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# Intro to Mathematical Biology MATH 345

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This 22 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 345 at College of William and Mary taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/231147/math-345-college-of-william-and-mary in Mathematics (M) at College of William and Mary.

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Date Created: 10/29/15

First epidemic model SIR modelKermackiMckendrick 1927 S 7551 1 51 7 a1 R a1 Assumptions 1 Total population is a constant N A average infective makes contact sufficient to transmit in fection With N others per unit time raction a of infectives leave the infective class per unit time or the average time of recovering from the disease is 104 N Basic reproduction number R0 04 330 04 Not used in last lecture Example A disease is introduced by two visitors into a town with 1200 inhabitants An average infective is in contact with 04 inhabitant per day The average duration of the infective period is 6 days and recovered infectives are immune against reinfection How many inhabitants would have to be immunized to avoid an epidemic 0N 04 so 3 041200 13000 m 003 means an aver age infective is in contact with 003 of all inhabitants per day or 16 per day Dimension of 3 is per person per day and dimension of or is per day dS d dR dT BS dT SI oz dT 04 Variable Dimension Parameter Dimension t 739 S p N p I p R p or 7 1 p1r1 S I R Nondnmensnonalization u v w t on N N N du d2 dw R y ZR 1 1 dt Ouv cu on 0 dt Qualitative analysis u R0uv v Rou 1v Behavior 1 If R0 lt 1 then 220 thus 10 is decreasing to zero no epidemics 2 If R0 gt 1 then 220 thus It is increasing initially if uO m 1 and is decreasing to zero as t gt oo epidemics occurs d2 1 2 1 Rou W ut Rio nut 2 220 uo Rio mum The final size of susceptible is uooi uoo Rio In uoo m 1 The fall of Aztecs In the Spanish invasion of South America in 1520 the Aztecs were devastated by a smallpox epidemic introduced by one of Cortez men Smallpox is a lethal disease Take N 1000 3 01 and the infective period for smallpox is two weeks 1 Find R0 2 Compare the final size of epidemics in the two cases i initially the whole population is susceptible ii initially population is 70 immune http www pbs orgconquistadorscortescortesh00 html SIS model disease With no Immunity 6gt S i SI a1 1 7 51 7 a1 Since SI N then it can be reduced to one equation 1 NiaIi 12 A logistic equation if R2 i gt 1 in this case a 1t tends to the carrying capacity and an endemic occurs If R lt 1 then epidemic Will never occur since 1t tends to 0 monotonically Other compartmental models aw e Exposed Infected but not yet Infectious CarrIer Infectious but not sIck SIR endemic including birth and death EbN BSI d8 BSI aI dI CI d RzaI dR d7 d7 d7 b birth rate d disease unrelated death rate c disease related death rate All new born are in the susceptible class so there is no vertical transmission parent to new born eg AIDS Nondimensionalized version assuming b d I R UZNUNwNtO I b7 dw or b d b d u l u ROU1 v R0u 1v 0 w dt ab dt dt ab ab 8 Basic reproductive number R0 f fb Equilibrium uv 10 disease free uvR61aLbl 351 endemic When R0 lt 1 the disease will die and disease free equilibrium is stable when R0 gt 1 the disease will stay in the population disease free equilibrium is unstable and endemic equilibrium is stable Vectorborne infectious diseases httpwwwicdc goVncidod dvbid CDC DIVISIOn of VectoriBorne Infectious Diseases West Nile Virus Transmission Cycle was mum 1mmquot xu 1 Criss cross infections example malaria Sl susceptible human 1 infected human 82 susceptible mosquito 1 infected mosquito N1 total human constant N2 total mosquito constant dsl 31 dIl 31 I I I I dT aplNl 2 71 1 dT aplNl 2 71 1 dSQ 11 db 1 ap2 S272I2b2N2 d2321 ap2 S2 V2I2 d212 d7 N1 d7 N1 a biting rate of mosquito 191192 probability of infection 11 Univariate difference equation Nn1 fNn Recap of mathematical tools see Appendix A1 A2 Linear equation Nn1 ANn solution Nn 2 NOW Equilibrium solution of N fN Periodic solution Nn 2 Nn for some p p is the period Linearization at equilibrium N linearized equation Nn1 N f NNn N stable if f N lt 1 unstable if f N gt 1 Behavior of linearized sequence If f N gt 1 exponential growth monotonically unstable If 0 lt f N lt 1 exponential decay monotonically stable If f N lt 1 growing oscillation oscillatorin unstable If 1 lt f N lt O decaying oscillation oscillatorily stable Graphing tools can be implemented via Matlab plotting the recursive sequence cobwebbing bifurcation diagram Bifurcation a qualitative change in the mathematical system and it can be reflected from the asymptotic behavior of solutions 1 Number of equilibrium changes 2 Stability of equilibria changes Bifurcation diagram of xn1 fxnu the curve of equilibria in mm space Use solid curve for stable equilibria and dotted curve for unstable equilibria Example 1 Saddle node blue sky bifurcation xn1 2 mm u mg 2 Transcritical bifurcation xn1 2 mm uxn mg 3 Pitchfork bifurcation xn1 2 mm iixn 323 all these three satisfying f 13 1 at the bifurcation point 4 Period doubling bifurcation xn1 mle mm at r 3 f 13 1 but ffa 1 so that the equilibrium becomes an unstable one and a stable period two solution emerges form the bifurcation before bifurcation stable pattern is a period one equilibrium after bifurcation a stable pattern is period two solution 23371 1 3371 Example Beverton Holt with harvesting xn1 2 consider the bifurcation with parameter h 3 Biological modeling Malthus model Nn1 ANn solution Nn NOW based on assumption the ratio of parents to offsprihgs is a constant n n1 gt Nonlinear model density dependent Nn1 ANnSNn SUV is the survival rate of the population which depends on the size of population and ASUV is the per capita reproduction rate How to determine SN 1 From certain assumptions to derive the mathematical forms 2 Available data suggests the shape of function then find func tions with such shape curve fitting see Lecture 2 New assumption the ratio NtNHl is an increasing function of the population size the parents have fewer children if the current population is larger Principle of modeling Everything should be made as simple as possible but not simpler Einstein Our choice the ratio NnNn1 is a increasing linear function Beverton Holt model Nnil i 1 1RNn or Nn1 R137 Nn R K 1 TNT Beverton Holt recruitment curve R maximum growth rate per capita K carrying capacity when the population size is over K it decreasgs 33m 1n Nondimensionalized equation xn1 Paradiso Purgatory and Inferno model xloii l 51 d 15d O1d x1n 13201 1 075d 1 d O1d 13201 13301 1 075d 05d 021 d 13301 x1O 100 x2O 2 200 x3O 300 Shorter form 5501 1 A 5501 f0 100 200 300 The solution A3301 1 A2330quot 2 A3330quot 3 Anx0 here Aquot is the power of matrix A which is not easy to calculate but easy for Matlab Observation from Matlab simulation d 02 1 All population grow 2 The total population grows exponentially logP has a linear growth 3 The fraction of each population in the total population tends to a constant d 08 1 All population decay 2 The total population decays exponentially 3 The fraction of each population in the total population tends to a constant A stage Structured model 3201 1 lt 025 01515 gt 5101 13O200O Observation exponential growth the numbers ofjuveniles and adults are almost same after a few time units 100 n100 nlOO For 5130 lt 100 then A 100 gt 11 100 100 100 AltlOOgt 11ltIOOgtI so the matrix power could bejust the power of a scalar number Linear algebra can explain all these Math 211 A crash course of linear algebra The numbers A satisfying Ax 2 Am are called eigenvalues The corresponding M72 0 is called eigenvector associated with the eigenvalue An n x n matrix has exactly n eigenvalues they could be same but with different eigenvectors If v is an eigenvector so is co c is a constant Example 2 x 2 matrix Find the eigenvalues and eigenvectors of A if g gt Eigenvalue and eigenvector of 2 x 2 matrix a b a A a a A b a O c d y y 39 c d A y O 39 solve a Ad A bc 2 O characteristic equation for eigen values A199 Example Fibonacci model Eigenvalues eigenvectors for higher dimensional matrices use Matlab to solve Solution of 5101 1 Axn A is a k x k matrix k c dfvl CQAEL UQ ckAka Z cikgbvi i1 where M are eigenvalues vi are eigenvectors and a are con stants The eigenvalues can be ordered so that gt1 2 gt2 2 2 Ak and A1 is called dominant eigenvalue 3301 m clk lvl for large n

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