SEMINARIO

TRANSDISCIPLINARIO

CINVESTAV

11 JUNIO 2012

Contar Q(t) = Cantidad al tiempo t

Q(t + h) = Q(t) + N[t,t + h]− M[t,t + h]+ + E[t,t + h]− S[t,t + h]

Si E[t,t + h] = 0 y S[t,t + h] = 0

Q(t + h) = Q(t) + N[t,t + h]− M[t,t + h]

Q(t + h) = Q(t) + βhQ(t) − µhQ(t) Ecuación en diferencias Contar

Q(t + h) − Q(t) = (β − µ)hQ(t)

Haciendo r = (β − µ)

Q(t + h) − Q(t) = rQ(t) h Bajo la suposición de muestras muy grandes

Q'(t) = rQ(t) Ecuación diferencial Modelo de compartimentos

SS I R Aedes aegypti

Virus del Dengue Distribución de Dengue mundial (2006) Cinche de agua Trypanosoma cruzi Niño enfermo de Chagas Corazón carcomido Culex asociado a WNV Culex quinquefasciatus Distribución de WNV en América Distribución del Culex pipiens Artículos fundamentales

1. W. O. Kermack and A. G. McKendrick, A Contributions to the Matematical Theory of Epidemics. Proc. Roy. Soc. A. Vol. 141, pp. 94-122 (1933).

2. Vito Volterra, Variations and Fluctuations of Population Size in Coexisting Animal Species. Acad. Lincei. 2, pp. 31-113 (1926).

3. Lotka, A. J., Elements of Physical Biology, Williams & Wilkins, Baltimore. (1925). Reprinted under the title Elements of Mathematical Biology. Dover, N. Y. (1956) .

4. Ross, R. The Prevention of Malaria. Murray, London (1911).

5. Hodgkin, A. L. and Huxley, A. F., A Quantitative Description of Membrane Current and its Application to Conduction and Exitation in Nerve. J. Physiol (London) 117, pp. 100-544 (1952)

6. Kolmogorov, A., Petrovsky, I. and Piskounov, N. Etude del L’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Universitè d’Etat à Moscow. (1937)

Artículos fundamentales

7. Turing , A. M. The Chemical Basis of Morphogenesis. Phil. Trans. Roy. Soc. Lond. B237, pp. 37-72. (1952)

8. Lanchester, F. W. Aircraft in Warfare: The Dawn of the Fourt Arm, Constable and Colin.

Modelling Chagas Disease

Modelling Chagas Disease

Gustavo Cruz Pacheco1, Lourdes Esteva2, Cristobal Vargas3 1IIMAS-UNAM, 2Facultad de Ciencias-UNAM, 3 CINVESTAV-IPN Modelling Chagas Disease

Abstract

I Formulation of a model for the dynamics of Chagas disease considering vectors, humans, transmitters (mammals) and non-transmitters (birds). I Main objective: to assess the effectiveness of disease control measures.

I Sensitivity analysis of the basic reproductive number R0 with respect to epidemiological and demographic parameters. Modelling Chagas Disease

Introduction

I Chagas disease (American trypanosomiasis) is a life-threatening illness caused by the protozoan parasite, Trypanosoma cruzi (T. cruzi). I Main mode of transmission: Bites of triatomine bug (kissing bugs). I Also spread through blood transfusion, organ transplantation, and from mother to fetus (1-10 %). I All species of Triatominae are capable to transmit Chagas disease. I Most important: Triatoma infestans, Rhodnius prolixus, Triatoma dimidiata, Triatoma brasiliensis, and Panstrongylus megistus. Modelling Chagas Disease

Kissing bug Modelling Chagas Disease

T. cruzi Modelling Chagas Disease

I T. cruzi has been found in more than 100 mammalian species in Latin America and south of U.S. I Dogs and cats are important link in the transmission to humans. I Birds and reptiles are resistant to T. cruzi infection. I Birds may be important sources of blood meals for triatomines Modelling Chagas Disease

Clinical manifestations

I Early acute phase: I Mild symptoms (local swelling at the site of infection) I Antiparasitic treatment with benznidazole and nifurtimox with 60-90 % cure rates. I Length: 4-8 weeks. I Chronic phase: I 60-80 % of chronically infected are asymptomatic through their lifetime. I 20-40 % of chronically infected eventually develop life-threatening heart and digestive system disorders. I Antiparasitic treatments delay or prevent the development of disease symptoms. I Duration: 10-20 years. Modelling Chagas Disease

Early symptoms Modelling Chagas Disease

Rotten heart Modelling Chagas Disease

I Chagas disease is endemic in poor, rural areas of Latin America. I Argentina, Brazil, Chile, Paraguay and Uruguay signed the Cono Sur Iniciative in 1991. I Simultaneous campaign to stop the transmission of Chagas disease: I Eliminating domestic T. infestant, and other species, I Screening of blood donors. I T. infestans has been eliminated from most of Chile, Brazil and Uruguay, and some regions of Argentina and Paraguay. I Despite campaigns as Cono Sur Iniciative, Chagas disease is consider the most important vector borne infection in Latin America. I 16-18 millions of persons infected with T. cruzi, 20,000 deaths each year, and 100 millions at risk. Modelling Chagas Disease

Control measures

I Control measures are limited: I no vaccines are available. I drugs are effective only in the acute and early chronic phase of infection, and have adverse effects.

I Control measures: I elimination of the vector, I screening blood donors, I treatment to patients in the acute phase, I zooprophylaxis: to attract vectors to domestic animals in which the pathogen cannot amplify (a dead-end host). Rcommended by WHO since 1982. Modelling Chagas Disease

I Chagas disease is most common among people who live in substandard housing in rural and semi rural areas. I Most cases are acquired by exposure to insects in domestic or peridomestic cycles, or by congenital transmission. I We focus on the transmission of Chagas infection taking into account the domestic structure in rural villages, where animals and humans are in continuous contact. Modelling Chagas Disease

Mathematical models

I Effect of demographic factors and blood transmission. Busenberg, S. N. and Vargas, C. in O. Arino, D. E. Axelrod and M. Kimmel, eds. Dekker, New York 1991 pp. 283–295. I Vector dynamics and blood transfusion. Velasco-Hern´andez, J.X (1994). Theor. Pop. Biol. 46 (1): 1-31. I Age-dependent infectivity. Inaba, H. and Sekine, H. (2004) Math. Biosc. 190 (1): 39-69m. I Role of domestic animals in Chagas transmissios. Cohen J.E. and G¨urtlerR. E. (2001). Science 293: 694-698. Modelling Chagas Disease

Variables

I Humans.

Nh = Sh + Ia + Ic • Sh susceptibles, Ia acute infectious, Ic chronic infectious. I Transmitters. Mammals that transmit the infection. Nt = St + It • St susceptibles, It infectious. I Non-transmitters. Animals that do not transmit the infection.

Nnt I Vectors. Triatomines. Nv = Sv + Iv • Sv susceptibles, Iv infectious. Modelling Chagas Disease

Populations Dynamics

I Non-transmitter population N¯nt . I Human and transmitter populations:

I Constant recruitment rates Λh,Λt

I Mortality rates µh, µt I Vector population:

I Recruitment rate: φhbhNh + φt bt Nt + φnt bnt N¯nt

I bh, bt , bnt number of triatomine bites per day in humans, transmitters, and non-transmitters.

I φh, φt , φnt egg-production rates due to blood meals from human, transmitter, and non-transmitters.

I Mortality rate µv . Modelling Chagas Disease

Epidemiological Parameters

I Infection rates per susceptible human and transmitter: b β b β h h I , t t I N v N v βh, βt transmission probabilities from vector susceptible humans and transmitters. I Infection rate per susceptible vector: bhαaIa + bhαc Ic + bt αt It N αa, αc , αt transmission probabilities from infective humans and transmitters to susceptible vectors. Modelling Chagas Disease

I p proportion of newborns that are acute infected due to vertical transmission. I γ proportion of acute infectious that become chronic infectious. I q proportion of cured acute infectious that return to the susceptible class. I σ disease-induced death rate in the chronic stage. Modelling Chagas Disease

The model

dIa Λh bhβh = p Ic + (Nh − Ia − Ic )Iv − (γ + µh)Ia dt Nh N dI c = (1 − q)γI − (µ + σ)I dt a h c dI b β t = t t (N − I )I − µ I dt N t t v t t dI b α I + b α I + b α I v = h a a h c c t t t (N − I ) − µ I dt N v v v v

Sh = Nh − Ia − Ic , St = Nt − It , Sv = Nv − Iv Modelling Chagas Disease

The population size of the three species satisfy the equations:

dN h = Λ − σI − µ N dt h c h h dN t = Λ − µ N dt t t t dN v = φ b N + φ b N + φ b N¯ − µ N . dt h h h t t t nt nt nt v v Modelling Chagas Disease

Basic reproductive number R0

I In the absence of disease:

Λh Λt φhbhN¯h + φt bt N¯t + φnt bnt N¯nt N¯h = N¯t = N¯v = µh µt µv

s b2β HN¯ N¯ b2β α N¯ N¯ R = h h h v + t t t t v 0 2 2 µv KN¯ µv µt N¯

H = [(µh + σ)αa + (1 − q)γαc ]

K = (γh + µh)(σ + µh) − p(1 − q)γµh Modelling Chagas Disease

s µ ¯ ¯ ¶2 µ ¯ ¯ ¶2 2 Nh + Nnt 2 Nt + Nnt R0 = R + R h N¯ t N¯

s b2β HN¯ N¯ R = h h v h h 2 µv K(N¯h + N¯nt ) secondary infections in human-vector cycle s b2β α N¯ N¯ R = t t t v t t 2 µv µt (N¯t + N¯nt ) secondary infections in transmitter-vector cycle Modelling Chagas Disease

Results (σ = 0)

I Disease free equilibrium P0 = (0, 0, 0, 0) I P0 is g.a.s. if R0 ≤ 1, unstable if R0 > 1 I Lyapunov function

V = b1Ia + b2Ic + b3It + Iv

bhαaN¯v bhαc (1 − q)γN¯v b1 = + KN¯ µhKN¯

pbhαaN¯v (γ + µh)bhαc N¯v b2 = + KN¯ µhKN¯

bt αt N¯v b3 = µt N¯ Modelling Chagas Disease

Endemic equilibrium

I P1 = (¯Ia,¯Ic ,¯It ,¯Iv )

µhbhβhN¯h¯Iv (1 − q)γ bt βt N¯t¯Iv ¯Ia = ¯Ic = ¯Ia ¯It = NK¯ µh + K1¯Iv µh bt βt¯Iv + µt N¯

¯ ¯2 ¯ p(Iv ) = a1Iv + a2Iv + a3

K1 = bhβh(µh + (1 − q)γ)

I P1 is feasible if R0 > 1

I Linear analysis: P1 is l.a.s. if R0 > 1 Modelling Chagas Disease

I q = 0, p = 0. Lyapunov function:

Sh Ia U = c1(Sh − S¯h − S¯h ln ) + c2(Ia − ¯Ia − ¯Ia ln ) S¯h ¯Ia

Ic + c3(Ic − ¯Ic − ¯Ic ln ) ¯Ic

St It + c4(St − S¯t − S¯t ln ) + c5(It − ¯It − ¯It ln ) S¯t ¯It

Sv Iv + c6(Sv − S¯v − S¯v ln ) + c7(Iv − ¯Iv − ¯Iv ln ), S¯v ¯Iv

S¯h = Nh − ¯Ia − ¯Ic , S¯t = Nt − ¯It , S¯v = Nv − ¯Iv

I P1 is g.a.s. Modelling Chagas Disease

Control measures. Reducing vector population

I Reducing vector population by insecticides (or other means) ⇒ vector mortality µv increases.

I Increasing µv by a factor θ > 1 ⇒ R0 decreases by a factor 1/θ.

I Insecticide application every 3 months ⇒ life span /µv decreases from 6 months to 3 months ⇒ µv increases twice, and R0 decreases by half. I The disease could be controlled by mere application of R insecticide every three months, if 0 < 1. 2 Modelling Chagas Disease

Control measures. Screening and treatment

I Early treatment and cure of acute infected ⇒ q increases by a factor θ > 1, θq ≤ 1.

∂R R (¯q) ≈ R (q) + 0 (1 − θ)q 0 0 ∂q · ¸ Q = R0(q) 1 − 2 (θ − 1)q 2R0 (q)

2 2 (Nh + Nnt ) qγ(pαaµh + αc (µh + γ)) Q = Rh 2 N K(µhαa + (1 − q)γαc ) · ¸ Q I θ = 1/q. Minimum value R0 1 − 2 (1 − q) 2R0 (q) Modelling Chagas Disease

Control measures. Zooprophylaxis

I Population of non- transmitters Nnt increases to N¯nt = θNnt , θ > 1.

∂R0(Nnt ) R0(N¯nt ) ≈ R0(Nnt ) + (θ − 1)Nnt ∂Nnt

∂R0 [(φnt bnt − 2φhbh)Nh + (φnt bnt − 2φt bt )Nt − φnt bnt Nnt ] = R0 ∂Nnt 2NN¯v

I R0(Nnt ) has a unique local maximum positive when

(φnt bnt − 2φhbh)Nh + (φnt bnt − 2φt bt )Nt > 0 I Depending on the density, biting rate, and egg-production rate from blood meals of the populations involved, R0 increases or decreases. Modelling Chagas Disease

parameter value sources p 2 % − 10 % Kirchhoff, 2010 −1 bh 0.04 day Cohen, 2001 −1 bt 0.1 day Cohen, 2001 −1 bnt 0.12 day Cohen, 2001 βh 0.0009 Rabinovich, 1999 βt 0.0009 Rabinovich, 1999 αa 0.03 Cohen, 2001 αc 0.03 Cohen, 2001 αt 0.49 Gurtler, 1996 1/γ 28-56 days Bern, 2007 q 0.04 % 1µh 65 years 1/µt 10 years 1/µnt 7 years 1/µv 7 months Rabinovich, 1972 Table 1. Parameters in the numerical simulations of Chagas model. Modelling Chagas Disease

N¯h human population 5000 N¯t transmitter population 2/5 × Nh N¯nt non-transmitter population 3/5 × Nh N¯v vector population 100 × Nh Modelling Chagas Disease

6

R (θ µ ) 5 0 v R (θ q) 0 0 R (θ N ) 0 nt 4

3

2 Basic reproductive number R

1

0 0 5 10 15 20 25 θ

Figure 1. Evolution of R0 as a function of the control variables θµv , θq, and θNnt , 1 ≤ θ ≤ 5. Initial R0 = 5,8 Modelling Chagas Disease

1.4 R (θ µv) h R (θ q) 1.2 h R (θ N ) h nt h 1

0.8

0.6

0.4 Basic reproductive number R

0.2

0 0 5 10 15 20 25 θ

Figure 2. Evolution of Rh as a function of the control variables θµv , θq, and θNnt , 1 ≤ θ ≤ 5. Initial R0 = 1,3 Modelling Chagas Disease

12 R (θ µv) t R (θ q) t 10 R (θ N ) t nt t

8

6

4 Basic reproductive number R

2

0 0 5 10 15 20 25 θ

Figure 3. Evolution of Rt as a function of the control variables θµv , θq, and θNnt , 1 ≤ θ ≤ 5. Initial R0 = 11,4 Modelling Chagas Disease

10

9.5

9 0 8.5

8

7.5

7

6.5 Basic reproductive number R 6

5.5

5 0 5 10 15 20 25 θ

Figure 4. Evolution of R0 as a function of θNnt with bnt = 0,5 and other parameters as in Table 1. Modelling Chagas Disease

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3 infectious human proportion I (θ µv)/Nh h 0.2 I (θ q)/N h h 0.1 I (θ N )/N h nt h

0 0 5 10 15 20 25 θ

Figure 5.Endemic human proportions Ih/Nh with Ih = Ia + Ic as a function of the control variables θµv , θq, and θNnt . Modelling Chagas Disease

1

0.9

0.8

0.7

0.6

0.5

0.4

I (θ µv)/Nt 0.3 t

infectious transmitter proportion I (θ q)/N 0.2 t t I (θ N )/N t nt t 0.1

0 0 5 10 15 20 25 θ

Figure 6.Endemic transmitter proportions It /Nt as a function of the control variables θµv , θq, and θNnt . Modelling Chagas Disease

1

I (θ µv)/N 0.9 v v I (θ q)/N v v 0.8 I (θ N )/N v nt v 0.7

0.6

0.5

0.4

0.3 infectious vector proportion

0.2

0.1

0 0 5 10 15 20 25 θ

Figure 7. Endemic vector proportions Iv /Nv as a function of the control variables θµv , θq, and θNnt . Modelling Chagas Disease

6 x 10 7

6

5

I 4 v N v

3 vector population

2

1

0 0 5 10 15 20 25 θ

Figure 8. Population size of triatomines and infected triatomines as a function of the number of non-transmiters Nnt . Modelling Chagas Disease

Relative rate of decrease

p − p η = i f pi

pi and pf human infectious at the start and end of the control.

θ 2 3 4 5 Increment of mortality µv 0.16 0.37 0.58 0.80 Increment of treatment q 0.005 0.009 0.01 0.02 Increment of non-transmitters Nnt -0.003 −0,0008 0.003 0.008

I Elimination of vectors is by far the best way to control the endemic levels, followed by early detection and treatment of acute cases. I Zooprophylaxis has no major impact in reducing the disease. Modelling Chagas Disease

Conclusions

I The model corroborates that reduction of vectors is the most effective measure to control the disease. I Early detection and treatment of disease is very important, however, the model shows that close proximity to domestic animals may prevent this treatment as a good control measure. I Increasing the number of non-transmitter animals at home, decreases very slightly the proportions of infected humans and animals. Under certain conditions, ths measures increases the number of new infections. I Reducing the population of triatomines, and keeping domestic animals out of the dormitories in order to reduce animal transmission seems to be best way to decrease the risk of human infections as mention in (Cohen, 2001). West Nile Virus

Multispecies Interactions in West Nile Virus

GUSTAVO CRUZ-PACHECO IIMAS, UNAM

LOURDES ESTEVA Facultad de Ciencias,UNAM

CRISTOBAL´ VARGAS Departamento de Control Autom´atico,CINVESTAV-IPN West Nile Virus

Introduction

• West Nile Virus (WNV) is an arbovirus transmitted by mosquito’s bites to different species of birds and mammals, including humans.

• Transmission cycle is between mosquitoes and birds. Mammals can not transmit the virus.

• Mosquitoes present vertical transmission. West Nile Virus

• WNV was detected the first time in the american continent in 1999 in New York City. (www.cdc.gov/ncidod/dvbid/westnile/).

• Since then, it has spread to almost all of the United States and parts of Canada.

• The virus have been found in serological tests in birds and horses in Latin America and Carribean islands.

• Low incidence of WNV in these regions. • Possible explanations: • Cross immunity with other flavivirus like dengue. • Mutations of the virus in its way to south (Hayes, 2005) West Nile Virus

Important questions

• The role of different species of birds and mosquitoes in WNV outbreaks. • The effect of the interaction among the bird and mosquito species in the emergency and prevalence of WNV.

[ Turell, M.J. et al. (2005). An update on the potential of north American mosquitoes (Diptera: Culicidae) to transmit West Nile virus. J. Med Entomol. 42: 57-62. ] [ Komar, N. (2003). West Nile virus: epidemiology and ecology in North America. Adv. Virus Res. 61:185-234. ] West Nile Virus

The basic model

• Na, Nv bird and mosquito populations.

• Sa(t), Sv (t) susceptible birds and mosquitoes.

• Ra(t) recovered birds.

• µv mortality rate of mosquitoes.

• µa mortality rate of birds.

• Λa recruitment rate of bird population. • b biting rate.

• βa, βv transmission probabilities.

• γa birds recovered rate.

• αa disease mortality rate. West Nile Virus

dSa bβa = Λa − Iv Sa − µaSa dt Na dIa bβa = Iv Sa − (γa + µa + αa)Ia dt Na dR a = γ I − µ R dt a a a a dSv bβv = µv Sv − IaSv − µv Sv dt Na dIv bβv = IaSv − µv Iv dt Na dN a = Λ − µ N − α I dt a a a a a

Sa + Ia + Ra = Na, Sv + Iv = Nv . West Nile Virus

Basic Reproductive Number R0

s mbβv bβa R0 = · γa + µa + αa µv

N m = v Na

Results

• If R0 < 1 ⇒ disease-free equilibrium E0 g.a.s.

• If R0 > 1 ⇒ endemic equilibrium E1 l.a.s. West Nile Virus

name βa βv γa αa µa µv m R0 Blue Jay 1.0 0.68 0.26 0.15 .0002 .06 5 5.89

Common Grackle 1.0 0.68 0.33 .07 .0001 .06 5 6.97 House Finch 1.0 0.32 0.18 0.14 .0003 .06 5 4.57 American Crow 1.0 0.5 0.31 0.19 .0002 .06 5 4.58 House Sparrow 1.0 0.53 0.33 0.1 .0002 .06 5 5.08

Ring-billed Gull 1.0 0.28 0.22 0.1 .0003 .06 5 4.28 Black-billed Magpie 1.0 0.36 0.33 0.16 .0001 .06 5 3.92 Fish Crow 1.0 0.26 0.36 .06 .0002 .06 5 3.60

Epidemiological and demographic parameters. Units days−1. West Nile Virus

0.5 Blue Jay 0.45 American Crow House Sparrow

0.4

0.35

0.3

0.25

0.2

proportion of infected birds 0.15

0.1

0.05

0 0 5 10 15 20 25 30 35 40 45 50 days

Temporal course of infected birds. sa = 1, ia = 0, iv = 0.001, na = 1. West Nile Virus

0.002 Blue Jay American Crow House Sparrow

0.0015

0.001 proportion of infected birds 0.0005

0 0 200 400 600 800 1000 1200 1400 days Asymptotic behavior of solutions. West Nile Virus

N species of birds

• Ni Total population of i bird species

• Si Susceptible population of i bird species

• Ii Infectious population of i bird species

dSi bβi = Λi − Si Iv − µi Si dt N1 + ... + Nn dIi bβi = Si Iv − (γi + µi + αi )Ii dt N1 + ... + Nn Xn dIv bβvi Ii = (Nv − Iv ) − µv Iv dt N1 + ... + Nn i=1 dN i = Λ − µ N − α I dt i i i i i i = 1, .., n. West Nile Virus

a Basic Reproductive Number R0

r X a n 2 ¡ ¢ R = R N¯i /N¯ 0 i=1 i

with s 2 ¡ ¢ b βi βvi Ri = Nv /N¯ µv (γi + µi + αi ) the basic reproductive number of the i bird species, i = 1, ..., n.

N¯i = Λi /µi , N¯ = N¯1 + N¯2 + ... + N¯n West Nile Virus

Results

a • Si R0 < 1 ⇒ the disease free equilibrium is g.a.s. a • Assume αi = 0, i = 1, ..n. If R0 > 1 ⇒ exists a unique endemic equilibrium g.a.e.

a • R0 is the average of the basic reproductive numbers of the bird species weighted by their corresponding population proportion with respect to the total number of birds.

• The interplay between the competence of the birds species to transmit the disease and their corresponding population density will determine the evolution of the disease. West Nile Virus

Bird βv γa αa µa Common Grackle 0.68 0.33 0.07 0.0001 Northern Flicker 0.06 1.0 0.0 0.0003

Data from 1. Komar, N. (2003). West Nile virus: epidemiology and ecology in North America. Adv. Virus Res. 61:185-234 2. The University of Michigan Museum of Zoology, 2004. Animal Diversity Web. http://www.animaldiversity.ummz.umich.edu. West Nile Virus

Vector b βa µv Ae.albopictus 0.1 0.86 0.07 Cx.pipiens 0.5 0.88 0.07

Data from: Savage et al. (1993). Host-feeding patterns of Aedes albopictus (Diptera:Culicidae) at a temperate North American site. J. Med. Entomol. 30: 27-34. Turell, et al. (2005). An update on the potential of north American mosquitoes (Diptera: Culicidae) to transmit West Nile virus. J. Med Entomol. 42: 57-62. Wonham et al. (2004). An epidemiological model for West Nile virus: invasion analysis and control applications. Proc. R. Soc. Lond. B 266:565-570. West Nile Virus

0.01 Northern Flicker 0.009 Common Grackle

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0 0 20 40 60 80 100

Temporal course of the proportion of infected birds. For Northern ¯ Flicker R01 = 0.41, and N1 = 1000. For Common Grackle, ¯ R02 = 2.2, and N2 = 100. In this case R0 = 0.76. West Nile Virus

0.01 Northern Flicker 0.009 Common Grackle

0.008

0.007

0.006

0.005

0.004

0.003 proportion of infected birds

0.002

0.001

0 0 100 200 300 400 500 600 days

Temporal course of the proportion of infected birds. For Northern ¯ Flicker, R01 = 0.37, and N1 = 1000. For Common Grackle, ¯ R02 = 1.95, and N2 = 400. In this case R0 = 1.2. West Nile Virus

a Dependence of R0 with respect to parameters.

a • R0 depends linearly on b, βi , βv , Nv . a 2 ∂(R0 ) 1 ¡ a 2 2¢ • = −2(R0 ) + Ri . ∂Ni N • ρ = Ra/Ra: i i 0√ a • If ρi > 2, R0 increases when Ni increases √ a • Si ρi < 2, R0 decreases when Ni increases • Many bites are wasted in birds poorly efficient to transmit the virus. • If the population of birds keeps growing, the basic reproduction number decreases. West Nile Virus

N species of mosquitoes

• Nvi Total population of i mosquito species

• Ivi Infectious population of i mosquito species

Xn dSa Nvi bi βai = Λa − SaIvi − µaSa dt Nv + Nv + ... + Nv Na i=1 1 2 n Xn dIa Nvi bi βai = SaIvi − (γa + µa + αa)Ia dt Nv + Nv + ... + Nv Na i=1 1 2 n

dIvi bi βvi Ia = (Nvi − Ivi ) − µvi Ivi dt Na dN a = Λ − µ N − α I dt a a a a a i = 1, .., n. West Nile Virus

r Xn v 2 Nvi R0 = Ri i=1 Nv s 2 bi βai βvi Nvi R0i = µvi (γa + µa + αa) Na

Nv = Nv1 + Nv2 + .... + Nvn

v • The disease-free equilibrium is g.a.s. if R0 < 1. v • Assume α = 0. If R0 > 1 there exists a unique endemic equilibrium g.a.s.. West Nile Virus

∂(Rv )2 1 ¡ ¢ 0 = −(Rv )2 + 2R2 . 0 0i ∂Nvi Nv

v v 1 I • R increases if σ = R0 /R > √ . 0 i 0 2

v v 1 I • R decreases if σ = R0 /R < √ . 0 i 0 2 1 I • If σi > √ , R0 will keep increasing when the mosquito 2 population increases, since R0i becomes bigger. West Nile Virus n species of birds and m species of mosquitoes

Xm dSi Nvj bj βij = Λi − Si Ivj − µi Si dt (Nv + ···Nv ) (N1 + ... + Nn) j=1 1 m Xm dIi Nvj bj βij = Si Ivj − (γi + µi + αi )Ii dt (Nv + ···Nv ) (N1 + ... + Nn) j=1 1 m Xn dIvj bj βvji = (Nvj − Ivj )Ii − µvj Ivj (1) dt N1 + ... + Nn i=1 dN i = Λ − µ N − α I dt i i i i i West Nile Virus

Basic Reproductive number R0

• Spectral ratio of the Next Generation Operator Φ associated to the disease-free equilibrium Si = N¯i , Ii = 0, i = 1, ...n,

Ivj = 0, j = 1, .., m   0 G1   Φ =   G2 0  West Nile Virus

 ¯ ¯  b1β11Nv1 N1 bmβ1mNvm N1  ···   γ1 + µ1 + α1 γ1 + µ1 + α1       b β N N¯ b β N N¯   1 21 v1 2 m 2m vm 2   ···   γ2 + µ2 + α2 γ2 + µ2 + α2    1   G1 = ¯   , NNv  ·····     ·····     ·····   ¯ ¯   b1βn1Nv1 Nn bmβnmNvm Nn   ···  γn + µn + αn γn + µn + αn West Nile Virus

  b1βv11 Nv1 b1βv1n Nv1  ···   µv1 µv1       b β N b β N   2 v21 v2 2 v2n v2   ···   µv2 µv2    1   G2 =   , N¯  ·····     ·····     ·····  b β N b β N   m vm1 vm m vmn vm   ···  µvm µvm

¯ ¯ ¯ ¯ with N = N1 + ···Nn, Ni = Λi /µi , and Nv = Nv1 + ··· + Nvm . West Nile Virus

0.85

0.8

0.75 0 R

0.7

0.65

0 500 1000 1500 2000 2500 3000 3500 Common Grackle population

R0 for different population sizes of Common Grackle species. West Nile Virus

0.7

0.6

0.5

0.4 0 R 0.3

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 Northern Flicker population x 10

R0 for different population sizes of Northern Flicker species. West Nile Virus

10

9 * C. pipiens x A. albopictus 8

7

6

0 5 R

4

3

2

1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 Mosquitoes population x 10

v R0 for different mosquito population sizes. West Nile Virus

Conclusions

a v • R0 and R0 estimate the importance of birds and mosquitoes on the prevalence of WNV.

• The competence and abundance of each species determines its role on the maintenance of the disease.

• Species not affected by the disease if isolated, can become infected if they share the space with a large enough competent species.

• The transmission of WNV is sensible to variation on populations density. West Nile Virus

Distribution of Cx. pipiens in North America. Data taken from http://svs.gsfc.nasa.gov/vs/a000000/a002500/a002565/. West Nile Virus

• Culex species that are efficient transmitter of WNV are not common in Mexico. http://www.cenave.gob.mx/von/.

• Common mosquitoes species in M´exico(for example, Ae. aegypti) are not very competent transmitters .

• A possible explanation of the low prevalence of WNV in Mexico.