Impact of reduction in contact time activity of infected individuals on the dynamics and control of directly transmitted respiratory infections in SIR models.

This paper aims to study the impact of using an educational strategy on reducing the efforts needed to control respiratory transmitted infections represented by SIR models, taking into account heterogeneity in contacts between infected and non-infected individuals. Therefore, a new incidence function, in which the difference in contact time activity between infected and non-infected individuals is taken into account, is formulated. Equilibrium and stability analyses of the model have been carried out. The model has been extended to include the effect of herd immunity and the analysis showed that the higher the percent reduction P ˆ r in the contact-activity time of infected individuals is, the lower the critical vaccination coverage level p c required to eliminate the infection is, and therefore, the lower the infection's minimum elimination effort is. Another extension of the basic model to include a control strategy based on treating infected individuals at rate α with a maximum capacity treatment I has been considered. The equilibrium analysis showed the existence of multiple subcritical and supercritical endemic equilibria, while the stability analysis showed that the model exhibits a Hopf bifurcation. Simulations showed that the higher the maximum treatment capacity I is, the lower the value of the critical reduction in infected individuals' time activity P r ⋆ , at which a Hopf bifurcation is generated, is. Simulations with parameter values corresponding to the case of influenza A have been carried out.

Mathematical analysis of an SIR respiratory infection model with sex and gender disparity: special reference to influenza A.

The aim of this work is to study the impact of sex and gender disparity on the overall dynamics of influenza A virus infection and to explore the direct and indirect effect of influenza A mass vaccination. To this end, a deterministic SIR model has been formulated and throughly analysed, where the equilibrium and stability analyses have been explored. The impact of sex disparity (i.e., disparity in susceptibility and in recovery rate between females and males) on the disease outcome (i.e., the basic reproduction number R0 and the endemic prevalence of influenza in females and males) has been investigated. Mathematical and numerical analyses show that sex and gender disparities affect on the severity as well as the endemic prevalence of infection in both sexes. The analysis shows further that the efficacy of the vaccine for both sexes (e1&e2) and the response of the gender to mass-vaccination campaigns Ψ play a crucial role in influenza A containment and elimination process, where they impact significantly on the protection ratio as well as on the direct, indirect and total effect of vaccination on the burden of infection.

Modeling the Diet Dynamics of Children: the Roles of Socialization and the School Environment.

Childhood obesity is a health emergency in many parts of the world including the U.S. and, consequently, identifying local, regional or national intervention models capable, of altering the dynamics of obesity at scales that make a difference remains a challenge. The fact that consumption of healthful foods among most youth has yet to meet recommended nutritional standards highlights a lack of effective policies aimed at addressing the epidemic of obesity. Mathematical models are used to evaluate the roles of socialization and school environment on the diet dynamics of children. Data suggest that standard nutrition education programs may have, at best, minimal impact in altering diet dynamics at the population-level. Inclusion of peer influence (model as contagion) reinforced by the use of culturally-sensitive school menus (environmental disruption) may prove capable of modifying obesity enhancing diet dynamics; altering the diets of a significant (critical) proportion of youngsters. A framework is introduced to explore the value of behavior-based interventions and policies that account for the sociocultural environments of at risk communities. These models capture carefully choreographed scenarios to account for the fact that when dealing with diet-dynamics systems, thinking additively is not enough as it cannot account for the power of nonlinear effects.

Mathematical Analysis of an SIQR Influenza Model with Imperfect Quarantine.

The identification of mechanisms responsible for recurrent epidemic outbreaks, such as age structure, cross-immunity and variable delays in the infective classes, has challenged and fascinated epidemiologists and mathematicians alike. This paper addresses, motivated by mathematical work on influenza models, the impact of imperfect quarantine on the dynamics of SIR-type models. A susceptible-infectious-quarantine-recovered (SIQR) model is formulated with quarantined individuals altering the transmission dynamics process through their possibly reduced ability to generate secondary cases of infection. Mathematical and numerical analyses of the model of the equilibria and their stability have been carried out. Uniform persistence of the model has been established. Numerical simulations show that the model supports Hopf bifurcation as a function of the values of the quarantine effectiveness and other parameters. The upshot of this work is somewhat surprising since it is shown that SIQR model oscillatory behavior, as shown by multiple researchers, is in fact not robust to perturbations in the quarantine regime.

Modeling eating behaviors: The role of environment and positive food association learning via a Ratatouille effect.

Eating behaviors among a large population of children are studied as a dynamic process driven by nonlinear interactions in the sociocultural school environment. The impact of food association learning on diet dynamics, inspired by a pilot study conducted among Arizona children in Pre-Kindergarten to 8th grades, is used to build simple population-level learning models. Qualitatively, mathematical studies are used to highlight the possible ramifications of instruction, learning in nutrition, and health at the community level. Model results suggest that nutrition education programs at the population-level have minimal impact on improving eating behaviors, findings that agree with prior field studies. Hence, the incorporation of food association learning may be a better strategy for creating resilient communities of healthy and non-healthy eaters. A Ratatouille effect can be observed when food association learners become food preference learners, a potential sustainable behavioral change, which in turn, may impact the overall distribution of healthy eaters. In short, this work evaluates the effectiveness of population-level intervention strategies and the importance of institutionalizing nutrition programs that factor in economical, social, cultural, and environmental elements that mesh well with the norms and values in the community.

Modeling the role of healthcare access inequalities in epidemic outcomes.

Urban areas, with large and dense populations, offer conditions that favor the emergence and spread of certain infectious diseases. One common feature of urban populations is the existence of large socioeconomic inequalities which are often mirrored by disparities in access to healthcare. Recent empirical evidence suggests that higher levels of socioeconomic inequalities are associated with worsened public health outcomes, including higher rates of sexually transmitted diseases (STD's) and lower life expectancy. However, the reasons for these associations are still speculative. Here we formulate a mathematical model to study the effect of healthcare disparities on the spread of an infectious disease that does not confer lasting immunity, such as is true of certain STD's. Using a simple epidemic model of a population divided into two groups that differ in their recovery rates due to different levels of access to healthcare, we find that both the basic reproductive number (R0) of the disease and its endemic prevalence are increasing functions of the disparity between the two groups, in agreement with empirical evidence. Unexpectedly, this can be true even when the fraction of the population with better access to healthcare is increased if this is offset by reduced access within the disadvantaged group. Extending our model to more than two groups with different levels of access to healthcare, we find that increasing the variance of recovery rates among groups, while keeping the mean recovery rate constant, also increases R0 and disease prevalence. In addition, we show that these conclusions are sensitive to how we quantify the inequalities in our model, underscoring the importance of basing analyses on appropriate measures of inequalities. These insights shed light on the possible impact that increasing levels of inequalities in healthcare access can have on epidemic outcomes, while offering plausible explanations for the observed empirical patterns.

Case fatality models for epidemics in growing populations.

The asymptotically homogeneous SIR model of Thieme (1992) for growing populations, with incidence depending in a general way on total population size, is reconsidered with respect to other parameterizations that give clear insight into epidemiological relevant relations and thresholds. One important feature of the present approach is case fatality as opposed to differential mortality. Although case fatality models and differential mortality models are equivalent via a transformation in parameter space, the underlying ideas and the dynamic behaviors are different, e.g. the basic reproduction number depends on differential mortality but not on case fatality. The persistent distributions and exponents of growth of infected solutions are computed and discussed in terms of the parameters. The notion of asymptotically exponentially growing state (as opposed to stationary state or exponential solution) coined by Thieme is interpreted in terms of stability theory. Of some interest are limiting cases of models without recovery where two infected solutions exist.

Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis.

The aim of this paper is to study the impact of introducing a partially protective vaccine on the dynamics of infection in SIRS models where primary and secondary infections are distinguished. We investigate whether a public health strategy based solely on vaccinating a proportion of newborns can lead to an effective control of the disease. In addition to carrying out the qualitative analysis, the findings are further explained by numerical simulations. The model exhibits backward bifurcation for certain values of the parameters. In these cases the standard basic reproduction number (obtained by inspection of the uninfected state) is not significant. The key threshold is the reinfection level which depends on the relative transmissibility (susceptibility) of secondary, with respect to primary, infected (susceptible) individuals and the relative loss of immunity of vaccinated, with respect to recovered, individuals. If one or all of these ratios decrease, then the threshold increases which increases the possibility to contain the infection by vaccination. The analysis shows further that symptomatic infections can be eliminated by vaccination solely.

Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009.

BACKGROUND: In many parts of the world, the exponential growth rate of infections during the initial epidemic phase has been used to make statistical inferences on the reproduction number, R, a summary measure of the transmission potential for the novel influenza A (H1N1) 2009. The growth rate at the initial stage of the epidemic in Japan led to estimates for R in the range 2.0 to 2.6, capturing the intensity of the initial outbreak among school-age children in May 2009. METHODS: An updated estimate of R that takes into account the epidemic data from 29 May to 14 July is provided. An age-structured renewal process is employed to capture the age-dependent transmission dynamics, jointly estimating the reproduction number, the age-dependent susceptibility and the relative contribution of imported cases to secondary transmission. Pitfalls in estimating epidemic growth rates are identified and used for scrutinizing and re-assessing the results of our earlier estimate of R. RESULTS: Maximum likelihood estimates of R using the data from 29 May to 14 July ranged from 1.21 to 1.35. The next-generation matrix, based on our age-structured model, predicts that only 17.5% of the population will experience infection by the end of the first pandemic wave. Our earlier estimate of R did not fully capture the population-wide epidemic in quantifying the next-generation matrix from the estimated growth rate during the initial stage of the pandemic in Japan. CONCLUSIONS: In order to quantify R from the growth rate of cases, it is essential that the selected model captures the underlying transmission dynamics embedded in the data. Exploring additional epidemiological information will be useful for assessing the temporal dynamics. Although the simple concept of R is more easily grasped by the general public than that of the next-generation matrix, the matrix incorporating detailed information (e.g., age-specificity) is essential for reducing the levels of uncertainty in predictions and for assisting public health policymaking. Model-based prediction and policymaking are best described by sharing fundamental notions of heterogeneous risks of infection and death with non-experts to avoid potential confusion and/or possible misuse of modelling results.

On the eradicability of infections with partially protective vaccination in models with backward bifurcation.

The SIS model of Hadeler and Castillo-Chavez [9] with a constant transfer rate of susceptibles into a partially protected state has been modified to take into account vaccination at birth. The model shows backward bifurcation (existence of multiple endemic stationary states) for certain values of parameters. Parameter values ensuring the existence and nonexistence of endemic equilibria have been discussed. Local and global stability of equilibria have been investigated. The minimum effort required to eradicate the infection has been determined.

The minimum effort required to eradicate infections in models with backward bifurcation.

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (mu/alpha) this model shows backward bifurcations, where mu is the death rate and alpha is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant beta. This eradication effort is explicitly expressed in terms of alpha,r, and mu As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R0 is smaller than the eradication effort for r > 1 + (mu/alpha) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.