SIR-SVS epidemic models with continuous and impulsive vaccination strategies

Vaccination is important for the control of some infectious diseases. This paper considers two SIR-SVS epidemic models with vaccination, where it is assumed that the vaccination for the newborns is continuous in the two models, and that the vaccination for the susceptible individuals is continuous and impulsive, respectively. The basic reproduction numbers of two models, determining whether the disease dies out or persists eventually, are all obtained. For the model with continuous vaccination for the susceptibles, the global stability is proved by using the Lyapunov function. Especially for the endemic equilibrium, to prove the negative definiteness of the derivative of the Lyapunov function for all the feasible values of parameters, it is expressed in three different forms for all the feasible values of parameters. For the model with pulse vaccination for the susceptibles, the global stability of the disease free periodic solution is proved by the comparison theorem of impulsive differential equations. At last, the effect of vaccination strategies on the control of the disease transmission is discussed, and two types of vaccination strategies for the susceptible individuals are also compared.     

Vaccination in density-dependent epidemic models

This paper examines the effect of vaccination for an epidemic model where the death rate depends on the number of individuals in the population. The basic model which is described is based on measles or other childhood diseases in developing countries or viral diseases such as rabies in animal populations. An equilibrium analysis of the model and the local stability of small perturbations about the equilibrium values are discussed. The biological implications of these results are examined and similar results presented for modifications of the basic model.     

The reinfection threshold

Thresholds in transmission are responsible for critical changes in infectious disease epidemiology. The epidemic threshold indicates whether infection invades a totally susceptible population. The reinfection threshold indicates whether self-sustained transmission occurs in a population that has developed a degree of partial immunity to the pathogen (by previous infection or vaccination). In models that combine susceptible and partially immune individuals, the reinfection threshold is technically not a bifurcation of equilibria as correctly pointed out by Breban and Blower. However, we show that a branch of equilibria to a reinfection submodel bifurcates from the disease-free equilibrium as transmission crosses this threshold. Consequently, the full model indicates that levels of infection increase by two orders of magnitude and the effect of mass vaccination becomes negligible as transmission increases across the reinfection threshold.     

Analytical threshold and stability results on age-structured epidemic models with vaccination

This paper examines mathematical models for common childhood diseases such as measles and rubella and in particular the use of such models to predict whether or not an epidemic pattern of regular recurrent disease incidence will occur. We use age-structured compartmental models which divide the population amongst whom the disease is spreading into classes and use partial differential equations to model the spread of the disease. This paper is particularly concerned with an analytical investigation of the effects of different types of vaccination schemes. We examine possible equilibria and determine the stability of small oscillations about these equilibria. The results are important in predicting the long-term overall level of incidence of disease, in designing immunisation programs and in describing the variations of the incidence of disease about this equilibrium level.     

Some threshold and stability results for epidemic models with a density-dependent death rate.

Most classical models for infectious diseases assume that the birth and death rates of individuals and the meeting rates between susceptible and infected individuals do not depend on the total number of individuals in the population. While these assumptions are valid in some situations they are less valid in others. For example, for diseases in animal an insects populations competition for scarce resources might well mean that the death rate depends on the number of individuals. The present paper examines two epidemic models where the death rate is density dependent. For each model the possible equilibrium levels of disease incidence are determined and the stability of these equilibrium levels to small perturbations is discussed. The biological interpretation of these results is presented together with the results of some numerical simulations.     

SVIR epidemic models with vaccination strategies

Vaccination is important for the elimination of infectious diseases. To finish a vaccination process, doses usually should be taken several times and there must be some fixed time intervals between two doses. The vaccinees (susceptible individuals who have started the vaccination process) are different from both susceptible and recovered individuals. Considering the time for them to obtain immunity and the possibility for them to be infected before this, two SVIR models are established to describe continuous vaccination strategy and pulse vaccination strategy (PVS), respectively. It is shown that both systems exhibit strict threshold dynamics which depend on the basic reproduction number. If this number is below unity, the disease can be eradicated. And if it is above unity, the disease is endemic in the sense of global asymptotical stability of a positive equilibrium for continuous vaccination strategy and disease permanence for PVS. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before this is neglected, this condition disappears and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.     

A discrete-time model with vaccination for a measles epidemic.

A discrete-time, age-independent SIR-type epidemic model is formulated and analyzed. The effects of vaccination are also included in the model. Three mathematically important properties are verified for the model: solutions are nonnegative, the population size is time-invariant, and the epidemic concludes with all individuals either remaining susceptible or becoming immune (a property typical of SIR models). The model is applied to a measles epidemic on a university campus. The simulated results are in good agreement with the actual data if it is assumed that the population mixes nonhomogeneously. The results of the simulations indicate that a rate of immunity greater than 98% may be required to prevent an epidemic in a university population. The model has applications to other contagious diseases of SIR type. Furthermore, the simulated results of the model can easily be compared to data, and the effects of a vaccination program can be examined.     

Population dynamic interference among childhood diseases.

Epidemiologists usually study the interaction between a host population and one parasitic infection. However, different parasite species effectively compete, in an ecological sense, for the same finite group of susceptible hosts, so there may be an indirect effect on the population dynamics of one disease due to epidemics of another. In human populations, recovery from any serious infection is normally preceded by a period of convalescence, during which infected individuals stay at home and are effectively shielded from exposure to other infectious diseases. We present a model for the dynamics of two infectious diseases, incorporating a temporary removal of susceptibles. We use this model to explore population-level consequences of a temporary insusceptibility in childhood diseases, the dynamics of which are partly driven by differences in contact rates in and out of school terms. Significant population dynamic interference is predicted and cannot be dismissed in the limited case-study data available for measles and whooping cough in England before the vaccination era.     

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.     

Epidemiology of bacterial meningitis in Lombardy Region in the period 2000-2006

Epidemiological study of bacterial meningitis, particularly those forms susceptible to vaccination, is an indispensable tool in choosing vaccination strategies. Lombardy Region, where approximately 20% of the Italian population resides, has conducted an in-depth study on invasive meningococcal diseases and pneumococcal meningitis using available health statistics and performing a control of single cases to achieve complete knowledge of the phenomenon. The results, relating to the period 2000-2006, indicate limited incidence rates both for the general population and childhood age groups that do not justify hypotheses of a universal offering of the specific vaccines. However, the healthcare system must equip itself with additional tools for the identification of the serogroups and serotypes responsible for disease as a further support for possible choices modifying the current vaccination policies.     

Uniqueness of Nash equilibrium in vaccination games

One crucial condition for the uniqueness of Nash equilibrium set in vaccination games is that the attack ratio monotonically decreases as the vaccine coverage level increasing. We consider several deterministic vaccination models in homogeneous mixing population and in heterogeneous mixing population. Based on the final size relations obtained from the deterministic epidemic models, we prove that the attack ratios can be expressed in terms of the vaccine coverage levels, and also prove that the attack ratios are decreasing functions of vaccine coverage levels. Some thresholds are presented, which depend on the vaccine efficacy. It is proved that for vaccination games in homogeneous mixing population, there is a unique Nash equilibrium for each game.     

Stochastic two-group models with transmission dependent on host infectivity or susceptibility

ABSTRACT

Stochastic epidemic models with two groups are formulated and applied to emerging and re-emerging infectious diseases. In recent emerging diseases, disease spread has been attributed to superspreaders, highly infectious individuals that infect a large number of susceptible individuals. In some re-emerging infectious diseases, disease spread is attributed to waning immunity in susceptible hosts. We apply a continuous-time Markov chain (CTMC) model to study disease emergence or re-emergence from different groups, where the transmission rates depend on either the infectious host or the susceptible host. Multitype branching processes approximate the dynamics of the CTMC model near the disease-free equilibrium and are used to estimate the probability of a minor or a major epidemic. It is shown that the probability of a major epidemic is greater if initiated by an individual from the superspreader group or by an individual from the highly susceptible group. The models are applied to Severe Acute Respiratory Syndrome and measles.     

Differential susceptibility epidemic models

We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either constant or proportional to the total population. We derive an explicit formula for the reproductive number of infection for each model by investigating the local stability of the infection-free equilibrium. We further prove that the infection-free equilibrium of each model is globally asymptotically stable by qualitative analysis of the dynamics of the model system and by utilizing an appropriately chosen Liapunov function. We show that if the reproductive number is greater than one, then there exists a unique endemic equilibrium for all of the DS models studied in this paper. We prove that the endemic equilibrium is locally asymptotically stable for the models with no disease-induced mortality and the models with contact numbers proportional to the total population. We also provide sufficient conditions for the stability of the endemic equilibrium for other situations. We briefly discuss applications of the DS models to optimal vaccine strategies and the connections between the DS models and predator-prey models with multiple prey populations or host-parasitic interaction models with multiple hosts are also given.This research was partially supported by the Department of Energy under contracts W-7405-ENG-36 and the Applied Mathematical Sciences Program KC-07-01-01.     

Erratic flu vaccination emerges from short-sighted behavior in contact networks

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.     

Individual Energetics and the Equilibrium Demography of Structured Populations

This paper considers the relationship between the demographic mechanisms of population control and the energetics of the individuals who comprise the population. We examine the equilibrium properties of a class of structured population models in which individuals compete for some environmental resource (such as food) and demonstrate that population demography is independent of the nature of the feedback loop which establishes the equilibrium state. We thus derive general insights into the influence exerted by individual energetic and allocation strategies on population average demographic characteristics. We show that models of individual energetics which produce apparently very similar predictions at the individual level can result in very different behaviour at the population level. In addition, we observe that different models of individual mortality can imply marked differences in population demography and that the common assumption of constant mortality can be responsible for potentially unrealistic model behaviour. Our results emphasise the substantial data requirements for parameterising and testing individual-based models.     

The impact of vaccine side effects on the natural history of immunization programmes: an imitation-game approach

When the incidence and prevalence of most common vaccine preventable childhood infectious diseases are constantly low, as is the case in many industrialized countries, the incidence of vaccine-associated side effects might become a key determinant in vaccine demand. We study an SIR transmission model with dynamic vaccine demand based on an imitation mechanism where the perceived risk of vaccination is modelled as a function of the incidence of vaccine side effects. The model shows some important differences compared to previous game dynamic models of vaccination, and allows noteworthy inferences as regards both the past and future lifetime of vaccination programmes. In particular it is suggested that a huge disproportion between the perceived risk of disease and vaccination is necessary in order to achieve high coverages. This disproportion is further increased in highly industrialised countries. Such considerations represent serious challenges for future vaccination programmes.     

Exciting chaos with noise: unexpected dynamics in epidemic outbreaks

 In this paper, we identify a mechanism for chaos in the presence of noise. In a study of the SEIR model, which predicts epidemic outbreaks in childhood diseases, we show how chaotic dynamics can be attained by adding stochastic perturbations at parameters where chaos does not exist apriori. Data recordings of epidemics in childhood diseases are still argued as deterministic chaos. There also exists noise due to uncertainties in the contact parameters between those who are susceptible and those who are infected, as well as random fluctuations in the population. Although chaos has been found in deterministic models, it only occurs in parameter regions that require a very large population base or other large seasonal forcing. Our work identifies the mechanism whereby chaos can be induced by noise for realistic parameter regions of the deterministic model where it does not naturally occur. Received: 13 October 2000 / Revised version: 15 May 2001 / Published online: 7 December 2001     

Do Antenatal Parasite Infections Devalue Childhood Vaccination?

On a global basis, both potent vaccine efficacy and high vaccine coverage are necessary to control and eliminate vaccine-preventable diseases. Emerging evidence from animal and human studies suggest that neglected tropical diseases (NTDs) significantly impair response to standard childhood immunizations. A review of efficacy and effectiveness studies of vaccination among individuals with chronic parasitic infections was conducted, using PUBMED database searches and analysis of data from the authors'' published and unpublished studies. Both animal models and human studies suggest that chronic trematode, nematode, and protozoan infections can result in decreased vaccine efficacy. Among pregnant women, who in developing countries are often infected with multiple parasites, soluble parasite antigens have been shown to cross the placenta and prime or tolerize fetal immune responses. As a result, antenatal infections can have a significant impact on later vaccine responses. Acquired childhood parasitic infections, most commonly malaria, can also affect subsequent immune response to vaccination. Additional data suggest that antiparasite therapy can improve the effectiveness of several human vaccines. Emerging evidence demonstrates that both antenatal and childhood parasitic infections alter levels of protective immune response to routine vaccinations. Successful antiparasite treatment may prevent immunomodulation caused by parasitic antigens during pregnancy and early childhood and may improve vaccine efficacy. Future research should highlight the varied effects that different parasites (alone and in combination) can have on human vaccine-related immunity. To optimize vaccine effectiveness in developing countries, better control of chronic NTDs may prove imperative.     

The impact of imitation on vaccination behavior in social contact networks

Previous game-theoretic studies of vaccination behavior typically have often assumed that populations are homogeneously mixed and that individuals are fully rational. In reality, there is heterogeneity in the number of contacts per individual, and individuals tend to imitate others who appear to have adopted successful strategies. Here, we use network-based mathematical models to study the effects of both imitation behavior and contact heterogeneity on vaccination coverage and disease dynamics. We integrate contact network epidemiological models with a framework for decision-making, within which individuals make their decisions either based purely on payoff maximization or by imitating the vaccination behavior of a social contact. Simulations suggest that when the cost of vaccination is high imitation behavior may decrease vaccination coverage. However, when the cost of vaccination is small relative to that of infection, imitation behavior increases vaccination coverage, but, surprisingly, also increases the magnitude of epidemics through the clustering of non-vaccinators within the network. Thus, imitation behavior may impede the eradication of infectious diseases. Calculations that ignore behavioral clustering caused by imitation may significantly underestimate the levels of vaccination coverage required to attain herd immunity.     

Do Maternal Knowledge and Attitudes towards Childhood Immunizations in Rural Uganda Correlate with Complete Childhood Vaccination?

Improving childhood vaccination coverage and timeliness is a key health policy objective in many developing countries such as Uganda. Of the many factors known to influence uptake of childhood immunizations in under resourced settings, parents’ understanding and perception of childhood immunizations has largely been overlooked. The aims of this study were to survey mothers’ knowledge and attitudes towards childhood immunizations and then determine if these variables correlate with the timely vaccination coverage of their children. From September to December 2013, we conducted a cross-sectional survey of 1,000 parous women in rural Sheema district in southwest Uganda. The survey collected socio-demographic data and knowledge and attitudes towards childhood immunizations. For the women with at least one child between the age of one month and five years who also had a vaccination card available for the child (N = 302), the vaccination status of this child was assessed. 88% of these children received age-appropriate, on-time immunizations. 93.5% of the women were able to state that childhood immunizations protect children from diseases. The women not able to point this out were significantly more likely to have an under-vaccinated child (PR 1.354: 95% CI 1.018–1.802). When asked why vaccination rates may be low in their community, the two most common responses were “fearful of side effects” and “ignorance/disinterest/laziness” (44% each). The factors influencing caregivers’ demand for childhood immunizations vary widely between, and also within, developing countries. Research that elucidates local knowledge and attitudes, like this study, allows for decisions and policy pertaining to vaccination programs to be more effective at improving child vaccination rates.