Math 19. Lecture 1Course OrientationT. JudsonFall 20051 Modeling in the Biological Sciences• Mathematics consists of the study and development of methods ofpredictions.• The goal of biology is to find useful and verifiable descriptions andexplanations of phenomena in the natural world.2 Population growthThe rate of growth of population is proportional to the size of the population,dPdt= kP,P (0) = P0.This is a good model for a population that grows without constraints.3 Predator-Prey ModelAssumptions for how a population of rabbits, R, and foxes, F , inter act.1. If no foxes are present, the rabbits repro duce at a rate proportional totheir population, and they are not affected by overcrowding.2. The foxes eat the rabbits, and the rate at which the rabbits are eatenis proportional to the rate at which the rabbits and the foxes interact.3. Without rabbits to eat, the fox population declines at a rate propor-tional to itself.14. The rate at which foxes are born is proportional to the number ofrabbits eaten by foxes, which by the second assumption, is proportionalto the rate at which the foxes and rabbits interact.The systemdRdt= αR − βR FdFdt= −γF + δRFis a model of how a predator population interacts with a prey population.4 The Evolutionary Development of HIV-1How d oes th e HIV-1 virus evolve in the human body? Once infected withthe HIV-1 virus, it can be years before an HIV-positive patient exhibits thefull symptoms of AIDS. Does the virus lie dormant in patient’s body? Thiswas thought to be the case at one time.Here is a model th at tells us otherwise. Due to transcription errorsduring the replication process, quasispecies of the virus are created. Thesequasispecies are populations of closely related but distinct viral genomes of2the virus. We can base a mathematical model of this process on the followingassumptions.1. The virus can kill C D4-positive T -helper cells.2. The continual evolution of new resistant viral mutants enables thetotal viral population to evade elimination by the immune system.3. Subpopu lations of CD4-positive T-helper cells specific to a particularviral direct immunological attack against that strain.4. Each mutant can kill all CD4 cells, regardless of their specificity to aparticular mutant.5. Immunological responses to the virus are characterized by a specificresponse to individual strains and a non-specific general response thatacts against all strains.We assign the following variables.• z is the population of nonspecific CD4-positive T-helper cells.• viis the population of a virus strain, where i = 1, 2, . . . , n.• xiis the population to strain-specific CD4-positive T-helper cells,where i = 1, 2, . . . , n.We are now ready to construct out basic model. We do this in terms ofsystems of equations.• Each str ain of the virus will grow at a ratedvidt= rvi− szvi− pxivi,where r, s, and p are constants.• The population of nons pecific CD4-positive T-helper cells will behaveaccording to the equationdzdt= k′v − uvz,where v =Pviand k′and u are constants. Note that the immunecells are produced at a rate k′v proportional to the density of antigens.The term uvz tells the rate at which the virions are destroyed.3• The population of specific CD4-positive T-helper cells will behave ac-cording to the equationdxidt= kvi− uvxi,where k and u are constants.The problem with modeling such a system is th at n keeps increasing.We do not know how to solve s uch a system. One way to deal with thisdifficulty is to consider a finite systemdxidt= kvi− uvxidzdt= k′v − uvzdvidt= rvi− szvi− pxivi+ M (v),where M(v) is a term representing the appearance of new viral strains.Homework• NoneReadings and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001.• P. Blanchard, R. Devaney, and G. Hall. Differential Equations, secondedition. Brooks/Cole, Pacific Grove, CA, 2002, pp. 4–8, 11-13.• M. Nowak, R. May, and R. Anderson. “The evolutionary dynamics ofHIV-1 quasispecies and the development of immunodeficiency disease,”AIDS, 1990, Vol 4, No
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