Building matrix population models when individuals are non-identifiable.

Matrix Population Models (MPM) are among the most widely used tools in ecology and evolution. These models consider the life cycle of an individual as composed by states to construct a matrix containing the likelihood of transitions between these states as well as sexual and/or asexual per-capita offspring contributions. When individuals are identifiable one can parametrize an MPM based on survival and fertility data and average development times for every state, but some of this information is absent or incomplete for non-cohort data, or for cohort data when individuals are not identifiable. Here we introduce a simple procedure for the parameterization of an MPM that can be used with cohort data when individuals are non-identifiable; among other aspects our procedure is a novelty in that it does not require information on stage development (or stage residence) times, which current procedures require to be estimated externally, and it is a frequent source of error. We exemplify the procedure with a laboratory cohort dataset from Eratyrus mucronatus (Reduviidae, Triatominae). We also show that even if individuals are identifiable and the duration of each stage is externally estimated with no error, our procedure is simpler to use and yields the same MPM parameter estimates.

Perspectives on the role of mobility, behavior, and time scales in the spread of diseases.

The dynamics, control, and evolution of communicable and vector-borne diseases are intimately connected to the joint dynamics of epidemiological, behavioral, and mobility processes that operate across multiple spatial, temporal, and organizational scales. The identification of a theoretical explanatory framework that accounts for the pattern regularity exhibited by a large number of host-parasite systems, including those sustained by host-vector epidemiological dynamics, is but one of the challenges facing the coevolving fields of computational, evolutionary, and theoretical epidemiology. Host-parasite epidemiological patterns, including epidemic outbreaks and endemic recurrent dynamics, are characteristic to well-identified regions of the world; the result of processes and constraints such as strain competition, host and vector mobility, and population structure operating over multiple scales in response to recurrent disturbances (like El Niño) and climatological and environmental perturbations over thousands of years. It is therefore important to identify and quantify the processes responsible for observed epidemiological macroscopic patterns: the result of individual interactions in changing social and ecological landscapes. In this perspective, we touch on some of the issues calling for the identification of an encompassing theoretical explanatory framework by identifying some of the limitations of existing theory, in the context of particular epidemiological systems. Fostering the reenergizing of research that aims at disentangling the role of epidemiological and socioeconomic forces on disease dynamics, better understood as complex adaptive systems, is a key aim of this perspective.

The role of mobility and health disparities on the transmission dynamics of Tuberculosis.

BACKGROUND: The transmission dynamics of Tuberculosis (TB) involve complex epidemiological and socio-economical interactions between individuals living in highly distinct regional conditions. The level of exogenous reinfection and first time infection rates within high-incidence settings may influence the impact of control programs on TB prevalence. The impact that effective population size and the distribution of individuals' residence times in different patches have on TB transmission and control are studied using selected scenarios where risk is defined by the estimated or perceive first time infection and/or exogenous re-infection rates. METHODS: This study aims at enhancing the understanding of TB dynamics, within simplified, two patch, risk-defined environments, in the presence of short term mobility and variations in reinfection and infection rates via a mathematical model. The modeling framework captures the role of individuals' 'daily' dynamics within and between places of residency, work or business via the average proportion of time spent in residence and as visitors to TB-risk environments (patches). As a result, the effective population size of Patch i (home of i-residents) at time t must account for visitors and residents of Patch i, at time t. RESULTS: The study identifies critical social behaviors mechanisms that can facilitate or eliminate TB infection in vulnerable populations. The results suggest that short-term mobility between heterogeneous patches contributes to significant overall increases in TB prevalence when risk is considered only in terms of direct new infection transmission, compared to the effect of exogenous reinfection. Although, the role of exogenous reinfection increases the risk that come from large movement of individuals, due to catastrophes or conflict, to TB-free areas. CONCLUSIONS: The study highlights that allowing infected individuals to move from high to low TB prevalence areas (for example via the sharing of treatment and isolation facilities) may lead to a reduction in the total TB prevalence in the overall population. The higher the population size heterogeneity between distinct risk patches, the larger the benefit (low overall prevalence) under the same "traveling" patterns. Policies need to account for population specific factors (such as risks that are inherent with high levels of migration, local and regional mobility patterns, and first time infection rates) in order to be long lasting, effective and results in low number of drug resistant cases.

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.

Per capita incidence of sexually transmitted infections increases systematically with urban population size: a cross-sectional study.

OBJECTIVES: Rampant urbanisation rates across the globe demand that we improve our understanding of how infectious diseases spread in modern urban landscapes, where larger and more connected host populations enhance the thriving capacity of certain pathogens. METHODS: A data-driven approach is employed to study the ability of sexually transmitted diseases (STDs) to thrive in urban areas. The conduciveness of population size of urban areas and their socioeconomic characteristics are used as predictors of disease incidence, using confirmed-case data on STDs in the USA as a case study. RESULTS: A superlinear relation between STD incidence and urban population size is found, even after controlling for various socioeconomic aspects, suggesting that doubling the population size of a city results in an expected increase in STD incidence larger than twofold, provided that all other socioeconomic aspects remain fixed. Additionally, the percentage of African-Americans, income inequalities, education and per capita income are found to have a significant impact on the incidence of each of the three STDs studied. CONCLUSIONS: STDs disproportionately concentrate in larger cities. Hence, larger urban areas merit extra prevention and treatment efforts, especially in low-income and middle-income countries where urbanisation rates are higher.

The role of residence times in two-patch dengue transmission dynamics and optimal strategies.

The reemergence and geographical dispersal of vector-borne diseases challenge global health experts around the world and in particular, dengue poses increasing difficulties in the Americas, due in part to explosive urban and semi-urban growth, increases of within and between region mobility, the absence of a vaccine, and the limited resources available for public health services. In this work, a simple deterministic two-patch model is introduced to assess the impact of dengue transmission dynamics in heterogeneous environments. The two-patch system models the movement (e.g. urban versus rural areas residence times) of individuals between and within patches/environments using residence-time matrices with entries that budget within and between host patch relative residence times, under the assumption that only the human budgets their residence time across regions. Three scenarios are considered: (i) resident hosts in Patch i visit patch j, where i≠j but not the other way around, a scenario referred to as unidirectional motion; (ii) symmetric bi-directional motion; and (iii) asymmetric bi-directional motion. Optimal control theory is used to identify and evaluate patch-specific control measures aimed at reducing dengue prevalence in humans and vectors at a minimal cost. Optimal policies are computed under different residence-matrix configurations mentioned above as well as transmissibility scenarios characterized by the magnitude of the basic reproduction number. Optimal patch-specific polices can ameliorate the impact of epidemic outbreaks substantially when the basic reproduction number is moderate. The final patch-specific epidemic size variation increases as the residence time matrix moves away from the symmetric case (asymmetry). As expected, the patch where individuals spend most of their time or in the patch where transmissibility is higher tend to support larger patch-specific final epidemic sizes. Hence, focusing on intervention that target areas where individuals spend "most" time or where transmissibility is higher turn out to be optimal. Therefore, reducing traffic is likely to take a host-vector system into the world of manageable outbreaks.

Resource consumption, sustainability, and cancer.

Preserving a system's viability in the presence of diversity erosion is critical if the goal is to sustainably support biodiversity. Reduction in population heterogeneity, whether inter- or intraspecies, may increase population fragility, either decreasing its ability to adapt effectively to environmental changes or facilitating the survival and success of ordinarily rare phenotypes. The latter may result in over-representation of individuals who may participate in resource utilization patterns that can lead to over-exploitation, exhaustion, and, ultimately, collapse of both the resource and the population that depends on it. Here, we aim to identify regimes that can signal whether a consumer-resource system is capable of supporting viable degrees of heterogeneity. The framework used here is an expansion of a previously introduced consumer-resource type system of a population of individuals classified by their resource consumption. Application of the Reduction Theorem to the system enables us to evaluate the health of the system through tracking both the mean value of the parameter of resource (over)consumption, and the population variance, as both change over time. The article concludes with a discussion that highlights applicability of the proposed system to investigation of systems that are affected by particularly devastating overly adapted populations, namely cancerous cells. Potential intervention approaches for system management are discussed in the context of cancer therapies.

SIS and SIR Epidemic Models Under Virtual Dispersal.

We develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general n-patch SIS model whose basic reproduction number [Formula: see text] is computed as a function of a patch residence-time matrix [Formula: see text]. Our analysis implies that the resulting n-patch SIS model has robust dynamics when patches are strongly connected: There is a unique globally stable endemic equilibrium when [Formula: see text], while the disease-free equilibrium is globally stable when [Formula: see text]. Our further analysis indicates that the dispersal behavior described by the residence-time matrix [Formula: see text] has profound effects on the disease dynamics at the single patch level with consequences that proper dispersal behavior along with the local environmental risk can either promote or eliminate the endemic in particular patches. Our work highlights the impact of residence-time matrix if the patches are not strongly connected. Our framework can be generalized in other endemic and disease outbreak models. As an illustration, we apply our framework to a two-patch SIR single-outbreak epidemic model where the process of disease invasion is connected to the final epidemic size relationship. We also explore the impact of disease-prevalence-driven decision using a phenomenological modeling approach in order to contrast the role of constant versus state-dependent [Formula: see text] on disease dynamics.

Adaptive human behavior in epidemiological models.

The science and management of infectious disease are entering a new stage. Increasingly public policy to manage epidemics focuses on motivating people, through social distancing policies, to alter their behavior to reduce contacts and reduce public disease risk. Person-to-person contacts drive human disease dynamics. People value such contacts and are willing to accept some disease risk to gain contact-related benefits. The cost-benefit trade-offs that shape contact behavior, and hence the course of epidemics, are often only implicitly incorporated in epidemiological models. This approach creates difficulty in parsing out the effects of adaptive behavior. We use an epidemiological-economic model of disease dynamics to explicitly model the trade-offs that drive person-to-person contact decisions. Results indicate that including adaptive human behavior significantly changes the predicted course of epidemics and that this inclusion has implications for parameter estimation and interpretation and for the development of social distancing policies. Acknowledging adaptive behavior requires a shift in thinking about epidemiological processes and parameters.

A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness.

The study of the dynamics of human infectious disease using deterministic models is typically carried out under the assumption that a critical mass of individuals is available and involved in the transmission process. However, in the study of animal disease dynamics where demographic considerations often play a significant role, this assumption must be weakened. Models of the dynamics of animal populations often naturally assume that the presence of a minimal number of individuals is essential to avoid extinction. In the ecological literature, this a priori requirement is commonly incorporated as an Allee effect. The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion in as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.

Discrete epidemic models with arbitrary stage distributions and applications to disease control.

W.O. Kermack and A.G. McKendrick introduced in their fundamental paper, A Contribution to the Mathematical Theory of Epidemics, published in 1927, a deterministic model that captured the qualitative dynamic behavior of single infectious disease outbreaks. A Kermack­McKendrick discrete-time general framework, motivated by the emergence of a multitude of models used to forecast the dynamics of epidemics, is introduced in this manuscript. Results that allow us to measure quantitatively the role of classical and general distributions on disease dynamics are presented. The case of the geometric distribution is used to evaluate the impact of waiting-time distributions on epidemiological processes or public health interventions. In short, the geometric distribution is used to set up the baseline or null epidemiological model used to test the relevance of realistic stage-period distribution on the dynamics of single epidemic outbreaks. A final size relationship involving the control reproduction number, a function of transmission parameters and the means of distributions used to model disease or intervention control measures, is computed. Model results and simulations highlight the inconsistencies in forecasting that emerge from the use of specific parametric distributions. Examples, using the geometric, Poisson and binomial distributions, are used to highlight the impact of the choices made in quantifying the risk posed by single outbreaks and the relative importance of various control measures.

Challenges in cybersecurity: Lessons from biological defense systems.

Defending against novel, repeated, or unpredictable attacks, while avoiding attacks on the 'self', are the central problems of both mammalian immune systems and computer systems. Both systems have been studied in great detail, but with little exchange of information across the different disciplines. Here, we present a conceptual framework for structured comparisons across the fields of biological immunity and cybersecurity, by framing the context of defense, considering different (combinations of) defensive strategies, and evaluating defensive performance. Throughout this paper, we pose open questions for further exploration. We hope to spark the interdisciplinary discovery of general principles of optimal defense, which can be understood and applied in biological immunity, cybersecurity, and other defensive realms.

SPREADING SPEEDS AND TRAVELING WAVES FOR NON-COOPERATIVE INTEGRO-DIFFERENCE SYSTEMS.

The study of spatially explicit integro-difference systems when the local population dynamics are given in terms of discrete-time generations models has gained considerable attention over the past two decades. These nonlinear systems arise naturally in the study of the spatial dispersal of organisms. The brunt of the mathematical research on these systems, particularly, when dealing with cooperative systems, has focused on the study of the existence of traveling wave solutions and the characterization of their spreading speed. Here, we characterize the minimum propagation (spreading) speed, via the convergence of initial data to wave solutions, for a large class of non cooperative nonlinear systems of integro-difference equations. The spreading speed turns out to be the slowest speed from a family of non-constant traveling wave solutions. The applicability of these theoretical results is illustrated through the explicit study of an integro-difference system with local population dynamics governed by Hassell and Comins' non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This paper is dedicated to Avner Friedman as we celebrate his immense contributions to the fields of partial differential equations, integral equations, mathematical biology, industrial mathematics and applied mathematics in general. His leadership in the mathematical sciences and his mentorship of students and friends over several decades has made a huge difference in the personal and professional lives of many, including both of us.

Optimal control for pandemic influenza: the role of limited antiviral treatment and isolation.

The implementation of optimal control strategies involving antiviral treatment and/or isolation measures can reduce significantly the number of clinical cases of influenza. Pandemic-level control measures must be carefully assessed specially in resource-limited situations. A model for the transmission dynamics of influenza is used to evaluate the impact of isolation and/or antiviral drug delivery measures during an influenza pandemic. Five pre-selected control strategies involving antiviral treatment and isolation are tested under the "unlimited" resource assumption followed by an exploration of the impact of these "optimal" policies when resources are limited in the context of a 1918-type influenza pandemic scenario. The implementation of antiviral treatment at the start of a pandemic tends to reduce the magnitude of epidemic peaks, spreading the maximal impact of an outbreak over an extended window in time. Hence, the controls' timing and intensity can reduce the pressures placed on the health care infrastructure by a pandemic reducing the stress put on the system during epidemic peaks. The role of isolation strategies is highlighted in this study particularly when access to antiviral resources is limited.

Static behavioral effects on gonorrhea transmission dynamics in a MSM population.

An SIS/SAS model of gonorrhea transmission in a population of highly active men-having-sex-with-men (MSM) is presented in this paper to study the impact of safe behavior on the dynamics of gonorrhea prevalence. Safe behaviors may fall into two categories-prevention and self-awareness. Prevention will be modeled via consistent condom use and self-awareness via STD testing frequency. Stability conditions for the disease free equilibrium and endemic equilibrium are determined along with a complete analysis of global dynamics. The control reproductive number is used as a means for measuring the effect of changes to model parameters on the prevalence of the disease. We also find that appropriate intervention would be in the form of a multifaceted approach at overall risk reduction rather than tackling one specific control individually.

Transmission dynamics and underreporting of Kala-azar in the Indian state of Bihar.

"Kala-azar" (or Indian Visceral Leishmaniasis) is a vector-borne infectious disease affecting communities in tropical and subtropical areas of the world. Bihar, a state in India, has one of the highest prevalence and mortality reported levels of Kala-azar. Yet, the magnitude of the problem is difficult to assess because most cases are handled by private health providers who are not required to and do not report them to the Ministry of Health. The impact of underreporting using district-level reported incidence data from the state of Bihar is the main goal of this manuscript. We derive expressions for, and compute estimates of Kala-azar's reproduction numbers, an indirect measure of disease prevalence, and levels of underreporting for the 21 most affected districts of Bihar. The average reproduction number (number of secondary cases generated per infective) estimates for Bihar range from 1.3 (2003) to 2.1 (2005) with some districts' estimates with mean values lower than one. Model estimates (using available data and a model-derived expression) show that the proportion of underreported cases declined from an average of 88% in 2003 to 73% in 2005. However, eight districts in 2003 and five districts in 2005 had more than 90% levels of underreporting. Model estimates are used to generate underreporting adjusted incidence rates. The analysis finds that reported data misidentify four of the eight (2003) and three of the nine (2005) districts classified as high-risk. In fact, seven (2003) and five (2005) of the most affected Kala-azar districts had been classified as low-risk when only reported incidence data were used.

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.

Using Inverse Problem Methods with Surveillance Data in Pneumococcal Vaccination.

The design and evaluation of epidemiological control strategies is central to public health policy. While inverse problem methods are routinely used in many applications, this remains an area in which their use is relatively rare, although their potential impact is great. We describe methods particularly relevant to epidemiological modeling at the population level. These methods are then applied to the study of pneumococcal vaccination strategies as a relevant example which poses many challenges common to other infectious diseases. We demonstrate that relevant yet typically unknown parameters may be estimated, and show that a calibrated model may used to assess implemented vaccine policies through the estimation of parameters if vaccine history is recorded along with infection and colonization information. Finally, we show how one might determine an appropriate level of refinement or aggregation in the age-structured model given age-stratified observations. These results illustrate ways in which the collection and analysis of surveillance data can be improved using inverse problem methods.

Impact of Relative Residence Times in Highly Distinct Environments on the Distribution of Heavy Drinkers.

Alcohol consumption is a function of social dynamics, environmental contexts, individuals' preferences and family history. Empirical surveys have focused primarily on identification of risk factors for high-level drinking but have done little to clarify the underlying mechanisms at work. Also, there have been few attempts to apply nonlinear dynamics to the study of these mechanisms and processes at the population level. A simple framework where drinking is modeled as a socially contagious process in low- and high-risk connected environments is introduced. Individuals are classified as light, moderate (assumed mobile), and heavy drinkers. Moderate drinkers provide the link between both environments, that is, they are assumed to be the only individuals drinking in both settings. The focus here is on the effect of moderate drinkers, measured by the proportion of their time spent in "low-" versus "high-" risk drinking environments, on the distribution of drinkers.A simple model within our contact framework predicts that if the relative residence times of moderate drinkers is distributed randomly between low- and high-risk environments then the proportion of heavy drinkers is likely to be higher than expected. However, the full story even in a highly simplified setting is not so simple because "strong" local social mixing tends to increase high-risk drinking on its own. High levels of social interaction between light and moderate drinkers in low-risk environments can diminish the importance of the distribution of relative drinking times on the prevalence of heavy drinking.