Mathematical Analysis of Epidemiological Models III Jan Medlock - PowerPoint PPT Presentation

Intro Computing R 0 Complex models Mathematical Analysis of Epidemiological Models III Jan Medlock Clemson University Department of Mathematical Sciences 27 July 2009

Intro Computing R 0 Complex models What is R 0 ? Basic Reproduction Number Net Reproductive Rate “the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is susceptible” (Anderson & May, 1991)

Intro Computing R 0 Complex models Why is R 0 important? • For a wholly susceptible host population, R 0 > 1 pathogen can invade. R 0 < 1 pathogen cannot invade. • When a pathogen is present in the population, often, but not always, R 0 < 1 pathogen will die out of the population.

Intro Computing R 0 Complex models The effective reproduction number, R If the population is not wholly susceptible, then we have R , the effective reproduction number. • Pathogen already present • Vaccinated population

Intro Computing R 0 Complex models How to compute R 0 ? • Heuristic methods • Systematic method P. van den Driessche & James Watmough, 2002, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Mathematical Biosciences , 180: 29–48.

Intro Computing R 0 Complex models Example model for STI M S M E M I M R F S F E F I F R d M S = ω M M R − β M F I d F S = ω F F R − β F M I F M S M F S d t d t d M E = β M F I d F E = β F M I F M S − τ M M E M F S − τ F F E d t d t d M I d F I = τ M M E − γ M M I d t = τ F F E − γ F F I d t d M R d F R = γ M M I − ω M M R = γ F F I − ω M F R d t d t

Intro Computing R 0 Complex models Procedure Decide which states are infected We need to decide which states are infected and which are uninfected. In the STI model, Infected: M E , F E , M I , F I Uninfected: M S , F S , M R , F R

Intro Computing R 0 Complex models Procedure Find disease-free equilibrium (or other equilibrium) Set d x d t = 0 for all model state variables to find equilibrium. Also, for disease-free equilibrium, there are no infected people.

Intro Computing R 0 Complex models Procedure Find disease-free equilibrium (or other equilibrium) 0 0 0 = ω M M R − β M F M S 0 = ω F F R − β F M F S 0 0 0 = β M F M S − τ M 0 0 = β F M F S − τ F 0 0 = τ M 0 − γ M 0 0 = τ F 0 − γ F 0 0 = γ M 0 − ω M M R 0 = γ F 0 − ω F F R M S = F S = P 2 M E = F E = M I = F I = M R = F R = 0 M = F = P 2

Intro Computing R 0 Complex models Procedure Decide which terms are new infections From the right-hand sides of the equations for the infected states, decide which terms represent new infections, F . The remainder are −V . d x d t = F − V F is the rate of production of new infections. V is the transition rates between states.

Intro Computing R 0 Complex models Procedure Decide which terms are new infections d M E = β M F I F M S − τ M M E d t d F E = β F M I M F S − τ F F E d t d M I = τ M M E − γ M M I d t d F I d t = τ F F E − γ F F I F I     β M F M S τ M M E M I M F S τ F F E β F     F = V =  ,     − τ M M E + γ M M I  0       − τ F F E + γ F F I 0

Intro Computing R 0 Complex models Procedure Take derivatives at equilibrium d F 1 d F 1 d V 1 d V 1     · · · · · · d x 1 d x n d x 1 d x n F = d F V = d V . . . .  . .   . .  d x = d x = . . . .         d F n d F n d V n d V n · · · · · · d x 1 d x n d x 1 d x n These are the rates for new infections and transitions near the equilibrium.

Intro Computing R 0 Complex models Procedure Take derivatives at equilibrium At the disease-free equilibrium, M S = F S = M = F = P 2 , M E = F E = M I = F I = M R = F R = 0 F I M S       β M F M S 0 0 0 β M 0 0 0 β M F M I F S 0 0 0 β F M F S 0 0 β F 0 β F       F =  , F = M  =       0 0 0 0 0 0 0 0 0           0 0 0 0 0 0 0 0 0     τ M M E τ M 0 0 0 0 0 0 τ F F E τ F     V =  , V =     − τ M M E + γ M M I − τ M 0 γ M 0        − τ F F E + γ F F I 0 0 − τ F γ F

Intro Computing R 0 Complex models Procedure Find V − 1 V − 1 gives the times spent in each state. In general, finding the inverse is difficult by hand, but computer algebra (Sage, Maple, Mathematica) takes care of that.  1  0 0 0 τ M 1 0 0 0   V − 1 = τ F    1 1  0 0   γ M γ M   1 1 0 0 γ F γ F

Intro Computing R 0 Complex models Procedure Find FV − 1 FV − 1 gives the total production of new infections over the course of an infection. 1  0 0 0   0 0 0 β M  τ M 1 0 0 0 0 0 β F 0 V − 1 =   τ F F =   ,    1 1 0 0 0 0 0 0   γ M γ M  0 0 0 0 1 1 0 0 γ F γ F β M β M 0 0   γ F γ F β F β F 0 0 FV − 1 =   γ M γ M     0 0 0 0   0 0 0 0

Intro Computing R 0 Complex models Procedure Find ρ ( FV − 1 ) The largest eigenvalue λ 0 gives the fastest growth of the infected population. FV − 1 � N → λ N � 0 v 0 for large N . So R 0 = λ 0 .  β M β M  0 0 γ F γ F β F β F FV − 1 = 0 0   γ M γ M   0 0 0 0   0 0 0 0 � � � � � β F β M β F β M β F β M σ ( FV − 1 ) = 0 , , − = ⇒ R 0 = γ M γ F γ M γ F γ M γ F

Intro Computing R 0 Complex models Alternative interpretation If we had chosen only F E & F I to be infected states, then R 0 = β F β M γ M γ F

Intro Computing R 0 Complex models More complex models Flu S I R d S a d t = − λ a S a 17 d I a λ a = σ a � d t = λ a S a − ( γ a + ν a ) I a , φ a α β α I α , N α = 1 d R a d t = γ a I a , for a = 1 , . . . , 17

Intro Computing R 0 Complex models More complex models Flu • I a are infected states • Equilibrium is everyone susceptible, with given age structure • New-infection term is λ a S a , so F = λ ⊗ S , V = ( γ + ν ) ⊗ I • Then σ ⊗ S �� � � β T F = V = diag ( γ + ν ) ⊗ φ, N • And �� σ ⊘ ( γ + ν ) ⊗ S � � FV − 1 = β T ⊗ φ N

Intro Computing R 0 Complex models More complex models Flu Putting in parameter values from the pandemics, we get 1918 R 0 = 1 . 2 1957 R 0 = 1 . 3 0 . 02 Proportion infected 1957 1918 0 . 01 0 . 00 0 60 120 180 240 300 360 Time (days)