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1.
When facing incursion of a major livestock infectious disease, the decision to implement a vaccination programme is made at the national level. To make this decision, governments must consider whether the benefits of vaccination are sufficient to outweigh potential additional costs, including further trade restrictions that may be imposed due to the implementation of vaccination. However, little consensus exists on the factors triggering its implementation on the field. This work explores the effect of several triggers in the implementation of a reactive vaccination-to-live policy when facing epidemics of foot-and-mouth disease. In particular, we tested whether changes in the location of the incursion and the delay of implementation would affect the epidemiological benefit of such a policy in the context of Scotland. To reach this goal, we used a spatial, premises-based model that has been extensively used to investigate the effectiveness of mitigation procedures in Great Britain. The results show that the decision to vaccinate, or not, is not straightforward and strongly depends on the underlying local structure of the population-at-risk. With regards to disease incursion preparedness, simply identifying areas of highest population density may not capture all complexities that may influence the spread of disease as well as the benefit of implementing vaccination. However, if a decision to vaccinate is made, we show that delaying its implementation in the field may markedly reduce its benefit. This work provides guidelines to support policy makers in their decision to implement, or not, a vaccination-to-live policy when facing epidemics of infectious livestock disease.  相似文献   

2.
Ring vaccination can be a highly effective control strategy for an emerging disease or in the final phase of disease eradication, as witnessed in the eradication of smallpox. However, the impact of behavioural dynamics on the effectiveness of ring vaccination has not been explored in mathematical models. Here, we analyze a series of stochastic models of voluntary ring vaccination. Contacts of an index case base vaccinating decisions on their own individual payoffs to vaccinate or not vaccinate, and they can also imitate the behaviour of other contacts of the index case. We find that including imitation changes the probability of containment through ring vaccination considerably. Imitation can cause a strong majority of contacts to choose vaccination in some cases, or to choose non-vaccination in other cases-even when the equivalent solution under perfectly rational (non-imitative) behaviour yields mixed choices. Moreover, imitation processes can result in very different outcomes in different stochastic realizations sampled from the same parameter distributions, by magnifying moderate tendencies toward one behaviour or the other: in some realizations, imitation causes a strong majority of contacts not to vaccinate, while in others, imitation promotes vaccination and reduces the number of secondary infections. Hence, the effectiveness of ring vaccination can depend significantly and unpredictably on imitation processes. Therefore, our results suggest that risk communication efforts should be initiated early in an outbreak when ring vaccination is to be applied, especially among subpopulations that are heavily influenced by peer opinions.  相似文献   

3.
Certain theories suggest that it should be difficult or impossible to eradicate a vaccine-preventable disease under voluntary vaccination: Herd immunity implies that the individual incentive to vaccinate disappears at high coverage levels. Historically, there have been examples of declining coverage for vaccines, such as MMR vaccine and whole-cell pertussis vaccine, that are consistent with this theory. On the other hand, smallpox was globally eradicated by 1980 despite voluntary vaccination policies in many jurisdictions. Previous modeling studies of the interplay between disease dynamics and individual vaccinating behavior have assumed that infection is transmitted in a homogeneously mixing population. By comparison, here we simulate transmission of a vaccine-preventable SEIR infection through a random, static contact network. Individuals choose whether to vaccinate based on infection risks from neighbors, and based on vaccine risks. When neighborhood size is small, rational vaccinating behavior results in rapid containment of the infection through voluntary ring vaccination. As neighborhood size increases (while the average force of infection is held constant), a threshold is reached beyond which the infection can break through partially vaccinated rings, percolating through the whole population and resulting in considerable epidemic final sizes and a large number vaccinated. The former outcome represents convergence between individually and socially optimal outcomes, whereas the latter represents their divergence, as observed in most models of individual vaccinating behavior that assume homogeneous mixing. Similar effects are observed in an extended model using smallpox-specific natural history and transmissibility assumptions. This work illustrates the significant qualitative differences between behavior–infection dynamics in discrete contact-structured populations versus continuous unstructured populations. This work also shows how disease eradicability in populations where voluntary vaccination is the primary control mechanism may depend partly on whether the disease is transmissible only to a few close social contacts or to a larger subset of the population.  相似文献   

4.
Busse JW  Walji R  Wilson K 《PloS one》2011,6(8):e22737

Background

Parents who choose to selectively vaccinate or avoid vaccination for their children may do so at risk of compromising relations with their family physician or pediatrician. Groups that are associated with reduced rates of pedicatic vaccination, such as parents who access naturopathic care, may be particularly vulnerable to this issue.

Methodology/Principal Findings

In March through September 2010, we administered a 26-item cross-sectional survey to 129 adult patients, all of whom were parents with children ≤16 years of age, presenting for naturopathic care in Ontario, Canada. Ninety-five parents completed the survey (response rate 74%), and only 50.5% (48 of 95) reported that their children had received all recommended vaccines. Most parents (50.5%; 48 of 95) reported feeling pressure to vaccinate from their allopathic physician and, of those who discussed vaccination with their physician, 25.9% (21 of 81) were less comfortable continuing care as a result. Five percent (4 of 81) of respondents were advised by their physician that their children would be refused care if they decided against vaccination. In our adjusted generalized linear model, feeling pressure to vaccinate (odds ratio [OR] = 3.07; 95% confidence interval [CI] = 1.14 to 8.26) or endorsing a naturopathic physician as their most trusted source of information regarding vaccination (OR = 3.57; 95% CI = 1.22 to 10.44) were associated with greater odds of having a partially vaccinated or unvaccinated child. The majority (69.6%; 32 of 46) of parent''s with partially vaccinated or unvaccinated children reported a willingness to re-consider this decision.

Conclusions/Significance

Use of naturopathic care should be explored among parents in order to identify this high-risk group and engage them in discussion regarding pediatric vaccination to encourage evidence-based, shared decision making. Physicians should ensure that discussions regarding vaccination are respectful, even if parents are determined not to vaccinate their children.  相似文献   

5.
The effectiveness of seasonal influenza vaccination programs depends on individual-level compliance. Perceptions about risks associated with infection and vaccination can strongly influence vaccination decisions and thus the ultimate course of an epidemic. Here we investigate the interplay between contact patterns, influenza-related behavior, and disease dynamics by incorporating game theory into network models. When individuals make decisions based on past epidemics, we find that individuals with many contacts vaccinate, whereas individuals with few contacts do not. However, the threshold number of contacts above which to vaccinate is highly dependent on the overall network structure of the population and has the potential to oscillate more wildly than has been observed empirically. When we increase the number of prior seasons that individuals recall when making vaccination decisions, behavior and thus disease dynamics become less variable. For some networks, we also find that higher flu transmission rates may, counterintuitively, lead to lower (vaccine-mediated) disease prevalence. Our work demonstrates that rich and complex dynamics can result from the interaction between infectious diseases, human contact patterns, and behavior.  相似文献   

6.

Introduction

The Chief Medical Officer for England recommends that healthcare workers have a seasonal influenza vaccination in an attempt to protect both patients and NHS staff. Despite this, many healthcare workers do not have a seasonal influenza vaccination. Social network analysis is a well-established research approach that looks at individuals in the context of their social connections. We examine the effects of social networks on influenza vaccination decision and disease dynamics.

Methods

We used a social network analysis approach to look at vaccination distribution within the network of the Lancaster Medical School students and combined these data with the students’ beliefs about vaccination behaviours. We then developed a model which simulated influenza outbreaks to study the effects of preferentially vaccinating individuals within this network.

Results

Of the 253 eligible students, 217 (86%) provided relational data, and 65% of responders had received a seasonal influenza vaccination. Students who were vaccinated were more likely to think other medical students were vaccinated. However, there was no clustering of vaccinated individuals within the medical student social network. The influenza simulation model demonstrated that vaccination of well-connected individuals may have a disproportional effect on disease dynamics.

Conclusions

This medical student population exhibited vaccination coverage levels similar to those seen in other healthcare groups but below recommendations. However, in this population, a lack of vaccination clustering might provide natural protection from influenza outbreaks. An individual student’s perception of the vaccination coverage amongst their peers appears to correlate with their own decision to vaccinate, but the directionality of this relationship is not clear. When looking at the spread of disease within a population it is important to include social structures alongside vaccination data. Social networks influence disease epidemiology and vaccination campaigns designed with information from social networks could be a future target for policy makers.  相似文献   

7.
We use the reinfection SIRI epidemiological model to analyze the impact of education programs and vaccine scares on individuals decisions to vaccinate or not. The presence of the reinfection provokes the novelty of the existence of three Nash equilibria for the same level of the morbidity relative risk instead of a single Nash equilibrium as occurs in the SIR model studied by Bauch and Earn (PNAS 101:13391–13394, 2004). The existence of three Nash equilibria, with two of them being evolutionary stable, introduces two scenarios with relevant and opposite features for the same level of the morbidity relative risk: the low-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a low probability; and the high-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a high probability. We introduce the evolutionary vaccination dynamics for the SIRI model and we prove that it is bistable. The bistability of the evolutionary dynamics indicates that the damage provoked by false scares on the vaccination perceived morbidity risks can be much higher and much more persistent than in the SIR model. Furthermore, the vaccination education programs to be efficient they need to implement a mechanism to suddenly increase the vaccination coverage level.  相似文献   

8.
Breban R 《PloS one》2011,6(12):e28300
Both pandemic and seasonal influenza are receiving more attention from mass media than ever before. Topics such as epidemic severity and vaccination are changing the way in which we perceive the utility of disease prevention. Voluntary influenza vaccination has been recently modeled using inductive reasoning games. It has thus been found that severe epidemics may occur because individuals do not vaccinate and, instead, attempt to benefit from the immunity of their peers. Such epidemics could be prevented by voluntary vaccination if incentives were offered. However, a key assumption has been that individuals make vaccination decisions based on whether there was an epidemic each influenza season; no other epidemiological information is available to them. In this work, we relax this assumption and investigate the consequences of making more informed vaccination decisions while no incentives are offered. We obtain three major results. First, individuals will not cooperate enough to constantly prevent influenza epidemics through voluntary vaccination no matter how much they learned about influenza epidemiology. Second, broadcasting epidemiological information richer than whether an epidemic occurred may stabilize the vaccination coverage and suppress severe influenza epidemics. Third, the stable vaccination coverage follows the trend of the perceived benefit of vaccination. However, increasing the amount of epidemiological information released to the public may either increase or decrease the perceived benefit of vaccination. We discuss three scenarios where individuals know, in addition to whether there was an epidemic, (i) the incidence, (ii) the vaccination coverage and (iii) both the incidence and the vaccination coverage, every influenza season. We show that broadcasting both the incidence and the vaccination coverage could yield either better or worse vaccination coverage than broadcasting each piece of information on its own.  相似文献   

9.
Several studies have found that some parents delay the age at which their children receive pediatric vaccines due to perception of higher vaccine risk at the recommended age of vaccination. This has been particularly apparently during the Measles-Mumps-Rubella scare in the United Kingdom. Under a voluntary vaccination policy, vaccine coverage in certain age groups is a potentially complex interplay between vaccinating behaviour, disease dynamics, and age-specific risk factors. Here, we construct an age-structured game dynamic model, where individuals decide whether to vaccinate according to imitation dynamics depending on age-dependent disease prevalence and perceived risk of vaccination. Individuals may be timely vaccinators, delayers, or non-vaccinators. The model exhibits multiple equilibria and a broad range of possible dynamics. For certain parameter regimes, the proportion of timely vaccinators and delayers oscillate in an anti-phase fashion in response to oscillations in infection prevalence. Under an exogenous change to the perceived risk of vaccination as might occur during a vaccine scare, the model can also capture an increase in delayer strategists similar in magnitude to that observed during the Measles-Mumps-Rubella vaccine scare in the United Kingdom. Our model also shows that number of delayers steadily increases with increasing severity of the scare, whereas it saturates to specific value with increases in duration of the scare. Finally, by comparing the model dynamics with and without the option of a delayer strategy, we show that adding a third delayer strategy can have a stabilizing effect on model dynamics. In an era where individual choice—rather than accessibility—is becoming an increasingly important determinant of vaccine uptake, more infectious disease models may need to use game theory or related techniques to determine vaccine uptake.  相似文献   

10.
Lee S  Rocha LE  Liljeros F  Holme P 《PloS one》2012,7(5):e36439
Decreasing the number of people who must be vaccinated to immunize a community against an infectious disease could both save resources and decrease outbreak sizes. A key to reaching such a lower threshold of immunization is to find and vaccinate people who, through their behavior, are more likely than average to become infected and to spread the disease further. Fortunately, the very behavior that makes these people important to vaccinate can help us to localize them. Earlier studies have shown that one can use previous contacts to find people that are central in static contact networks. However, real contact patterns are not static. In this paper, we investigate if there is additional information in the temporal contact structure for vaccination protocols to exploit. We answer this affirmative by proposing two immunization methods that exploit temporal correlations and showing that these methods outperform a benchmark static-network protocol in four empirical contact datasets under various epidemic scenarios. Both methods rely only on obtainable, local information, and can be implemented in practice. For the datasets directly related to contact patterns of potential disease spreading (of sexually-transmitted and nosocomial infections respectively), the most efficient protocol is to sample people at random and vaccinate their latest contacts. The network datasets are temporal, which enables us to make more realistic evaluations than earlier studies--we use only information about the past for the purpose of vaccination, and about the future to simulate disease outbreaks. Using analytically tractable models, we identify two temporal structures that explain how the protocols earn their efficiency in the empirical data. This paper is a first step towards real vaccination protocols that exploit temporal-network structure--future work is needed both to characterize the structure of real contact sequences and to devise immunization methods that exploit these.  相似文献   

11.
BackgroundRabies has been a serious public health threat in Flores Island, Indonesia since it was introduced in 1997. To control the disease, annual dog vaccination campaigns have been implemented to vaccinate all dogs free of charge. Nevertheless, the uptake rate of the vaccination campaigns has been low. The objective of this paper is to identify risk factors associated with the uptake of rabies control measures by individual dog owners in Flores Island.Conclusions/significanceDog owners in Flores Island had a high level of knowledge of rabies and its control, but this was not associated with uptake of the 2012 vaccination campaign. Geographical accessibility was one of the important factors influencing the vaccination uptake among dog owners. Targeted distribution of information on vaccination schedules and methods to catch and restrain dogs in those villages with poor accessibility may increase vaccination uptake in the future.  相似文献   

12.
In modeling individuals vaccination decision making, existing studies have typically used the payoff-based (e.g., game-theoretical) approaches that evaluate the risks and benefits of vaccination. In reality, whether an individual takes vaccine or not is also influenced by the decisions of others, i.e., due to the impact of social influence. In this regard, we present a dual-perspective view on individuals decision making that incorporates both the cost analysis of vaccination and the impact of social influence. In doing so, we consider a group of individuals making their vaccination decisions by both minimizing the associated costs and evaluating the decisions of others. We apply social impact theory (SIT) to characterize the impact of social influence with respect to individuals interaction relationships. By doing so, we propose a novel modeling framework that integrates an extended SIT-based characterization of social influence with a game-theoretical analysis of cost minimization. We consider the scenario of voluntary vaccination against an influenza-like disease through a series of simulations. We investigate the steady state of individuals’ decision making, and thus, assess the impact of social influence by evaluating the coverage of vaccination for infectious diseases control. Our simulation results suggest that individuals high conformity to social influence will increase the vaccination coverage if the cost of vaccination is low, and conversely, will decrease it if the cost is high. Interestingly, if individuals are social followers, the resulting vaccination coverage would converge to a certain level, depending on individuals’ initial level of vaccination willingness rather than the associated costs. We conclude that social influence will have an impact on the control of an infectious disease as they can affect the vaccination coverage. In this respect, our work can provide a means for modeling the impact of social influence as well as for estimating the effectiveness of a voluntary vaccination program.  相似文献   

13.

Purpose

The quadrivalent HPV vaccination was approved for use in males ages 9 to 26 in 2009 and recommended for routine administration in 2011. The purpose of this study was to uncover predictable commonalities amongst parents who chose to vaccinate their 11–17 year old sons against HPV.

Methods

We compiled data from a U.S. national sample of 779 parents with sons 11–17 years old using a web-based survey to gather information about behavioral and sociodemographic factors which predicted receipt of 1 or more HPV vaccine doses based on parental report. Predictors were first modeled individually for univariable associations. Significant predictors (p<0.10) were combined in a multivariable model.

Results

In the adjusted model, independent predictors included receipt of flu vaccination, health insurance coverage and sexual health topic discussions with sons. Sons who had received a flu shot in the last two years more frequently received at least one dose of the vaccine (OR 1.82; 95% CI 1.45–2.26). Sons covered by private health insurance had decreased odds of HPV vaccination (OR 0.56 95% CI 0.37–0.83). Lastly, parents who had discussed sexual health topics with their sons were more likely to vaccinate (OR 1.61; 95% CI 1.37–1.89).

Conclusions

Male vaccination rates in the U.S. have increased, but males continue to be under-immunized. Utilization of health care is an important factor in HPV vaccine uptake; therefore, health care providers should use every contact as an opportunity to vaccinate. Communication about sexual health topics may provide a forum for parents and health care providers to have conversations about HPV vaccination as those more comfortable discussing these topics may also be more comfortable discussing HPV vaccination.  相似文献   

14.
Despite a high prevalence of oncogenic human papilloma virus (HPV) infection and cervical cancer mortality, HPV vaccination is not currently available in Mali. Knowledge of HPV and cervical cancer in Mali, and thereby vaccine readiness, may be limited. Research staff visited homes in a radial pattern from a central location to recruit adolescent females and males aged 12–17 years and men and women aged ≥18 years (N = 51) in a peri-urban village of Bamako, Mali. Participants took part in structured interviews assessing knowledge, attitudes, and practices related to HPV, cervical cancer, and HPV vaccination. We found low levels of HPV and cervical cancer knowledge. While only 2.0% of respondents knew that HPV is a sexually transmitted infection (STI), 100% said they would be willing to receive HPV vaccination and would like the HPV vaccine to be available in Mali. Moreover, 74.5% said they would vaccinate their child(ren) against HPV. Men were found to have significantly greater autonomy in the decision to vaccinate themselves than women and adolescents (p = 0.005), a potential barrier to be addressed by immunization campaigns. HPV vaccination would be highly acceptable if the vaccine became widely available in Bamako, Mali. This study demonstrates the need for a significant investment in health education if truly informed consent is to be obtained for HPV vaccination. Potential HPV vaccination campaigns should provide more information about HPV and the vaccine. Barriers to vaccination, including the significantly lower ability of the majority of the target population to autonomously decide to get vaccinated, must also be addressed in future HPV vaccine campaigns.  相似文献   

15.
The SARS-CoV-2 pandemic is a major concern all over the world and, as vaccines became available at the end of 2020, optimal vaccination strategies were subjected to intense investigation. Considering their critical role in reducing disease burden, the increasing demand outpacing production, and that most currently approved vaccines follow a two-dose regimen, the cost-effectiveness of delaying the second dose to increment the coverage of the population receiving the first dose is often debated. Finding the best solution is complex due to the trade-off between vaccinating more people with lower level of protection and guaranteeing higher protection to a fewer number of individuals. Here we present a novel extended age-structured SEIR mathematical model that includes a two-dose vaccination schedule with a between-doses delay modelled through delay differential equations and linear optimization of vaccination rates. By maintaining the minimum stock of vaccines under a given production rate, we evaluate the dose interval that minimizes the number of deaths. We found that the best strategy depends on an interplay between the vaccine production rate and the relative efficacy of the first dose. In the scenario of low first-dose efficacy, it is always better to vaccinate the second dose as soon as possible, while for high first-dose efficacy, the best strategy of time window depends on the production rate and also on second-dose efficacy provided by each type of vaccine. We also found that the rate of spread of the infection does not affect significantly the thresholds of the best window, but is an important factor in the absolute number of total deaths. These conclusions point to the need to carefully take into account both vaccine characteristics and roll-out speed to optimize the outcome of vaccination strategies.  相似文献   

16.
Chronic hepatitis B virus (HBV) infection may lead to liver cirrhosis, chronic liver disease, and liver cancer. Immunization rates are suboptimal among Asian Americans/Pacific Islanders (AAPIs), who remain disproportionately affected by these illnesses. We investigated socioecological factors affecting HBV prevention among 316 Vietnamese Americans in Atlanta, Georgia. Social and community support of HBV vaccination was associated with screening (OR=1.69, 95% CI [1.21,2.38]), vaccination (OR=1.89, [1.27,2.81]), and intent to vaccinate (OR=1.77, [1.13,2.78]). Misconceptions decreased screening likelihood (OR=0.67, [0.46,0.99]) and vaccination (OR=0.55, [0.35,0.86]). Those able to pay for medical treatment (OR=1.23, [1.01,1.50]) were also more likely immunized, and greater transportation access (OR=1.42, [1.07,1.87]) was associated with greater intention to vaccinate. Multi-level factors facilitated HBV vaccination in this population. Tailored, culturally appropriate communication strategies will positively influence immunization uptake.  相似文献   

17.
Annual vaccination against seasonal influenza viruses is recommended for certain individuals that have a high risk for complications resulting from infection with these viruses. Recently it was recommended in a number of countries including the USA to vaccinate all healthy children between 6 and 59 months of age as well. However, vaccination of immunologically naïve subjects against seasonal influenza may prevent the induction of heterosubtypic immunity against potentially pandemic strains of an alternative subtype, otherwise induced by infection with the seasonal strains.Here we show in a mouse model that the induction of protective heterosubtypic immunity by infection with a human A/H3N2 influenza virus is prevented by effective vaccination against the A/H3N2 strain. Consequently, vaccinated mice were no longer protected against a lethal infection with an avian A/H5N1 influenza virus. As a result H3N2-vaccinated mice continued to loose body weight after A/H5N1 infection, had 100-fold higher lung virus titers on day 7 post infection and more severe histopathological changes than mice that were not protected by vaccination against A/H3N2 influenza.The lack of protection correlated with reduced virus-specific CD8+ T cell responses after A/H5N1 virus challenge infection. These findings may have implications for the general recommendation to vaccinate all healthy children against seasonal influenza in the light of the current pandemic threat caused by highly pathogenic avian A/H5N1 influenza viruses.  相似文献   

18.
Accumulation of beta-amyloid (Aβ) is an important pathological event in Alzheimer’s disease (AD). It is now well known that vaccination against fibrillar Aβ prevents amyloid accumulation and preserves cognitive function in transgenic mouse models. To study the effect of vaccination against generic oligomer epitopes, Aβ oligomers, islet amyloid polypeptide oligomers, random peptide oligomer (3A), and Aβ fibrils were used to vaccinate 3xTg-AD, which develop a progressive accumulation of plaques and cognitive impairment. Subcutaneous administration of these antigens markedly reduced total plaque load (Aβ burden) and improved cognitive function in the 3xTg-AD mouse brains as compared to controls. We demonstrated that vaccination with this nonhuman amyloid oligomer generated high titers of specifically antibodies recognizing Aβ oligomers, which in turn inhibited accumulation of Aβ pathology in mice. In addition to amyloid plaques, another hallmark of AD is tau pathology. It was found that there was a significant decline in the level of hyper-phosphorylated tau following vaccination. We have previously shown that immunization with 3A peptide improves cognitive function and clears amyloid plaques in Tg2576 mice, which provides a novel strategy of AD therapy. Here, we have shown that vaccination with 3A peptide in 3xTg-AD mice not only clears amyloid plaques but also extensively clears abnormal tau in brain.  相似文献   

19.
Zhejiang Province was a high endemicity for hepatitis B disease in the 1990''s. A number of measures implemented since then have begun to control and prevent hepatitis B. In 1992, hepatitis B vaccine came on the market. In 2002, hepatitis B vaccine was included in the national Expanded Programme on Immunization (EPI). Between 2007 and 2010, catch-up vaccination was implemented for children under 15. Since 2010, vaccination guidelines for high-risk groups have also been adopted. This study evaluated the impact of these control and prevention strategies on acute hepatitis B notification rates from 2005 through 2013. Data from the National Notifiable Disease Reporting System (NNDRS) revealed a steady downward trend in notification rates of acute hepatitis B. The most dramatic decline occurred among pre-adults, highlighting the benefits of EPI''s policy of universal vaccination for children. However, the highest notification rates occurred among young adults of lower socio-economic status. These findings indicate the strong need to vaccinate young adults at risk for HBV infection as well as to collect risk-factor information in the NNDRS for monitoring and following up persons with acute hepatitis B.  相似文献   

20.
Vaccination campaigns for common childhood diseases   总被引:1,自引:0,他引:1  
Mathematical models exist for vaccination programs for common childhood diseases such as measles, chickenpox, polio, rubella, and mumps. In compartmental models, the population is divided into compartments containing susceptible, infected, and immune individuals, and the model also takes account of the age structure in the population. An equilibrium analysis is performed on the resulting partial differential equations to determine the vaccination program that just eliminates the disease. The stability of the resulting equilibrium solutions is also examined. Particular attention is paid to multistage vaccination strategies that vaccinate given proportions of susceptible individuals at various ages. Finally, some numerical results are analyzed to see what these vaccination coverages mean for certain common childhood diseases.  相似文献   

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