eradication in another areas. Thus we can obtain deeper understanding of an effect of vaccination for better development of vaccination strategies to control avian influenza spread.     

The effect of integral conditions in certain equations modelling epidemics and population growth

Summary Models of epidemics that lead to delay differential equations often have subsidiary integral conditions that are imposed by the interpretation of these models. The neglect of these conditions may lead to solutions that behave in a radically different manner from solutions restricted to obey them. Examples are given of such behavior, including cases where periodic solutions may occur off the natural set defined by these conditions but not on it. A complete stability analysis is also given of a new model of a disease propagated by a vector where these integral conditions play an important role.This work was partially supported by N.S.F. Grant MCS 7903497     

Finding optimal vaccination strategies under parameter uncertainty using stochastic programming

We present a stochastic programming framework for finding the optimal vaccination policy for controlling infectious disease epidemics under parameter uncertainty. Stochastic programming is a popular framework for including the effects of parameter uncertainty in a mathematical optimization model. The problem is initially formulated to find the minimum cost vaccination policy under a chance-constraint. The chance-constraint requires that the probability that R(*) 相似文献   

Network structure, and vaccination strategy and effort interact to affect the dynamics of influenza epidemics

There is growing interest in understanding and controlling the spread of diseases through realistically structured host populations. We investigate how network structures, ranging from circulant, through small-world networks, to random networks, and vaccination strategy and effort interact to influence the proportion of the population infected, the size and timing of the epidemic peak, and the duration of the epidemic. We found these three factors, and their higher-order interactions, significantly influenced epidemic development and extent. Increasing vaccination effort (from 0% to 90%) decreased the number of hosts infected while increasing network randomness worked to increase disease spread. On average, vaccinating hosts based on degree (hubs) resulted in the smallest epidemics while vaccinating hosts with the highest clustering coefficient resulted in the largest epidemics. In a targeted test of five vaccination strategies on a small-world network (probability of rewiring edges=5%) with 10% vaccination effort we found that vaccinating hosts preferentially with high-clustering coefficients (similar to real-world strategies) resulted in twice the number of hosts infected as random vaccinations and nearly a 30-fold higher number of cases than our strategy targeting hubs (highest degree hosts). Our model suggests how vaccinations might be implemented to minimize the extent of an epidemic (e.g., duration and total number infected) as well as the timing and number of hosts infected at a given time over a wide range of structured host networks.     

Control of emerging infectious diseases using responsive imperfect vaccination and isolation

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.     

The saturating contact rate in marriage- and epidemic models

In this note we show how to derive, by a mechanistic argument, an expression for the saturating contact rate of individual contacts in a population that mixes randomly. The main assumption is that the individual interaction times are typically short as compared to the time-scale of changes in, for example, individual-type, but that the interactions yet make up a considerable fraction of the time-budget of an individual. In special cases an explicit formula for the contact rate is obtained. The result is applied to mathematical epidemiology and marriage models.     

Modeling the joint epidemics of TB and HIV in a South African township

We present a simple mathematical model with six compartments for the interaction between HIV and TB epidemics. Using data from a township near Cape Town, South Africa, where the prevalence of HIV is above 20% and where the TB notification rate is close to 2,000 per 100,000 per year, we estimate some of the model parameters and study how various control measures might change the course of these epidemics. Condom promotion, increased TB detection and TB preventive therapy have a clear positive effect. The impact of antiretroviral therapy on the incidence of HIV is unclear and depends on the extent to which it reduces sexual transmission. However, our analysis suggests that it will greatly reduce the TB notification rate.     

Estimating the immunity coverage required to prevent epidemics in a community of households

An estimation of the immunity coverage needed to prevent future outbreaks of an infectious disease is considered for a community of households. Data on outbreak size in a sample of households from one epidemic are used to derive maximum likelihood estimates and confidence bounds for parameters of a stochastic model for disease transmission in a community of households. These parameter estimates induce estimates and confidence bounds for the basic reproduction number and the critical immunity coverage, which are the parameters of main interest when aiming at preventing major outbreaks in the future. The case when individuals are homogeneous, apart from the size of their household, is considered in detail. The generalization to the case with variable infectivity, susceptibility and/or mixing behaviour is discussed more briefly. The methods are illustrated with an application to data on influenza in Tecumseh, Michigan.     

Lineage tracing reveals the phylodynamics,plasticity, and paths of tumor evolution

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Sensitivity of medial and lateral knee contact force predictions to frontal plane alignment and contact locations

Musculoskeletal models are increasingly used to estimate medial and lateral knee contact forces, which are difficult to measure in vivo. The sensitivity of contact force predictions to modeling parameters is important to the interpretation and implication of results generated by the model. The purpose of this study was to quantify the sensitivity of knee contact force predictions to simultaneous errors in frontal plane knee alignment and contact locations under different dynamic conditions. We scaled a generic musculoskeletal model for N = 23 subjects’ stature and radiographic knee alignment, then perturbed frontal plane alignment and mediolateral contact locations within experimentally-possible ranges of 10° to −10° and 10 to −10 mm, respectively. The sensitivity of first peak, second peak, and mean medial and lateral knee contact forces to knee adduction angle and contact locations was modeled using linear regression. Medial loads increased, and lateral loads decreased, by between 3% and 6% bodyweight for each degree of varus perturbation. Shifting the medial contact point medially increased medial loads and decreased lateral loads by between 1% and 4% bodyweight per millimeter. This study demonstrates that realistic measurement errors of 5 mm (contact distance) or 5° (frontal plane alignment) could result in a combined 50% BW error in subject specific contact force estimates. We also show that model sensitivity varies between subjects as a result of differences in gait dynamics. These results demonstrate that predicted knee joint contact forces should be considered as a range of possible values determined by model uncertainty.     

Knee medial and lateral contact forces in a musculoskeletal model with subject-specific contact point trajectories

Contact point (CP) trajectory is a crucial parameter in estimating medial/lateral tibio-femoral contact forces from the musculoskeletal (MSK) models. The objective of the present study was to develop a method to incorporate the subject-specific CP trajectories into the MSK model. Ten healthy subjects performed 45 s treadmill gait trials. The subject-specific CP trajectories were constructed on the tibia and femur as a function of extension-flexion using low-dose bi-plane X-ray images during a quasi-static squat. At each extension-flexion position, the tibia and femur CPs were superimposed in the three directions on the medial side, and in the anterior-posterior and proximal-distal directions on the lateral side to form the five kinematic constraints of the knee joint. The Lagrange multipliers associated to these constraints directly yielded the medial/lateral contact forces. The results from the personalized CP trajectory model were compared against the linear CP trajectory and sphere-on-plane CP trajectory models which were adapted from the commonly used MSK models. Changing the CP trajectory had a remarkable impact on the knee kinematics and changed the medial and lateral contact forces by 1.03 BW and 0.65 BW respectively, in certain subjects. The direction and magnitude of the medial/lateral contact force were highly variable among the subjects and the medial-lateral shift of the CPs alone could not determine the increase/decrease pattern of the contact forces. The suggested kinematic constraints are adaptable to the CP trajectories derived from a variety of joint models and those experimentally measured from the 3D imaging techniques.