Avian influenza in migratory birds and basic reproductive ratio

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1 Avian influenza in migratory birds and basic reproductive ratio Xiang-Sheng Wang Mprime Centre for Disease Modelling York University, Toronto (joint work with Jianhong Wu) Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 1

2 Outline The model Two kinds of basic reproductive ratio Numerical simulation The third basic reproductive ratio Conclusion and discussion Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 2

3 The ecological model Let S w and S b be bird populations at winter refuge site and summer birth site. S w(t) = [µ w + m wb (t)]s w (t) + α bw m bw (t τ bw )S b (t τ bw ); S b(t) = [µ b + m bw (t)]s b (t) + α wb m wb (t τ wb )S w (t τ wb ) + b(t)s b (1 S b K ). 1. µ w and µ b : natural death rates. 2. m bw and m wb : migration rates; α bw and α wb : survival probabilities. 3. τ bw and τ wb : time delay (flight duration from one site to another). 4. b(t) and K: logistic birth and capacity. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 3

4 The full model S w(t) = [µ w + m wb (t)]s w (t) + α bw m bw (t τ bw )S b (t τ bw ) β ws w I w S w + I w ; I w(t) = [µ w + m wb (t)]i w (t) + α bw m bw (t τ bw )I b (t τ bw )+ β ws w I w S w + I w µ i wi w ; S b(t) = [µ b + m bw (t)]s b (t) + α wb m wb (t τ wb )S w (t τ wb ) + b(t)s b (1 S b K ) β bs b I b S b + I b ; I b(t) = [µ b + m bw (t)]i b (t) + α wb m wb (t τ wb )I w (t τ wb )+ β bs b I b S b + I b µ i bi b. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 4

5 Three basic assumptions The migratory activity is insignificant during summer breeding season and winter refuge time; The population left in the winter refuge site (resp. summer breeding site) after spring (resp. autumn) migration is comparably negligible; The breeding activity does not occur during autumn and winter seasons. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 5

6 Notations According to seasonal activities of migratory birds, we divide a year by four seasons: t spring(t 1 ) t 1 summer(t 2 ) t 2 autumn(t 3 ) t 3 winter(t 4 ) t + T. spring migration begins spring migration ends autumn migration begins autumn migration ends t t 1 t 2 t 3 Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 6

7 Three basic assumptions Assumption 1: migration rates are piecewise constants m wb (t) = M wb 1 t <t<t 1 ; m bw (t) = M bw 1 t2 <t<t 3. Assumption 2: migration rates are sufficiently large ε := e M wbt 1 + e M bwt 3. Assumption 3: birth rate is also a piecewise constant function b(t) = b 1 t +τ wb <t<t 2. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 7

8 Three basic assumptions M bw > M wb > b > t τ wb t t +τ wb t 1 t 2 t 3 t +T τ wb t +T The quantity is sufficiently small. ε := e M wbt 1 + e M bwt 3 Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 8

9 The basic reproductive ratio A more general basic reproduction number can be defined as the number of new infections produced by a typical infective individual in a population at a DFE. van den Driessche, P. & Watmough, J. 22. Reproduction num- bers and sub-threshold endemic equilibria for compartmen- tal models of disease transmission. Math. Biosci. 18, More references on R : Diekmann et al., 199; Heffernan et al. 25; Wang & Zhao, 28; Diekmann et al., 21; Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 9

10 The basic reproductive ratio The ecological basic reproductive ratio R c = exp[(b µ b )(T 1 + T 2 τ wb )] exp[µ w (T 3 + T 4 τ bw )] α wb M wb M wb + µ w + b µ b α bw M bw M bw + µ b µ w The epidemiological basic reproductive ratio R p = exp[r b(t 1 + T 2 τ wb ) + r w (T 3 + T 4 τ bw )]α wb M wb α bw M bw, (M wb r w + r b )(M bw r b + r w ) where r b := β b µ i b µ b and r w := β w µ i w µ w are infection rates. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 1

11 Numerical simulations conditions birds disease case 1 R c < 1 extinct vanished case 2 R c > 1 and R p < 1 survived vanished case 3 R c > 1 and R p > 1 (Rp case 4 R c > 1 and R p > 1 (Rp small) survived persistant large) extinct vanished Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 11

12 Case 1: R c = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 12

13 Case 2: R c = and R p = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 13

14 Case 3: R c = and R p = susceptible birds t t 1 t 2 t 3 t T Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 14

15 Case 4: R c = and R p = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 15

16 Endemic periodic equilibrium or trivial equilibrium When R c > 1 and R p > 1 with small Rp, the birds will survive and the disease will persist. When R c > 1 and R p > 1 with large Rp, the disease will kill all the birds and finally the disease will vanish. We set R c = and increase R p until the trivial equilibrium is stable. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 16

17 infected birds susceptible birds Case 31: Rc = and Rp = years Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 17

18 Case 31: R c = and R p = susceptible birds t t 1 t 2 t 3 t T Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 18

19 Case 32: R c = and R p = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 19

20 Case 32: R c = and R p = susceptible birds t t 1 t 2 t 3 t T Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 2

21 Case 33: R c = and R p = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 21

22 Case 33: R c = and R p = susceptible birds t t 1 t 2 t 3 t T Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 22

23 Case 34: R c = and R p = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 23

24 Case 34: R c = and R p = susceptible birds t t 1 t 2 t 3 t T Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 24

25 Case 35: R c = and R p = susceptible birds Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 25

26 Derivation of the third basic reproductive ratio Assuming there exists an endemic periodic equilibrium, we have from the epidemiological model t2 t +τ wb β b S b (t) S b (t) + I b (t) dt + t +T t 3 +τ bw β w S w (t) S w (t) + I w (t) = (µ b + µ i b)(t 1 + T 2 τ wb ) + (µ w + µ i w)(t 3 + T 4 τ bw ) ln(α wb α bw ). Recall r b := β b µ i b µ b and r w := β w µ i w µ w are infection rates at summer birth site and winter refuge site respectively. The above equation can be written as t2 t +τ wb β b I b (t) S b (t) + I b (t) dt + t +T t 3 +τ bw β w I w (t) S w (t) + I w (t) = r b (T 1 + T 2 τ wb ) + r w (T 3 + T 4 τ bw ) + ln(α wb α bw ). Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 26

27 Derivation of the third basic reproductive ratio Going back to the ecological model, we then obtain the third basic reproductive ratio R e : R e = exp[(b µ b r b )(T 1 + T 2 τ wb ) (µ w + r w )(T 3 + T 4 τ bw )]M wb M bw. (M wb + µ w + r w + b µ b r b )(M bw + µ b + r b µ w r w ) Taking into account the formulas of R c and R p, it is readily seen that R e = Rc R p M wb (M wb µ + b ) (M wb + r)(m wb µ r + b ) M bw (M bw + µ) (M bw r)(m bw + µ + r), where r := r b r w and µ := µ b µ w. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 27

28 Conclusion and discussion The ecological basic reproductive ratio R c = exp[(b µ b )(T 1 + T 2 τ wb )] exp[µ w (T 3 + T 4 τ bw )] α wb M wb M wb + µ w + b µ b α bw M bw M bw + µ b µ w The epidemiological basic reproductive ratio R p = exp[r b(t 1 + T 2 τ wb ) + r w (T 3 + T 4 τ bw )]α wb M wb α bw M bw. (M wb r w + r b )(M bw r b + r w ) The third basic reproductive ratio (assuming r/m wb and r/m bw ) R e = Rc R p. Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 28

29 Conclusion and discussion conditions birds disease case 1 R c < 1 extinct vanished case 2 R c > 1 and R p < 1 survived vanished case 3 R c > 1 and 1 < R p < Rc (i.e. R e > 1) survived persistant case 4 R c > 1 and R p > Rc (i.e. R e < 1) extinct vanished Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 29

30 Future work Stopovers during migrations. Bourouiba, L., Wu, J., Newman, S., Takekawa, J., Natdorj, T., Batbayar, N., Bishop, C. M., Hawkes, L. A., Butler, P. J. & Wikelski, M. 21 Spatial dynamics of bar-headed geese migration in the context of H5N1. J. R. Soc. Interface 7, Gourley, S., Liu, R. & Wu, J. 21 Spatiotemporal distributions of migratory birds: patchy models with delay. SIAM J. Appl. Dyn. Syst. 9, Interaction between migratory birds and domestic poultry at stopovers. Bourouiba, L., Gourley, S., Liu, R. & Wu, J. 211 The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza. SIAM J. Appl. Math. 71, Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 3

31 Thank you! Avian influenza in migratory birds and basic reproductive ratio First Previous Next Last 31