Honoring the life and legacy of Fred Brauer.

The work of Fred Brauer (1932-2021) broke new ground in several areas of mathematical population biology, especially mathematical epidemiology and population management. This special issue reflects his legacy: the lines of inquiry he opened, the impact of his research and his books, and his mentoring of generations of young researchers. This dedication highlights milestones in his career and connects his work to the contributions in this issue.

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.

How heterogeneity in density dependence affects disease spread: when lifestyle matters.

People's lifestyles play a major role in disease risk. Some employment sectors and transport modes involve fixed exposures regardless of community size, while in other settings exposure tracks with population density. MERS-CoV, a coronavirus discovered in Saudi Arabia in 2012 closely related to those causing SARS and COVID-19, appears to need extended contact time for transmission, making some segments of a community at greater risk than others. We model mathematically how heterogeneity in contact rate structure impacts disease spread, using as a case study a MERS outbreak in two Saudi Arabian communities. We divide the at-risk population into segments with exposure rates either independent of population density or density-dependent. Analysis shows disease spread is minimized for intermediate size populations with a limited proportion of individuals in the density-independent group. In the case study, the high proportion of density-independent exposure may explain the historical outbreak's extinction in the larger city.

Impact of tetravalent dengue vaccination with screening, ADE, and altered infectivity on single-serotype dengue and Zika transmission.

Acquired immunity to a dengue virus serotype (whether by infection or the only licensed dengue vaccine) can produce antibody-dependent enhancement (ADE) in later infections with another dengue serotype, resulting in higher viral loads and more severe symptoms such as dengue hemorrhagic fever, unless the person already has immunity to multiple dengue serotypes. Screening to confirm dengue seropositivity is therefore recommended before vaccination. Recent studies suggest that the closely-related Zika virus may also interact with dengue through ADE. This study uses a mathematical model to evaluate the likely impact of imperfect screening and dengue vaccination on the spread of both viruses in a population where only one dengue serotype circulates, although the vaccine may take against any or all of the four recognized serotypes. Analysis focuses on the reproductive numbers of the viruses. Results indicate that vaccination increases the spread of Zika through induced ADE, while its impact on the spread of dengue depends on screening specificity and serotype-specific vaccine efficacies, as well as the intensity of ADE. Numerical analysis identifies the roles played by age-in and catch-up vaccination as well as screening characteristics and prior dengue exposure.

How influenza vaccination and virus interference may impact combined influenza-coronavirus disease burden.

Demand for influenza vaccine rose as countries prepared for the second COVID-19 wave over the winter months of 2020-2021. High coverage of the influenza vaccine can significantly reduce morbidity and mortality of the burden of influenza. Natural influenza infection creates short-term non-specific immunity against respiratory viruses (virus interference). We model two viral diseases, both of the SEIR type, to investigate whether the influenza vaccine increases the combined disease burden of influenza and COVID-19 in a dual outbreak. We show that the combined disease burden's behavior depends on virus interference factors and the proportion of the population vaccinated against influenza. Our results indicate that influenza vaccination only lowers the overall disease burden when net virus interference is relatively low.

Competition between obligate and facultative scavengers and infection: vulture-jackal-anthrax dynamics in Etosha National Park.

Different species of scavengers may compete for the same food in an ecosystem. This case study considers the competition between jackals and vultures in Etosha National Park in Namibia. While jackals are facultative scavengers, able to hunt for food if necessary, vultures are obligate scavengers wholly dependent on carcasses of animals like zebras for persistence. This competition may be further affected by outbreaks of infections such as anthrax, which temporarily increase the number of carcasses but lower the zebra population, acting in some ways as a third competitor. We use a dynamical system to model the interplay between competition dynamics and infection dynamics, and how it is affected by the nature of the competition: indirect (exploitative) or direct (interference). A bifurcation analysis using reproduction numbers shows how vultures' survival may depend on their direct competitive edge in reaching carcasses faster than jackals, and how the infection and the scavengers complicate each other's persistence. Vultures' interference causes a backward bifurcation which enables them to persist. One possible outcome is a "strange bedfellows" bistability in which anthrax and vultures persist only together, not apart, despite being competitors.

Impact of cattle on joint dynamics and disease burden of Japanese encephalitis and leptospirosis.

Japanese encephalitis (JE) is a mosquito-borne neglected tropical disease. JE is mostly found in rural areas where people usually keep cattle at home for their needs. Cattle in households reduce JE virus infections since they distract vectors and act as a dead-end host for the virus. However, the presence of cattle introduces risk of leptospirosis infections in humans. Leptospirosis is a bacterial disease that spreads through direct or indirect contact of urine of the infected cattle. Thus, cattle have both positive and negative impacts on human disease burden. This study uses a mathematical model to study the joint dynamics of these two diseases in the presence of cattle and to identify the net impact of cattle on the annual disease burden in JE-prevalent areas. Analysis indicates that the presence of cattle helps to reduce the overall disease burden in JE-prevalent areas. However, this reduction is dominated by the vector's feeding pattern. To the best of our knowledge, this is the first study to examine the joint dynamics of JE and leptospirosis.

A Mathematical Modeling Study: Assessing Impact of Mismatch Between Influenza Vaccine Strains and Circulating Strains in Hajj.

The influenza virus causes severe respiratory illnesses and deaths worldwide every year. It spreads quickly in an overcrowded area like the annual Hajj pilgrimage in Saudi Arabia. Vaccination is the primary strategy for protection against influenza. Due to the occurrence of antigenic shift and drift of the influenza virus, a mismatch between vaccine strains and circulating strains of influenza may occur. The objective of this study is to assess the impact of mismatch between vaccine strains and circulating strains during Hajj, which brings together individuals from all over the globe. To this end, we develop deterministic mathematical models of influenza with different populations and strains from the northern and southern hemispheres. Our results show that the existence and duration of an influenza outbreak during Hajj depend on vaccine efficacy. In this concern, we discuss four scenarios: vaccine strains for both groups match/mismatch circulating strains, and vaccine strains match their target strains and mismatch the other strains. Further, there is a scenario where a novel pandemic strain arises. Our results show that as long as the influenza vaccines match their target strains, there will be no outbreak of strain H1N1 and only a small outbreak of strain H3N2. Mismatching for non-target strains causes about 10,000 new H3N2 cases, and mismatching for both strains causes about 2,000 more new H1N1 cases and 6,000 additional H3N2 cases during Hajj. Complete mismatch in a pandemic scenario may infect over 342,000 additional pilgrims (13.75%) and cause more cases in their home countries.

Can scavengers save zebras from anthrax? A modeling study.

Namibia's Etosha National Park (ENP) is home to many different animals such as lions, jackals, hyenas, zebras, elephants, etc. Each year, grazing animals are infected and die from anthrax caused by the bacteria Bacillus anthracis. This increases the number of carcasses in the park, which serve as food for scavengers such as jackals. This study investigates the interplay between anthrax transmission in zebras and the scavenging of zebra carcasses in ENP, using a deterministic mathematical model to describe the population dynamics. We strive to answer the following research questions: Under what conditions can the presence of scavengers control anthrax outbreaks in zebra populations? Does carcass production by anthrax help or hurt scavengers in the long term? Standard qualitative analysis techniques distinguished outcomes (stable equilibria) using reproduction numbers as threshold quantities. We found that, when scavengers feed on anthrax-laden carcasses, the scavengers help the zebras, by eliminating potential infection zones for the zebras. In this way they reduce anthrax's spread by orders of magnitude. We also identify conditions under which the presence of anthrax benefits the scavengers, in terms of death-to-birth ratios for zebras, scavengers and anthrax.

Modeling anthrax-rabies interactions in zebra-jackal cycles.

Etosha National Park (ENP) is located in Namibia, where an annual anthrax outbreak (caused by Bacillus anthracis) occurs among grazing animals such as zebras. This increases the number of carcasses in ENP, allowing for scavengers such as jackals to feed off these carcasses. Carcasses provide a location of conspecific interaction between jackals and may be a means of disease transmission among the jackals. We are interested in studying how a disease in the zebra population may help to propagate a different disease (rabies) in the jackal population since the carcasses are providing a location of interaction between the jackals. We aim to answer the following research question: how do anthrax and rabies affect each other's ability to spread? Standard qualitative analysis techniques distinguished outcomes (stable equilibria) using reproduction numbers as threshold quantities. We found that rabies helps anthrax, and a little anthrax helps rabies invade, but a lot of anthrax prevents rabies by reducing the jackal population through its food source.

Decoys and Dilution: The Impact of Incompetent Hosts on Prevalence of Chagas Disease.

Biodiversity is commonly believed to reduce risk of vector-borne zoonoses. However, researchers already showed that the effect of biodiversity on disease transmission is not that straightforward. This study focuses on the effect of biodiversity, specifically on the effect of the decoy process (additional hosts distracting vectors from their focal host), on reducing infections of vector-borne diseases in humans. Here, we consider the specific case of Chagas disease and use mathematical population models to observe the impact on human infection of the proximity of chickens, which are incompetent hosts for the parasite but serve as a preferred food source for vectors. We consider three cases as the distance between the two host populations varies: short (when farmers bring chickens inside the home to protect them from predators), intermediate (close enough for vectors with one host to detect the presence of the other host type), and far (separate enclosed buildings such as a home and hen-house). Our analysis shows that the presence of chickens reduces parasite prevalence in humans only at an intermediate distance under the condition that the vector birth rate from feeding on chickens is sufficiently low.

Cost analysis of vaccination in tick-mouse transmission of Lyme disease.

Lyme disease is one of the most prevalent and fastest growing vector-borne bacterial illnesses in the United States, with over 25,000 new confirmed cases every year. Humans contract the bacterium Borrelia burgdorferi through the bite of the tick Ixodes scapularis. The tick can receive the bacterium from a variety of small mammal and bird species, but the white-footed mouse Peromyscus leucopus is the primary reservoir in the northeastern United States, especially near human settlement. The tick's life cycle and behavior depend greatly on the season, with different stages of tick biting at different times. Reducing the infection in the tick-mouse cycle may greatly lower human Lyme incidence in some areas. However, research on the effects of various mouse-targeted interventions is limited. One particularly promising method involves administering vaccine pellets to white-footed mice through special bait boxes. In this study, we develop and analyze a mathematical model consisting of a system of nonlinear difference equations to understand the complex transmission dynamics and vector demographics in both tick and mice populations. We evaluate to what extent vaccination of white-footed mice can affect Lyme incidence in I. scapularis, and under which conditions this method saves money in preventing Lyme disease. We find that, in areas with high human risk, vaccination can eliminate mouse-tick transmission of B. burgdorferi while saving money.

Impact of dogs with deltamethrin-impregnated collars on prevalence of visceral leishmaniasis.

Leishmaniasis is a vector borne zoonosis which is classified as a neglected tropical disease. Among the three most common forms of the disease, Visceral Leishmaniasis (VL) is the most threatening to human health, causing 20,000 to 30,000 deaths worldwide each year. Areas where VL is mostly endemic have unprotected dogs in community and houses. The "presence of dogs usually increases VL risk for humans since dogs are the principal reservoir host for the parasite of the disease. Based on this fact, most earlier studies consider culling dogs as a control measure for the spread of VL. A more recent control measure has been the use of deltamethrin-impregnated dog collars ( D I D C s) to protect both humans and dogs by putting D I D C s on dogs neck. The presence of dogs helps to grow the sandfly population faster by offering a more suitable blood-meal source. On the other hand, the presence of D I D C s on dogs helps to reduce sandfly population by the lethality of deltamethrin insecticide. This study brings an ecological perspective to this public health concern, aiming to understand the impact of an additional host (here, protected dogs) on disease risk to a primary host (here, humans). To answer this question, we compare two different settings: a community without dogs, and a community with dogs protected with D I D C . Our analysis shows the presence of protected dogs can reduce VL infection risk in humans. However, this disease risk reduction depends on dogs' tolerance for sandfly bites.

Coinfection, Altered Vector Infectivity, and Antibody-Dependent Enhancement: The Dengue-Zika Interplay.

Although dengue and Zika cocirculation has increased within the past 5 years, very little is known about its epidemiological consequences. To investigate the effect of dengue and Zika cocirculation on the spread of both pathogens, we create a deterministic dengue and Zika coinfection model, the first to incorporate altered infectivity of mosquitoes (due to coinfection). The model also addresses increased infectivity due to antibody-dependent enhancement (ADE) within the human population. Central to our analysis is the derivation and interpretation of the basic reproductive number and invasion reproductive number of both pathogens. In addition, we investigate how model parameters impact the persistence of each disease. Our results identify threshold conditions under which one disease facilitates the spread of the other and show that ADE has a greater impact on disease persistence than altered vector infectivity. This work highlights the importance of ADE and illustrates that while the endemic presence of dengue facilitates the spread of Zika, it is possible for high Zika prevalence to prevent the establishment of dengue.

Invasion reproductive numbers for periodic epidemic models.

There are many cases within epidemiology where infections compete to persist within a population. In studying models for such cases, one of the goals is to determine which infections can invade a population and persist when other infections are already resident within the population. Invasion reproductive numbers (IRN), which are tied to the stability of boundary endemic equilibria, can address this question. By reinterpreting resident infections epidemiologically, this study extends methods for finding IRNs to periodic systems, and presents some examples which illustrate the often complex computations required. Results identify conditions under which a simple time-average can be used to derive IRNs, and apply the methods to examine how seasonal fluctuations in influenza incidence facilitate the year-round persistence of bacterial respiratory infections.

Invasion reproductive numbers for discrete-time models.

Although invasion reproductive numbers (IRNs) are utilized frequently in continuous-time models with multiple interacting pathogens, they are yet to be explored in discrete-time systems. Here, we extend the concept of IRNs to discrete-time models by showing how to calculate them for a set of two-pathogen SIS models with coinfection. In our exploration, we address how sequencing events impacts the basic reproductive number (BRN) and IRN. As an illustrative example, our models are applied to rhinovirus and respiratory syncytial virus co-circulation. Results show that while the BRN is unaffected by variations in the order of events, the IRN differs. Furthermore, our models predict copersistence of multiple pathogen strains under cross-immunity, which is atypical of analogous continuous-time models. This investigation shows that sequencing events has important consequences for the IRN and can drastically alter competition dynamics.

Dynamics of two pathogens in a single tick population.

A mathematical model for a two-pathogen, one-tick, one-host system is presented and explored. The model system is based on the dynamics of Amblyomma americanum, Rickettsia parkeri, and Rickettsia amblyommatis. The goal of this model is to determine how long an invading pathogen, R. parkeri, persists within a tick population, A. americanum, in which a resident pathogen, R. amblyommatis, is already established. The numerical simulations of the model demonstrate the parameter ranges that allow for coexistence of the two pathogens. Sensitivity analysis highlights the importance of vector-borne, tick-to-host, transmission rates on the invasion reproductive number and persistence of the pathogens over time. The model is then applied to a case study based on a reclaimed swampland field site in south-eastern Virginia using field and laboratory data. The results pinpoint the thresholds required for persistence of both pathogens in the local tick population. However, R. parkeri, is not predicted to persist beyond 3 years. Understanding the persistence and coexistence of tick-borne pathogens will allow public health officials increased insight into tick-borne disease dynamics.

Ebola: Impact of hospital's admission policy in an overwhelmed scenario.

Infectious disease outbreaks sometimes overwhelm healthcare facilities. A recent case occurred in West Africa in 2014 when an Ebola virus outbreak overwhelmed facilities in Sierra Leone, Guinea and Liberia. In such scenarios, how many patients can hospitals admit to minimize disease burden? This study considers what type of hospital admission policy during a hypothetical Ebola outbreak can better serve the community, if overcrowding degrades the hospital setting. Our result shows that which policy minimizes loss to the community depends on the initial estimation of the control reproduction number, R0. When the outbreak grows extremely fast (R0 ≫ 1) it is better (in terms of total disease burden) to stop admitting patients after reaching the carrying capacity because overcrowding in the hospital makes the hospital setting ineffective at containing infection, but when the outbreak grows only a little faster than the system's ability to contain it (R0 ≳ 1), it is better to admit patients beyond the carrying capacity because limited overcrowding still reduces infection more in the community. However, when R0 is no more than a little greater than 1 (for our parameter values, 1.012), both policies result the same because the number of patients never exceeds the maximum capacity.

Effects of multiple transmission pathways on Zika dynamics.

Although the Zika virus is transmitted to humans primarily through the bite of infected female Aedes aegypti mosquitoes, it can also be sexually and vertically transmitted within both populations. In this study, we develop a new mathematical model of the Zika virus which incorporates sexual transmission in humans and mosquitos, vertical transmission in mosquitos, and mosquito to human transmission through bites. Analysis of this deterministic model shows that the secondary transmission routes of Zika increase the basic reproductive number ( R 0 ) of the virus by 5%, shift the peak time of an outbreak to occur 10% sooner, increase the initial growth of an epidemic, and have important consequences for control strategies and estimates of R 0 . Furthermore, sensitivity analysis show that the basic reproductive number is most sensitive to the mosquito biting rate and transmission probability parameters and reveal that the dynamics of juvenile mosquito stages greatly impact the peak time of an outbreak. These discoveries deepen our understanding of the complex transmission routes of ZIKV and the consequences that they may hold for public health officials.