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Find the nullclines and equilibrium points of the SIRS system. Using the SIR model, a. Derive the solution for I(S) from the SIR model. b. Derive the formula for R-naught involving 5* c. If the total number of people (in a population of size 100,000} who did not get infected by a virus by the end of an outbreak is 60,000, determine R-naught (using the formula from part b.). d. If the viral infection (from part c.) is contagious for a period of 4 days, determine the value of the parameter k. Use this value and the R-naught value to determine the contact rote. e. Find the minimum vaccination rate for this infectious disease. SIR models 1 Monday , March 30, 2020 1:04 PM systems of DE s Infections Disease Models ( SJR ) sit) to of susceptible Individuals at time t I ( t ) olb of infected R(t) olo of recovered N (t ) total pop N = S+ItR (assume constant ) N + N N SI Model I Let b # of contacts sufficient to transmit the virus per time per individual then by # of new infections pertime generated by a single infected individual the model is = - bs I At * I It = b SI 3-30 DE systems (SIR models) Page 1Equilibria Long-term state of the system Set S = O and I = O 6 = - b s I -> s zu VI phase plane Trajectonties 0 - bsI I = O V S Divide DE'S At S+ I = N - -1 d s AS at at as = 1 - Ids I ( s ) = - s + ( Analytic Soln I ( 0 ) = I . = - ( 0 ) + @ Note ' S t I = 1 => S = 1 - I C = I . d I = b ( 1 - I ) I logistic trough Model I ( s ) = - S + I . ( solve by separation of vary ) slope = - 1 3-30 DE systems (SIR models) Page 2 3-30 DE systems (SIR models) Page 3slope stable. I = Ous (unstable) 6 Un stadle S bS I ( - bstr ) = 0-7/st. 6 SIS Model S Since I = 1 -S, (I- 1-Y I b = 05 I rate of recover V = 0.2 ( loss of immunity ) r r fraction of infected group the) s* = b 6 2 - 0.4 05 recovers pertime SIS /R Model b 5 ' = - b s I tr I S I = b S I - r. I I E. g, If the any duration of infection M is 3days , then an average, Mortality , of the currently infected pop recovers each dry Heure r= 12 Eg- Values S ' = 0 = - b s I + rI I = 0 = bs I - vI 3-30 DE systems (SIR models) Page 4 3-30 DE systems (SIR models) Page 5