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 mathematical model to assess the impact of testing and isolation compliance on the transmission of COVID-19.

The COVID-19 pandemic has ravaged global health and national economies worldwide. Testing and isolation are effective control strategies to mitigate the transmission of COVID-19, especially in the early stage of the disease outbreak. In this paper, we develop a deterministic model to investigate the impact of testing and compliance with isolation on the transmission of COVID-19.

A contact tracing SIR model for randomly mixed populations.

Contact tracing is an important intervention measure to control infectious diseases. We present a new approach that borrows the edge dynamics idea from network models to track contacts included in a compartmental SIR model for an epidemic spreading in a randomly mixed population. Unlike network models, our approach does not require statistical information of the contact network, data that are usually not readily available.

Disease-Induced Hydra Effect with Overcompensatory Recruitment.

In ecological systems, the hydra effect is an increase in population size caused by an increase in mortality. This seemingly counterintuitive effect has been observed in several populations, including fish, blowflies, snails and plants, and has been modeled in both continuous and discrete time. A similar effect induced by disease has recently been observed empirically.

Evolution of an asymptomatic first stage of infection in a heterogeneous population.

Pathogens evolve different life-history strategies, which depend in part on differences in their host populations. A central feature of hosts is their population structure (e.g.

Backward bifurcation in within-host HIV models.

The activation and proliferation of naive CD4 T cells produce helper T cells, and increase the susceptible population in the presence of HIV. This may cause backward bifurcation. To verify this, we construct a simple within-host HIV model that includes the key variables, namely healthy naive CD4 T cells, helper T cells, infected CD4 T cells and virus.

Superinfection and the evolution of an initial asymptomatic stage.

Pathogens have evolved a variety of life-history strategies. An important strategy consists of successful transmission by an infected host before the appearance of symptoms, that is, while the host is still partially or fully asymptomatic. During this initial stage of infection, it is possible for another pathogen to superinfect an already infected host and replace the previously infecting pathogen.

Effect of feedback regulation on stem cell fractions in tissues and tumors: Understanding chemoresistance in cancer.

While resistance mutations are often implicated in the failure of cancer therapy, lack of response also occurs without such mutants. In bladder cancer mouse xenografts, repeated chemotherapy cycles have resulted in cancer stem cell (CSC) enrichment, and consequent loss of therapy response due to the reduced susceptibility of CSCs to drugs. A particular feedback loop present in the xenografts has been shown to promote CSC enrichment in this system.

Stochastic Model of Bovine Babesiosis with Juvenile and Adult Cattle.

A stochastic model for Bovine Babesiosis (BB) including ticks, and both juvenile and adult cattle is developed. This model is formulated by a system of continuous-time Markov chains (CTMCs) that is derived based on an extension of the deterministic ordinary differential equation model developed by Saad-Roy et al. (Bull Math Biol 77:514-547, 2015).

Age structured discrete-time disease models with demographic population cycles.

We use juvenile-adult discrete-time infectious disease models with intrinsically generated demographic population cycles to study the effects of age structure on the persistence or extinction of disease and the basic reproduction number, [Formula: see text]. Our juvenile-adult Susceptible-Infectious-Recovered (SIR) and Infectious-Salmon Anemia-Virus (ISA[Formula: see text] models share a common disease-free system that exhibits equilibrium dynamics for the Beverton-Holt recruitment function. However, when the recruitment function is the Ricker model, a juvenile-adult disease-free system exhibits a range of dynamic behaviours from stable equilibria to deterministic period population cycles to Neimark-Sacker bifurcations and deterministic chaos.

Modelling optimal timing and frequency of insecticide sprays for eradication or knockdown of closed populations of tsetse flies Glossina spp. (Diptera: Glossinidae).

A population model for tsetse species was used to assess the optimal number and spacing of airborne sprays to reduce or eradicate a tsetse population. It was found that the optimal spray spacing was determined by the time (days) from adult emergence to the first larviposition and, for safety, spacing was assigned to that duration minus 2 days. If sprays killed all adults, then the number of sprays required for eradication is determined by a simple formula.

Habitat fragmentation promotes malaria persistence.

Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence.

Estimating tsetse fertility: daily averaging versus periodic larviposition.

When computing mean daily fertility in adult female tsetse, the common practice of taking the reciprocal of the interlarval period (called averaged fertility) was compared with the method of taking the sum of the products of daily fertility and adult survivorship divided by the sum of daily survivorships (called periodic fertility). The latter method yielded a consistently higher measure of fertility (approximately 10% for tsetse) than the former method. A conversion factor was calculated to convert averaged fertility to periodic fertility.

A general theory for target reproduction numbers with applications to ecology and epidemiology.

A general framework for threshold parameters in population dynamics is developed using the concept of target reproduction numbers. This framework identifies reproduction numbers and other threshold parameters in the literature in terms of their roles in population control. The framework is applied to the analysis of single and multiple control strategies in ecology and epidemiology, and this provides new biological insights.

Demographic population cycles and ℛ in discrete-time epidemic models.

We use a general autonomous discrete-time infectious disease model to extend the next generation matrix approach for calculating the basic reproduction number, [Formula: see text], to account for populations with locally asymptotically stable period k cycles in the disease-free systems, where [Formula: see text]. When [Formula: see text] and the demographic equation (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove the local asymptotic stability of the disease-free period k cycle. That is, the disease goes extinct whenever [Formula: see text].

Host contact structure is important for the recurrence of Influenza A.

An important characteristic of influenza A is its ability to escape host immunity through antigenic drift. A novel influenza A strain that causes a pandemic confers full immunity to infected individuals. Yet when the pandemic strain drifts, these individuals will have decreased immunity to drifted strains in the following seasonal epidemics.

Reproduction numbers of infectious disease models.

This primer article focuses on the basic reproduction number, , for infectious diseases, and other reproduction numbers related to that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of determining whether or not the disease dies out. The next generation matrix method of calculating in a compartmental model is described and illustrated.

The effect of sexual transmission on Zika virus dynamics.

Zika virus is a human disease that may lead to neurological disorders in affected individuals, and may be transmitted vectorially (by mosquitoes) or sexually. A mathematical model of Zika virus transmission is formulated, taking into account mosquitoes, sexually active males and females, inactive individuals, and considering both vector transmission and sexual transmission from infectious males to susceptible females. Basic reproduction numbers are computed, and disease control strategies are evaluated.

Disease Extinction Versus Persistence in Discrete-Time Epidemic Models.

We focus on discrete-time infectious disease models in populations that are governed by constant, geometric, Beverton-Holt or Ricker demographic equations, and give a method for computing the basic reproduction number, [Formula: see text]. When [Formula: see text] and the demographic population dynamics are asymptotically constant or under geometric growth (non-oscillatory), we prove global asymptotic stability of the disease-free equilibrium of the disease models. Under the same demographic assumption, when [Formula: see text], we prove uniform persistence of the disease.

Estimation of Cross-Immunity Between Drifted Strains of Influenza A/H3N2.

To determine the cross-immunity between influenza strains, we design a novel statistical method, which uses a theoretical model and clinical data on attack rates and vaccine efficacy among school children for two seasons after the 1968 A/H3N2 influenza pandemic. This model incorporates the distribution of susceptibility and the dependence of cross-immunity on the antigenic distance of drifted strains. We find that the cross-immunity between an influenza strain and the mutant that causes the next epidemic is 88%.

Stability of Control Networks in Autonomous Homeostatic Regulation of Stem Cell Lineages.

Design principles of biological networks have been studied extensively in the context of protein-protein interaction networks, metabolic networks, and regulatory (transcriptional) networks. Here we consider regulation networks that occur on larger scales, namely the cell-to-cell signaling networks that connect groups of cells in multicellular organisms. These are the feedback loops that orchestrate the complex dynamics of cell fate decisions and are necessary for the maintenance of homeostasis in stem cell lineages.

A Mathematical Model of Anthrax Transmission in Animal Populations.

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number [Formula: see text] is calculated, and existence of a unique endemic equilibrium is established for [Formula: see text] above the threshold value 1. Using data from the literature, elasticity indices for [Formula: see text] and type reproduction numbers are computed to quantify anthrax control measures.

Estimation of Zika virus prevalence by appearance of microcephaly.

Background: There currently is a severe Zika Virus (ZIKV) epidemic in Brazil and other South American countries. Due to international travel, this poses severe public health risk of ZIKV importation to other countries. We estimate the prevalence of ZIKV in an import region by the time a microcephaly case is detected, since microcephaly is presently the most significant indication of ZIKV presence.

Analysis of a model of gambiense sleeping sickness in humans and cattle.

Human African Trypanosomiasis (HAT) and Nagana in cattle, commonly called sleeping sickness, is caused by trypanosome protozoa transmitted by bites of infected tsetse flies. We present a deterministic model for the transmission of HAT caused by Trypanosoma brucei gambiense between human hosts, cattle hosts and tsetse flies. The model takes into account the growth of the tsetse fly, from its larval stage to the adult stage.

A mathematical model of syphilis transmission in an MSM population.

Syphilis is caused by the bacterium Treponema pallidum subspecies pallidum, and is a sexually transmitted disease with multiple stages. A model of transmission of syphilis in an MSM population (there has recently been a resurgence of syphilis in such populations) that includes infection stages and treatment is formulated as a system of ordinary differential equations. The control reproduction number is calculated, and it is proved that if this threshold parameter is below one, syphilis dies out; otherwise, if it is greater than one, it is shown that there exists a unique endemic equilibrium and that for certain special cases, this equilibrium is globally asymptotically stable.