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Fred Brauer: The Man and His Mathematics

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Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 126))

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Abstract

When I first met Fred Brauer, he was warning a class of first-semester calculus students about the dangers of applying techniques without thinking. As an example, he wrote on the chalkboard the expression\( \frac{{\sin x}}{n} \). He then proceeded to cancel the n’s from numerator and denominator:

$$ \frac{{si\not{n}x}}{{\not{n}}} $$

, leaving six.

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References

  1. Fred Brauer and Shlomo Sternberg (1958). Local uniqueness, existence in the large, and the convergence of successive approximations, Amer. J. Math. 80: 421–430.

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  2. Fred Brauer and Shlomo Sternberg (1959). Errata to our paper “Local uniqueness, etc.”, Amer. J. Math. 81: 797.

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  3. Fred Brauer (1959). A note on uniqueness and convergence of successive approximations, Canad. Math. Bull. 2: 5–8.

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  9. Fred Brauer and A. C. Soudack (1978). Response of predator-prey systems to nutrient enrichment and proportional harvesting, Internat. J. Control 27(1): 65–86.

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  10. Fred Brauer and A. C. Soudack (1982). On constant effort harvesting and stocking in a class of predator-prey systems, J. Theor. Biol. 95(2): 247–252.

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  11. Fred Brauer (1987). A class of Volterra integral equations arising in delayed-recruitment population models, Natur. Resource Modeling 2(2): 259–278.

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  12. Herbert W. Hethcote and James A. Yorke (1984). Gonorrhea transmission dynamics and control. Lecture Notes in Biomathematics 56. New York: Springer-Verlag.

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  13. S. P. Blythe, Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1995). Models for sexually transmitted diseases with recruitment, in Mathematical Population Dynamics: Analyses of Heterogeneity (O. Arino, D. Axelrod, M. Kimmel, M. Langlais, eds.). Winnipeg: Wuerz Publishing Co. Vol. 1, pp. 197–207.

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  14. Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1996). Recruitment effects in heterosexually transmitted disease models, Int. J. App. Science & Computation 3: 78–90.

    Google Scholar 

  15. Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1997). Recruitment with a core group and its effect on the spread of a sexually transmitted disease, in Advances in Mathematical Population Dynamics — Molecules, Cells, and Man (O. Arino, D. Axelrod, M. Kimmel, eds.). Singapore: World Scientific Press. pp. 477–486.

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Papers by Fred Brauer

  1. Fred Brauer (1958). Singular self-adjoint boundary value problems for the differential equation Lx = λMx, Trans. Amer. Math. Soc. 88: 331–345.

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  2. Fred Brauer (1958). Spectral theory for the differential equation Lu = λMu, Can. J. Math. 10: 431–446.

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  3. Fred Brauer and Shlomo Sternberg (1958). Local uniqueness, existence in the large, and the convergence of successive approximations, Amer. J. Math. 80: 421–430; errata, same journal, 81: 797 (1959).

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  4. Fred Brauer (1959). A note on uniqueness and convergence of successive approximations, Can. Math. Bulletin 2: 5–8.

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  5. Fred Brauer (1959). Some results on uniqueness and successive approximations, Can. J. Math. 11: 527–533.

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  6. Fred Brauer (1960). Spectral theory for linear systems of differential equations, Pacific J. Math. 10: 17–34.

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  7. Fred Brauer (1961). Global behavior of solutions of ordinary differential equations, J. Math. Anal. Appl. 2: 145–158.

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  8. Fred Brauer (1962). Asymptotic equivalence and asymptotic behavior of linear systems, Mich. Math. J. 9: 33–43.

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  9. Fred Brauer (1963). Lyapunov functions and comparison theorems, in International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics (J. P. LaSalle & S. Lefschetz, eds.). New York: Academic Press. pp. 435–441.

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  10. Fred Brauer (1963). Bounds for solutions of ordinary differential equations, Proc. Amer. Math. Soc. 14: 36–43.

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  11. Fred Brauer (1963). On the asymptotic behavior of Bessel functions, Amer. Math. Monthly 70: 954–957.

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  12. Fred Brauer (1964). Nonlinear differential equations with forcing terms, Proc. Amer. Math. Soc. 15: 758–765.

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  13. Fred Brauer (1964). On the completeness of biorthogonal systems, Mich. Math. J. 11: 379–383; errata, same journal, 12: 127–128 (1965).

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  14. Fred Brauer (1965). Some refinements of Lyapunov’s second method, Can. J. Math. 17: 811–819.

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  15. Fred Brauer (1966). Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 14: 198–206.

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  16. Fred Brauer (1966). The use of comparison theorems for ordinary differential equations, in Stability Problems of Solutions of Diff. Equations (Proc. NATO Advanced Study Inst., Padua, 1965). Gubbio: Edizione “Oderisi”. pp. 29–50.

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  17. Fred Brauer (1966). The asymptotic behavior of perturbed nonlinear systems, in Stability Problems of Solution of Diff. Equations (Proc. NATO Advanced Study Inst., Padua, 1965). Gubbio: Edizione “Oderisi”. pp. 51–56.

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  18. Fred Brauer (1966). The solution of non-homogeneous systems of differential equations by undetermined coefficients, Can. Math. Bull. 9: 81–87.

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  19. Fred Brauer (1967). Green’s functions for singular ordinary differential operators, Can. J. Math. 19: 571–582.

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  20. Fred Brauer (1967). Perturbations of nonlinear systems of differential equations, II., J. Math. Anal. Appl. 17: 418–434.

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  21. Fred Brauer (1968). A class of nonlinear eigenvalue problems, in U.S.-Japan Seminar on Differential & Functional Equations. New York: W. A. Benjamin, Inc. pp. 429–433.

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  22. Fred Brauer (1968). Nonlinear perturbations of Sturm-Liouville boundary-value problems, J. Math. Anal. Appl. 22: 591–598.

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  23. Fred Brauer and James S. W. Wong (1969). On asymptotic behavior of perturbed linear systems, J. Diff. Equations 6: 142–153.

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  24. Fred Brauer and James S. W. Wong (1969). On the asymptotic relationships between solutions of two systems of ordinary differential equations, J. Diff. Equations 6: 527–543.

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  25. Fred Brauer and Aaron Strauss (1970). Perturbations of nonlinear systems of differential equations, III, J. Math. Anal. Appl. 31: 37–48.

    Article  MathSciNet  MATH  Google Scholar 

  26. Fred Brauer (1972). Perturbations of nonlinear systems of differential equations, IV, J. Math. Anal. Appl. 37(1): 214–222.

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  27. Fred Brauer (1972). A nonlinear variation of constants formula for Volterra equations, Math. Systems Theory 6: 226–235.

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  28. Fred Brauer (1972). A nonlinear predator-prey problem, Ordinary differential equations, Proc. 1971 NRL-MRC Conference (L. Weiss, ed.). New York: Academic Press. pp. 371–377.

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  29. Fred Brauer (1972). The nonlinear simple pendulum, Ann. Math. Monthly 79: 348–355.

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  30. Fred Brauer (1974). On the populations of competing species, Math. Biosci. 19: 299–306.

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  31. Fred Brauer (1975). On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal. 6: 312–317.

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  32. Fred Brauer and David A. Sánchez (1975). Constant rate population harvesting: equilibrium and stability, Theor. Pop. Biol. 8: 12–30.

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  33. Fred Brauer and David A. Sánchez (1975). Some models for population growth with harvesting, in Proc. International Conference on Differential Equations, Univ. of Southern Calif., Los Angeles, 1974 (H. A. Antosiewicz, ed.). New York: Academic Press. pp. 53–64.

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  34. Fred Brauer and David A. Sánchez (1976). Cosecha de poblaciones en competencia [Harvesting of competing populations], in Mathematical Notes and Symposia, Vol. 2: Ecuaciones Differenciales (Proc. Third Mexico-U.S. Symposium) (C. Imaz, ed.). Mexico City: Fondo de Cultura Económica. pp. 171–176.

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  35. Fred Brauer (1976). Constant rate harvesting of populations governed by Volterra integral equations, J. Math. Anal. Appl. 56: 18–27.

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  36. Fred Brauer (1976). Some applications of the theory of ordinary differential equations to population growth problems, Ann. Acad. Brasil de Ciencias 48: 369–385.

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  37. Fred Brauer (1976, October). Perturbations of the nonlinear renewal equation, Advances in Math. 22(1): 32–51.

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  38. Fred Brauer (1976). Destabilization of predator-prey systems under enrichment, Int. J. Control 23: 541–552.

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  39. Fred Brauer, A. C. Soudack and H. S. Jarosch (1976). Stabilization and destabilization of predator-prey systems under harvesting and nutrient enrichment, Int. J. Control 23: 553–573.

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  40. Fred Brauer (1977). Stability of some population models with delay, Math.Biosciences 33: 345–358.

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  41. Fred Brauer (1977). Periodic solutions of some ecological models, J. Theor. Biol. 69: 143–152.

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  42. Fred Brauer and A. C. Soudack (1978). Response of predator-prey systems to nutrient enrichment and proportional harvesting, Int. J. Control 27: 65–86.

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  43. Fred Brauer (1978). Asymptotic stability of a class of integro-differential equations, J. Differential Equations 28: 180–188.

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  44. Fred Brauer (1979). Harvesting strategies for population systems, Rocky Mountain J. Math. 19: 19–26.

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  45. Fred Brauer (1979). Decay rates for solutions of a class of differential-difference equations, SIAM J. Math. Anal. 10: 783–788.

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  46. Fred Brauer and A. C. Soudack (1979). Stability regions and transitions phenomena for harvested predator prey systems, J. Math. Biol. 7: 319–337.

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  47. Fred Brauer (1979). Boundedness of solutions of predator-prey systems, Theor. Pop. Biol. 15: 268–273.

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  48. Fred Brauer (1979). Characteristic return times for harvested population models with time-lag, Math. Biosci. 45: 295–311.

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  49. Fred Brauer and A. C. Soudack (1979). Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol. 8: 55–71.

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  50. Fred Brauer and A. C. Soudack (1981). Constant-rate stocking of predator-prey systems, J. Math. Biol. 11(1): 1–14.

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  51. Fred Brauer and A. C. Soudack (1981). Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol. 12: 101–114.

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  52. Fred Brauer and A. C. Soudack (1981). Constant-rate effort harvesting and stocking in a class of predator-prey systems, in Differential Equations and Applications in Ecology, Epidemics and Population Problems (Stavros N. Busenberg and Kenneth L. Cooke, eds.). New York: Academic Press. pp. 131–144.

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  53. Fred Brauer and A. C. Soudack (1982). On constant effort harvesting and stocking in a class of predator-prey systems, J. Theor. Biol. 95: 247–252.

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  54. Fred Brauer (1983). Constant-rate harvesting of age-structured populations, SIAM J. Math Anal. 14 (1983), 947–961.

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  55. Fred Brauer (1983). Nonlinear age-dependent population growth under harvesting, Int. J. Computers and Math. with Appl. 9: 345–352.

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  56. Fred Brauer (1984). The effect of harvesting on population systems, in Trends in Theory & Practice of Nonlinear Differential Equations (Proc., Arlington, TX, 1982, V. Lakshmikantham, ed.). New York: Marcel Dekker. pp. 81–89.

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  57. Fred Brauer (1984). Constant-yield harvesting of population systems, in Mathematical Ecology, Proc. Trieste 1982 (S. A. Levin and T. G. Hallam, eds.), Lec. Notes in Biomathematics 54. Berlin: Springer-Verlag. pp. 238–246.

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  58. Fred Brauer and A. C. Soudack (1985). Optimal harvesting in predator-prey systems, Int. J. Control 41(1) 111–128.

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  59. Fred Brauer and A. C. Soudack (1985). Mutualism models with nonlinear growth rates, Int. J. Control 41(6): 1601–1612.

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  60. Fred Brauer and Ma Zhien (1987). Stability of stage-structured population models, J. Math. Anal. Appl. 126: 301–315.

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  61. Fred Brauer (1987). A class of Volterra integral equations arising in delayed-recruitment population models, Nat. Res. Modeling 2(2): 259–278.

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  62. Fred Brauer (1987). Absolute stability in delay equations, J. Differential Equations 69: 185–191.

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  63. Fred Brauer (1987). A simplification of Taylor’s theorem, Amer. Math. Monthly 94: 453–455.

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  64. Fred Brauer (1987). Harvesting in delayed-recruitment population models, in Proc. Canadian Mathematical Society 1986 Seminar on Oscillation, Bifurcation and Chaos. Providence, RI: Amer. Math. Soc. pp. 317–327.

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  65. Fred Brauer, Stephen Ellner and M. D. Krom (1988). Modeling and management for seawater fishponds, in Proc. 1986 Trieste Research Conference on Mathematical Ecology (T. G. Hallam, S. A. Levin, and L. J. Gross, eds.). Teaneck, NJ: World Scientific Press. pp. 215–235.

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  66. Fred Brauer (1988). Coexistence and survival of invading species, in Proc. 1986 Trieste Research Conference on Mathematical Ecology (T. G. Hallam, S. A. Levin, and L. J. Gross, eds.). Teaneck, NJ: World Scientific Press. pp. 599–610.

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  67. Fred Brauer, David Rollins and A. C. Soudack (1988). Harvesting in population models with delayed recruitment and age-dependent mortality, Nat. Res. Modeling 3(1): 45–62.

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  68. Fred Brauer (1988). Some topics in population biology involving delay equations, 38 pp.; translated into Chinese by Ma Zhien. Xian, China: Department of Mathematics, Xian Jiaotong University. 58 pp.

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  69. Fred Brauer (1989). Epidemic models in populations of varying size, in Mathematical Approaches to Problems in Resource Management and Epidemiology (C. Castillo-Chávez, S. Levin, and C. Shoemaker, eds.), Lecture Notes in Biomathematics 81. Berlin: Springer-Verlag. pp. 109–123.

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  70. Fred Brauer (1989). Multi-species interactions and coexistence, in Proc. Int. Conf. on Theory and Applications of Differential Equations (A.R. Aftabizadeh, ed.). Athens, OH: Ohio Univ. Press. pp. 91–96.

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  71. Fred Brauer (1990). Models for the spread of universally fatal diseases, J. Math. Biology 28: 451–462.

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  72. Fred Brauer (1990). Some infectious disease models with population dynamics and general contact rates, J. Diff. Integral Equations 5: 827–836.

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  73. Fred Brauer (1991). Stability of equilibria in some infectious disease models, in Differential Equations: Stability and Control (S. Elaydi, ed.). New York: Marcel Dekker. pp. 53–62.

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  74. Fred Brauer and Hao Dun-Yuan (1991). Analysis of a characteristic equation, J. Integral Equations and Appl. 3: 239–254.

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  75. Fred Brauer (1991). Models for the spread of universally fatal diseases II, in Differential Equations Models in Biology, Epidemiology, and Ecology, Proc. Claremont 1990 (Stavros Busenberg and Mario Martelli, eds.), Lec. Notes in Biomathematics 92. Berlin: Springer-Verlag. pp. 57–69.

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  76. S. P. Blythe, Fred Brauer and Carlos Castillo-Chávez (1995). Demographic recruitment in sexually transmitted disease models, in Computational Medicine, Public Health and Biotechnology: Building a Man in the Machine (M. Witten, ed.). Singapore: World Scientific Press. pp. 1438–1457.

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  77. Fred Brauer (1995). Models for diseases with exposed period, Rocky Mountain J. Math. 25: 57–66.

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  78. S. P. Blythe, Fred Brauer, Carlos Castillo-Chávez, and Jorge X. Velasco-Hernández (1995). Models for sexually transmitted diseases with recruitment, in Mathematicl Population Dynamics: Analyses of Heterogeneity (O. Arino, D. Axelrod, M. Kimmel, and M. Langlais, eds.). Vol 1. Winnipeg: Wuerz Publishing Co. pp. 197–207.

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  79. Fred Brauer (1995). Models for diseases with vertical transmission and nonlinear population dynamics, Math. Biosci. 128: 13–24.

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  80. Fred Brauer (1996). Variable infectivity in communicable disease models, in Proc. First World Congress of Nonlinear Analysts, Tampa, Florida, 1992 (V. Lakshmikantham, ed.). Berlin: de Gruyter. Vol 4, pp. 3201–3210.

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  81. Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1996). Recruitment effects in heterosexually transmitted diseases, Int. J. App. Science & Computation 3: 78–90.

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  82. Fred Brauer and Carlos Castillo-Chávez (1995). Basic models in epidemiology, in Ecological Time Series (T.M. Powell and J.H. Steele, eds.). Chapman & Hall. pp. 410–447.

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  83. Fred Brauer, Jorge X. Velasco-Hernández and Carlos Castillo-Chávez (1996). Effects of treatment and prevalence-dependent recruitment on the dynamics of a fatal disease, IMA J. Math. Applied to Medicine and Biology 13: 175–192.

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  84. Fred Brauer (1996). A characteristic equation arising in models for diseases with vertical transmission and without immunity, in Differential Equations and Applications to Biology and to Industry (M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme, eds.). Singapore: World Scientific Press. pp. 41–48.

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  85. Fred Brauer (1996). A model for populations with variable maturation period, Dynamics of Continuous, Discrete, and Impulsive Systems 2: 41–50.

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  86. Fred Brauer, Carlos Castillo-Chávez and Jorge X. Velasco-Hernández (1997). Recruitment with a core group and its effect on the spread of a sexually transmitted disease, in Advances in Mathematical Population Dynamics: Molecules, Cells, and Man (O. Arino, D. Axelrod, and M. Kimmel, eds.). Singapore: World Scientific Press. pp. 477–486.

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  87. Fred Brauer (1997). Continuous and discrete delayed, recruitment population models, Dynamics of Continuous, Discrete and Impulsive Systems 3: 245–252.

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  88. Fred Brauer (1999). General recruitment models for sexually transmitted diseases, in Differential Equations with Applications to Biology (S. Ruan, G. S. K. Wolkowicz, and J. Wu, eds.), Fields Institute Communications No. 21. Providence, RI: Amer. Math. Soc. pp. 45–50.

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  89. Fred Brauer (1999). Continuous and discrete population models with age-dependent mortality, Dynamics of Continuous, Discrete, and Impulsive Systems 5: 107–113.

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  90. Fred Brauer (in press). Time lags in disease models with recruitment, Mathematical and Computer Modelling, to appear.

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  91. Fred Brauer (in press). Infectious disease models with chronological age structure, this volume.

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  92. Fred Brauer (in press). Basic ideas of mathematical epidemiology, this volume.

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  93. Fred Brauer (in press). Extensions of the basic models, this volume.

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  94. Fred Brauer and Pauline van den Driessche (in press). Models for the transmission of disease with immigration of infectives, to appear.

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  95. Fred Brauer (in press). What goes up must come down, eventually, American Mathematical Monthly, to appear.

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  96. Fred Brauer (in press). A model for an SI disease in an age-structured population, to appear.

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Books by Fred Brauer

  1. Fred Brauer and John A. Nohel (1967). Ordinary Differential Equations: A first course. New York: W. A. Benjamin, Inc. 1st ed. xvi + 457 pp. 2nd ed., 1973, ix + 470 pp.

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  2. Fred Brauer and John A. Nohel (1968). Elementary Differential Equations: Principles, problems, solutions. New York: W. A. Benjamin, Inc. xi + 222 pp.

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  3. Fred Brauer and John A. Nohel (1968). Problems and Solutions in Ordinary Differential Equations. New York: W. A. Benjamin, Inc. x + 267 pp.

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  4. Fred Brauer and John A. Nohel (1969). Qualitative Theory of Ordinary Differential Equations. New York: W. A. Benjamin, Inc. xi + 314 pp. Reprinted, Dover, 1989.

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  5. Fred Brauer, John A. Nohel and Hans Schneider (1970). Linear Mathematics. New York: W. A. Benjamin, Inc. xii + 347 pp.

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  6. Fred Brauer (1976). Some Stability and Perturbation Problems for Differential and Integral Equations, Monografías de Matemática no. 25. Rio de Janeiro: Instituo de Matemática Pura e Aplicada, iii + 163 pp.

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  7. Fred Brauer and John A. Nohel (1985). An Introduction to Differential Equations with Applications. New York: Harper & Row. xii + 620 pages.

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  8. Fred Brauer and Carlos Castillo-Chávez (2001). Mathematical Models in Population Biology and Epidemiology. (Texts in Applied Mathematics 40), Springer-Verlag. New York, (c) 2001, 416 pages, ISBN 0–387–98902–1.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019-0408, USA

    Christopher M. Kribs-Zaleta

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  1. Christopher M. Kribs-Zaleta
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Editor information

Editors and Affiliations

  1. Mathematical and Theoretical Biology Institute and member, Departments of Biometrics, Statistics and Theoretical and Applied Mechanics, Cornell University, 14853-7801, Ithaca, NY, USA

    Carlos Castillo-Chavez

  2. Department of Biomathematics, UCLA School of Medicine, 90095-1766, Los Angeles, CA, USA

    Sally Blower

  3. Department of Math and Stats, University of Victoria, V8W 3P4, Victoria, British Columbia, Canada

    Pauline van den Driessche

  4. Dept. of Microbiology and Immunology, University of Michigan Medical School, 48109-0620, Ann Arbor, MI, USA

    Denise Kirschner

  5. Department of Mathematics Howard, Howard University, 20059, Washington, DC, USA

    Abdul-Aziz Yakubu

  6. Biometrics Department, Cornell University, USA

    Abdul-Aziz Yakubu

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Kribs-Zaleta, C.M. (2002). Fred Brauer: The Man and His Mathematics. In: Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., Yakubu, AA. (eds) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory. The IMA Volumes in Mathematics and its Applications, vol 126. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0065-6_2

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