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.

Fred Brauer , Figure 1) made pioneering contributions to several areas of applied mathematics as well as several parts of the mathematical community.His researchcomprising at least 170 articles and ten books -opened new areas in differential equations, population management, and mathematical epidemiology.His dedication to mentoring other researchers touched lives and careers around the world.His intellectual curiosity, gentle good humor, and steadfast commitment remain key elements of his legacy through the many students he mentored.This special issue of The Journal of Biological Dynamics is dedicated to Fred and to honouring that legacy.
After completing undergraduate studies at the University of Toronto, Fred, whose father Richard and uncle Alfred were noted pure mathematicians, completed his doctorate at MIT under Norman Levinson, studying differential equations.Following his graduation in 1956, Fred held teaching positions at the University of Chicago and the University of British Columbia before joining the faculty of the University of Wisconsin in 1960, where he remained until his retirement in 1997.(During roughly this same timespan, Fred's brother George was a professor of mathematics at the University of Minnesota.)Fred and his wife Esther raised their three children David, Deborah, and Michael in Madison before retirement brought them back to Vancouver, where Fred held an honorary appointment at UBC.In this way UBC bookended his career, and while in Vancouver Fred remained very active in mathematical biology, publishing research, supervising students, and organizing workshops, especially under the auspices of the Pacific Institute for the Mathematical Sciences (PIMS) and the Mathematics of Information Technology and Complex Systems (MITACS).Fred had tremendous influence on the work of the MITACS group that was formed in 2003 as a result of the SARS epidemic.He designed and analyzed SARS models, taught graduate  summer schools, contributed to workshops and gave interviews to the media about that epidemic.
Fred's research concentrated in three areas.For about fifteen years following completion of his Ph.D., Fred investigated properties of differential equations.He is best known for results on different types of stability of solutions.In Madison, Fred formed a longtime collaboration with John Nohel, with whom he wrote several books, one of which, The Qualitative Theory of Ordinary Differential Equations, remains a recognized classic.It is still in print after more than five decades.Another department colleague, Hans Schneider, joined them to write another work unifying linear algebra and linear systems of differential equations.
Beginning in the early 1970s, Fred shifted his focus to applying systems of differential equations to model predator-prey systems under population management policies, called harvesting or stocking depending on the policy.This groundbreaking work, much of it in collaboration with A.C. Soudack or David Sanchez, opened a new area of research in mathematical population biology, to which many researchers have contributed in the decades that followed.This was at a time when mathematical biology was in its infancy, hence the work was particularly novel.During this time Fred also studied the generalization of ODE systems to systems of Volterra integral equations in order to analyze the effects of distributed delays.
Fred began working in epidemiology in the late 1980s, a focus which remained with him for the rest of his career.Some of his early work examined the interplay between disease transmission and risk behaviour.In collaboration with colleagues, he produced early studies of emerging disease outbreaks during the past two decades, such as SARS, the 2009 H1N1 influenza pandemic, and COVID-19, writing extensively on pandemic preparedness.Meanwhile, his solo papers revisited key epidemiological ideas such as final outbreak size, developing results as simply as possible to facilitate interpretation and wider application.
Fred Brauer also dedicated significant effort over many decades to the development and mentoring of new researchers.In addition to teaching and supervising students of his own, he reached out through textbooks and many years of visits to research training programs.
The graduate text he co-authored with Carlos Castillo-Chavez, Mathematical models in population biology and ecology, has been cited over 3800 times in scientific literature and is now in its second edition.A more recent book of lecture notes written with Pauline van den Driessche and Jianhong Wu has already been cited more than 1100 times.Another text, Dynamical Systems for Biological Modelling: An Introduction, coauthored with Christopher Kribs, discusses a variety of biological modelling topics in a way that is accessible to both mathematics and biology students.Fred was a frequent visitor to the Mathematical and Theoretical Biology Institute founded by Carlos, where over the span of a quarter-century hundreds of undergraduates from backgrounds historically underrepresented in the sciences learned to conduct research and went on to earn doctorates and later tenured faculty positions at the best universities in the U.S. At one point MTBI was more than doubling the annual number of math Ph.D.s awarded to Latinos in the U.S. Fred and Carlos's experiences at MTBI formed the foundation for the textbooks they wrote together.Fred's teaching and research around the world dates back to his earliest years in mathematical biology, and he also made repeat visits to the International Research Experience for Students: Population Dynamics and Complex Systems program in Bogotá, Colombia, where he taught and mentored students from around the world.
Fred's work continues to have a profound impact on the field of mathematical biology.He was inducted into the 2015 class of SIAM Fellows for his contributions to all the areas described above.In 2021 alone his work was cited over 1700 times.Far greater is the number of students and researchers around the world who have benefited from the generosity of his time and encouragement.
This special volume honoring Fred contains reprints of the seven papers that Fred published in The Journal of Biological Dynamics (TJBD).These are followed by five research papers (previously published in TJBD in 2022 and 2023) written by authors influenced by Fred's research, including several of Fred's coauthors.These papers formulate and analyze diverse models of infectious diseases, some of which apply generally and some of which are specific to particular diseases, for example,

Figure 1 .
Figure 1.Fred Brauer speaking at a conference in May 2018, encouraging students to have the confidence to reach out and engage with senior researchers.Photo by Zhisheng Shuai.