Analysis of serological surveys of antibodies to SARS-CoV-2 in the United States to estimate parameters needed for transmission modeling and to evaluate and improve the accuracy of predictions.

Seroprevalence studies can estimate proportions of the population that have been infected or vaccinated, including infections that were not reported because of the lack of symptoms or testing. Based on information from studies in the United States from mid-summer 2020 through the end of 2021, we describe proportions of the population with antibodies to SARS-CoV-2 as functions of age and time. Slices through these surfaces at arbitrary times provide initial and target conditions for simulation modeling. They also provide the information needed to calculate age-specific forces of infection, attack rates, and - together with contact rates - age-specific probabilities of infection on contact between susceptible and infectious people. We modified the familiar Susceptible-Exposed-Infectious-Removed (SEIR) model to include features of the biology of COVID-19 that might affect transmission of SARS-CoV-2 and stratified by age and location. We consulted the primary literature or subject matter experts for contact rates and other parameter values. Using time-varying Oxford COVID-19 Government Response Tracker assessments of US state and DC efforts to mitigate the pandemic and compliance with non-pharmaceutical interventions (NPIs) from a YouGov survey fielded in the US during 2020, we estimate that the efficacy of social-distancing when possible and mask-wearing otherwise at reducing susceptibility or infectiousness was 31% during the fall of 2020. Initialized from seroprevalence among people having commercial laboratory tests for purposes other than SARS-CoV-2 infection assessments on 7 September 2020, our age- and location-stratified SEIR population model reproduces seroprevalence among members of the same population on 25 December 2020 quite well. Introducing vaccination mid-December 2020, first of healthcare and other essential workers, followed by older adults, people who were otherwise immunocompromised, and then progressively younger people, our metapopulation model reproduces seroprevalence among blood donors on 4 April 2021 less well, but we believe that the discrepancy is due to vaccinations being under-reported or blood donors being disproportionately vaccinated, if not both. As experimenting with reliable transmission models is the best way to assess the indirect effects of mitigation measures, we determined the impact of vaccination, conditional on NPIs. Results indicate that, during this period, vaccination substantially reduced infections, hospitalizations and deaths. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics."

A celebration of Fred Brauer's legacy in mathematical biology.

Fred Brauer (1932-2021), one of the pioneers of mathematical population biology, shaped generations of researchers through his lines of research, his books which have become key references in the field, and his mentoring of junior researchers. This dedication reviews some of his work in population harvesting and epidemiological modeling, highlighting how this special collection reflects the impact of his legacy through both his research accomplishments and the formation of new researchers.

Implications for infectious disease models of heterogeneous mixing on control thresholds.

Mixing among sub-populations, as well as heterogeneity in characteristics affecting their reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. In this overview, we apply a linear algebraic approach to re-derive some well-known results pertaining to preferential within- and proportionate among-group contacts in compartmental models of pathogen transmission. We give results for the meta-population effective reproduction number ([Formula: see text]) assuming different levels of vaccination in the sub-populations. Specifically, we unpack the dependency of [Formula: see text] on the fractions of contacts reserved for individuals within one's own subgroup and, by obtaining implicit expressions for the partial derivatives of [Formula: see text], we show that these increase as this preferential-mixing fraction increases in any sub-population.

Analysis of metapopulation models of the transmission of SARS-CoV-2 in the United States.

During the COVID-19 pandemic, renewal equation estimates of time-varying effective reproduction numbers were useful to policymakers in evaluating the need for and impact of mitigation measures. Our objective here is to illustrate the utility of mechanistic expressions for the basic and effective (or intrinsic and realized) reproduction numbers, [Formula: see text] and related quantities derived from a Susceptible-Exposed-Infectious-Removed (SEIR) model including features of COVID-19 that might affect transmission of SARS-CoV-2, including asymptomatic, pre-symptomatic, and symptomatic infections, with which people may be hospitalized. Expressions from homogeneous host population models can be analyzed to determine the effort needed to reduce [Formula: see text] from [Formula: see text] to 1 and contributions of modeled mitigation measures. Our model is stratified by age, 0-4, 5-9, …, 75+ years, and location, the 50 states plus District of Columbia. Expressions from such heterogeneous host population models include subpopulation reproduction numbers, contributions from the above-mentioned infectious states, metapopulation numbers, subpopulation contributions, and equilibrium prevalence. While the population-immunity at which [Formula: see text] has captured the popular imagination, the metapopulation [Formula: see text] could be attained in an infinite number of ways even if only one intervention (e.g., vaccination) were capable of reducing [Formula: see text] However, gradients of expressions derived from heterogeneous host population models,[Formula: see text] can be evaluated to identify optimal allocations of limited resources among subpopulations. We illustrate the utility of such analytical results by simulating two hypothetical vaccination strategies, one uniform and other indicated by [Formula: see text] as well as the actual program estimated from one of the CDC's nationwide seroprevalence surveys conducted from mid-summer 2020 through the end of 2021.

Influence of demographically-realistic mortality schedules on vaccination strategies in age-structured models.

Because demographic realism complicates analysis, mathematical modelers either ignore demography or make simplifying assumptions (e.g., births and deaths equal). But human populations differ demographically, perhaps most notably in their mortality schedules. We developed an age-stratified population model with births, deaths, aging and mixing between age groups. The model includes types I and II mortality as special cases. We used the gradient approach (Feng et al., 2015, 2017) to explore the impact of mortality patterns on optimal strategies for mitigating vaccine-preventable diseases such as measles and rubella, which the international community has targeted for eradication. Identification of optimal vaccine allocations to reduce the effective reproduction number Rv under various scenarios is presented. Numerical simulations of the model with various types of mortality are carried out to ascertain the long-term effects of vaccination on disease incidence. We conclude that both optimal vaccination strategies and long-term effects of vaccination may depend on demographic assumptions.

Modeling the waning and boosting of immunity from infection or vaccination.

Immunity following natural infection or immunization may wane, increasing susceptibility to infection with time since infection or vaccination. Symptoms, and concomitantly infectiousness, depend on residual immunity. We quantify these phenomena in a model population composed of individuals whose susceptibility, infectiousness, and symptoms all vary with immune status. We also model age, which affects contact, vaccination and possibly waning rates. The resurgences of pertussis that have been observed wherever effective vaccination programs have reduced typical disease among young children follow from these processes. As one example, we compare simulations with the experience of Sweden following resumption of pertussis vaccination after the hiatus from 1979 to 1996, reproducing the observations leading health authorities to introduce booster doses among school-aged children and adolescents in 2007 and 2014, respectively. Because pertussis comprises a spectrum of symptoms, only the most severe of which are medically attended, accurate models are needed to design optimal vaccination programs where surveillance is less effective.

Assessing the effects of modeling the spectrum of clinical symptoms on the dynamics and control of Ebola.

Mathematical modelers have attempted to capture the dynamics of Ebola transmission and to evaluate the effectiveness of control measures, as well as to make predictions about ongoing outbreaks. Many of their models consider only infections with typical symptoms, but Ebola presents clinically in a more complicated way. Even the most common symptom, fever, is not experienced by 13% of patients. This suggests that infected individuals could be asymptomatic or have moderately symptomatic infections as reported during previous Ebola outbreaks. To account crudely for the spectrum of clinical symptoms that characterizes Ebola infection, we developed a model including moderate and severe symptoms. Our model captures the dynamics of the recent outbreak of Ebola in Liberia. Our estimate of the basic reproduction number is 1.83 (CI: 1.72, 1.86), consistent with the WHO response team's estimate using early outbreak case data. We also estimate the effectiveness of interventions using observations before and after their introduction. As the final epidemic size is linked to the timing of interventions in an exponential fashion, a simple empirical formula is provided to guide policy-making. It suggests that early implementation could significantly decrease final size. We also compare our model to one with typical symptoms by excluding moderate ones. The model with only typical symptoms overestimates the basic reproduction number and effectiveness of control measures, and exaggerates changes in peak size attributable to the timing of interventions. In addition, uncertainty about how moderate symptoms affect the basic reproduction number is considered, and PRCC (Partial rank correlation coefficient) is used to analyze the global sensitivity of relevant parameters. Possible control strategies are evaluated through numerical simulations and sensitivity analysis, indicating that simultaneously strengthening contact-tracing and effectiveness of isolation in hospital would be most effective. In this study, we show that asymptomatic Ebola infections may have implications for policy-making.

Intermittent Preventive Treatment (IPT): Its Role in Averting Disease-Induced Mortality in Children and in Promoting the Spread of Antimalarial Drug Resistance.

We develop an age-structured ODE model to investigate the role of intermittent preventive treatment (IPT) in averting malaria-induced mortality in children, and its related cost in promoting the spread of antimalarial drug resistance. IPT, a malaria control strategy in which a full curative dose of an antimalarial medication is administered to vulnerable asymptomatic individuals at specified intervals, has been shown to reduce malaria transmission and deaths in children and pregnant women. However, it can also promote drug resistance spread. Our mathematical model is used to explore IPT effects on drug resistance and deaths averted in holoendemic malaria regions. The model includes drug-sensitive and drug-resistant strains as well as human hosts and mosquitoes. The basic reproduction, and invasion reproduction numbers for both strains are derived. Numerical simulations show the individual and combined effects of IPT and treatment of symptomatic infections on the prevalence of both strains and the number of lives saved. Our results suggest that while IPT can indeed save lives, particularly in high transmission regions, certain combinations of drugs used for IPT and to treat symptomatic infection may result in more deaths when resistant parasite strains are circulating. Moreover, the half-lives of the treatment and IPT drugs used play an important role in the extent to which IPT may influence spread of the resistant strain. A sensitivity analysis indicates the model outcomes are most sensitive to the reduction factor of transmission for the resistant strain, rate of immunity loss, and the natural clearance rate of sensitive infections.

Constrained minimization problems for the reproduction number in meta-population models.

The basic reproduction number ([Formula: see text]) can be considerably higher in an SIR model with heterogeneous mixing compared to that from a corresponding model with homogeneous mixing. For example, in the case of measles, mumps and rubella in San Diego, CA, Glasser et al. (Lancet Infect Dis 16(5):599-605, 2016. https://doi.org/10.1016/S1473-3099(16)00004-9 ), reported an increase of 70% in [Formula: see text] when heterogeneity was accounted for. Meta-population models with simple heterogeneous mixing functions, e.g., proportionate mixing, have been employed to identify optimal vaccination strategies using an approach based on the gradient of the effective reproduction number ([Formula: see text]), which consists of partial derivatives of [Formula: see text] with respect to the proportions immune [Formula: see text] in sub-groups i (Feng et al. in J Theor Biol 386:177-187, 2015.  https://doi.org/10.1016/j.jtbi.2015.09.006 ; Math Biosci 287:93-104, 2017.  https://doi.org/10.1016/j.mbs.2016.09.013 ). These papers consider cases in which an optimal vaccination strategy exists. However, in general, the optimal solution identified using the gradient may not be feasible for some parameter values (i.e., vaccination coverages outside the unit interval). In this paper, we derive the analytic conditions under which the optimal solution is feasible. Explicit expressions for the optimal solutions in the case of [Formula: see text] sub-populations are obtained, and the bounds for optimal solutions are derived for [Formula: see text] sub-populations. This is done for general mixing functions and examples of proportionate and preferential mixing are presented. Of special significance is the result that for general mixing schemes, both [Formula: see text] and [Formula: see text] are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively.

Evaluations of Interventions Using Mathematical Models with Exponential and Non-exponential Distributions for Disease Stages: The Case of Ebola.

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511-1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.

Modeling Virus Coinfection to Inform Management of Maize Lethal Necrosis in Kenya.

Maize lethal necrosis (MLN) has emerged as a serious threat to food security in sub-Saharan Africa. MLN is caused by coinfection with two viruses, Maize chlorotic mottle virus and a potyvirus, often Sugarcane mosaic virus. To better understand the dynamics of MLN and to provide insight into disease management, we modeled the spread of the viruses causing MLN within and between growing seasons. The model allows for transmission via vectors, soil, and seed, as well as exogenous sources of infection. Following model parameterization, we predict how management affects disease prevalence and crop performance over multiple seasons. Resource-rich farmers with large holdings can achieve good control by combining clean seed and insect control. However, crop rotation is often required to effect full control. Resource-poor farmers with smaller holdings must rely on rotation and roguing, and achieve more limited control. For both types of farmer, unless management is synchronized over large areas, exogenous sources of infection can thwart control. As well as providing practical guidance, our modeling framework is potentially informative for other cropping systems in which coinfection has devastating effects. Our work also emphasizes how mathematical modeling can inform management of an emerging disease even when epidemiological information remains scanty. [Formula: see text] Copyright © 2017 The Author(s). This is an open access article distributed under the CC BY-NC-ND 4.0 International license .

Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals.

Antibiotic-resistant bacteria have posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study transmission dynamics of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in Lipstich et al. (Proc Natl Acad Sci 97(4):1938-1943, 2000) and Lipstich and Bergstrom (Infection control in the ICU environment. Kluwer, Boston, 2002) have provided valuable insights in understanding the transmission of antibiotic-resistant bacteria in a hospital. However, their results are limited to numerical simulations of a few different scenarios without analytical analyses of the models in broader parameter regions that are biologically feasible. Bifurcation analysis and identification of the global stability conditions can be very helpful for assessing interventions that are aimed at limiting nosocomial infections and stemming the spread of antibiotic-resistant bacteria. In this paper we study the global dynamics of the mathematical model of antibiotic resistance in hospitals considered in Lipstich et al. (2000) and Lipstich and Bergstrom (2002). The invasion reproduction number [Formula: see text] of antibiotic-resistant bacteria is derived, and the relationship between [Formula: see text] and two control reproduction numbers of sensitive bacteria and resistant bacteria ([Formula: see text] and [Formula: see text]) is established. More importantly, we prove that a backward bifurcation may occur at [Formula: see text] when the model includes superinfection, which is not mentioned in Lipstich and Bergstrom (2002). More specifically, there exists a new threshold [Formula: see text], such that if [Formula: see text], then the system can have two positive interior equilibria, which leads to an interesting bistable phenomenon. This may have critical implications for controlling the antibiotic-resistance in a hospital.

Dynamics of Predator-Prey Metapopulations with Allee Effects.

Allee effects increasingly are recognized as influential determinants of population dynamics, especially in disturbed landscapes. We developed a predator-prey metapopulation model to study the impact of an Allee effect on predator-prey. The model incorporates habitat destruction and predators with imperfect information about prey distribution. Criteria are established for the existence and stability of equilibria, and the possible existence of a limit cycle is discussed. Numerical bifurcation analysis of the model is carried out to examine the impact of Allee effects as well as other key processes on trophic dynamics. Inclusion of Allee effects produces a richer array of dynamics than earlier models in which it was absent. When prey interacts with generalist predators, Allee effects operate synergistically to depress prey populations. Allee effects are more likely to depress occupancy levels when destruction of habitat patches is moderate; at severe levels of destruction, Allee effects are swamped by demographic effects of habitat loss. Stronger Allee effects correspond to lower thresholds of predator colonization rates at which prey become extinct. We discuss implications of our model for conservation of rare species as well as pest management via biocontrol.

COMPUTATION OF ℛ IN AGE-STRUCTURED EPIDEMIOLOGICAL MODELS WITH MATERNAL AND TEMPORARY IMMUNITY.

For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting reinfection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ℛ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ℛ < 1 and unstable if ℛ > 1.

Intermediate disturbance in experimental landscapes improves persistence of beetle metapopulations.

Human-dominated landscapes often feature patches that fluctuate in suitability through space and time, but there is little experimental evidence relating the consequences of dynamic patches for species persistence. We used a spatially and temporally dynamic metapopulation model to assess and compare metapopulation capacity and persistence for red flour beetles (Tribolium castaneum) in experimental landscapes differentiated by resource structure, patch dynamics (destruction and restoration), and connectivity. High connectivity increased the colonization rate of beetles, but this effect was less pronounced in heterogeneous relative to homogeneous landscapes. Higher connectivity and faster patch dynamics increased extinction rates in landscapes. Lower connectivity promoted density-dependent emigration. Heterogeneous landscapes containing patches of different carrying capacity enhanced landscape-level occupancy probability. The highest metapopulation capacity and persistence was observed in landscapes with heterogeneous patches, low connectivity, and slow patch dynamics. Control landscapes with no patch dynamics exhibited rapid declines in abundance and approached extinction due to increased adult mortality in the matrix, higher pupal cannibalism by adults, and extremely low rates of exchange between remaining habitable patches. Our results highlight the role of intermediate patch dynamics, intermediate connectivity, and the nature of density dependence of emigration for persistence of species in heterogeneous landscapes. Our results also demonstrate the importance of incorporating local dynamics into the estimation of metapopulation capacity for conservation planning.

An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing.

The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.

Emerging disease dynamics in a model coupling within-host and between-host systems.

Epidemiological models and immunological models have been studied largely independently. However, the two processes (between- and within-host interactions) occur jointly and models that couple the two processes may generate new biological insights. Particularly, the threshold conditions for disease control may be dramatically different when compared with those generated from the epidemiological or immunological models separately. An example is considered in this paper for an environmentally driven infectious disease such as Toxoplasma gondii. The model explicitly couples the within-host and between-host dynamics. The within-host sub-system is linked to a contaminated environment E via an additional term g(E) to account for the increase in the parasite load V within a host due to the continuous ingestion of parasites from the contaminated environment. The parasite load V can also affect the rate of environmental contamination, which directly contributes to the infection rate of hosts for the between-host sub-system. When the two sub-systems are considered in isolation, the dynamics are standard and simple. That is, either the infection-free equilibrium is stable or a unique positive equilibrium is stable depending on the relevant reproduction number being less or greater than 1. However, when the two sub-systems are explicitly coupled, the full system exhibits more complex dynamics including backward bifurcations; that is, multiple positive equilibria exist with one of which being stable even if the reproduction number is less than 1. The biological implications of such bifurcations are illustrated using an example concerning the spread and control of toxoplasmosis.

Modeling rates of infection with transient maternal antibodies and waning active immunity: application to Bordetella pertussis in Sweden.

Serological surveys provide reliable information from which to calculate forces (instantaneous rates) of infection, but waning immunity and clinical consequences that depend on residual immunity complicate interpretation of results. We devised a means of calculating these rates that accounts for passively acquired maternal antibodies that decay or active immunity that wanes, permitting re-infection. We applied our method to pertussis (whooping cough) in Sweden, where vaccination was discontinued from 1979 to 1995. A national cross-sectional serosurvey of antibodies to pertussis toxin, which peak soon after infection and then decay, was conducted shortly after vaccination resumed. Together with age-specific contact rates in Finland, contemporary forces of infection enable us to evaluate the recent assertion that the probability of infection upon contact is age-independent. We find elevated probabilities among children, adolescents and young adults, whose contacts may be more intimate than others. Products of contact rates and probabilities of infection permit transmission modeling and estimation of the intrinsic reproduction number. In contrast to another recent estimate, ours approximates the ratio of life expectancy and age at first infection. Our framework is sufficiently general to accommodate more realistic sojourn distributions and additional lifetime infections.

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.

The physician CEO advantage and hospital performance during the COVID-19 pandemic: capacity utilization and patient satisfaction.

PURPOSE: Prior studies have shown that physician-led hospitals have several advantages over non-physician-led hospitals. This study seeks to test whether these advantages also extend to periods of extreme disruptions such as the COVID-19 pandemic, which affect bed availability and hospital utilization. DESIGN/METHODOLOGY/APPROACH: The authors utilize a bounded Tobit estimation to identify differences in patient satisfaction rates and in-hospital utilization rates of top-rated hospitals in the United States. FINDINGS: Among top-rated US hospitals, those that are physician-led achieve higher patient satisfaction ratings and are more likely to have higher utilization rates. RESEARCH LIMITATIONS/IMPLICATIONS: While the COVID-19 pandemic generated greater demand for inpatient beds, physician-led hospitals improved their hospitals' capacity utilization as compared with those led by non-physician leaders. A longitudinal study to show the change over the years and whether physician Chief Executive Officers (CEOs) are more likely to improve their hospitals' ratings than non-physician CEOs is highly recommended. PRACTICAL IMPLICATIONS: Recruiting and retaining physicians to lead hospitals, especially during disruptions, improve hospital's operating efficiency and enhance patient satisfaction. ORIGINALITY/VALUE: The paper reviews prior research on physician leadership and adds further insights into the crisis leadership literature. The authors provide evidence based on quantitative data analysis that during the COVID-19 pandemic, physician-led top-rated US hospitals experienced an improvement in operating efficiency.