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 e®ectiveness 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 e®ective only in the acute and early chronic phase of infection, and have adverse e®ects. 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 E®ect 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 GÄurtlerR. E. (2001). Science 293: 694-698. Modelling Chagas Disease Variables I Humans. Nh = Sh + Ia + Ic ² Sh susceptibles, Ia acute infectious, Ic chronic infectious. I Transmitters. Mammals that transmit the infection. Nt = St + It ² St susceptibles, It infectious. I Non-transmitters. Animals that do not transmit the infection. Nnt I Vectors. Triatomines. Nv = Sv + Iv ² Sv susceptibles, Iv infectious. Modelling Chagas Disease Populations Dynamics I Non-transmitter population N¹nt . I Human and transmitter populations: I Constant recruitment rates ¤h, ¤t I Mortality rates ¹h; ¹t I Vector population: I Recruitment rate: ÁhbhNh + Át bt Nt + Ánt bnt N¹nt I bh; bt , bnt number of triatomine bites per day in humans, transmitters, and non-transmitters. I Áh, Át , Ánt egg-production rates due to blood meals from human, transmitter, and non-transmitters. I Mortality rate ¹v . Modelling Chagas Disease Epidemiological Parameters I Infection rates per susceptible human and transmitter: b ¯ b ¯ h h I ; t t I N v N v ¯h, ¯t transmission probabilities from vector susceptible humans and transmitters. I Infection rate per susceptible vector: bh®aIa + bh®c Ic + bt ®t It N ®a, ®c ; ®t 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 ¤h bh¯h = p Ic + (Nh ¡ Ia ¡ Ic )Iv ¡ (γ + ¹h)Ia dt Nh N dI c = (1 ¡ q)γI ¡ (¹ + σ)I dt a h c dI b ¯ t = t t (N ¡ I )I ¡ ¹ I dt N t t v t t dI b ® I + b ® I + b ® I v = h a a h c c t t t (N ¡ I ) ¡ ¹ 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 = ¤ ¡ σI ¡ ¹ N dt h c h h dN t = ¤ ¡ ¹ N dt t t t dN v = Á b N + Á b N + Á b N¹ ¡ ¹ 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: ¤h ¤t ÁhbhN¹h + Át bt N¹t + Ánt bnt N¹nt N¹h = N¹t = N¹v = ¹h ¹t ¹v s b2¯ HN¹ N¹ b2¯ ® N¹ N¹ R = h h h v + t t t t v 0 2 2 ¹v KN¹ ¹v ¹t N¹ H = [(¹h + σ)®a + (1 ¡ q)γ®c ] K = (γh + ¹h)(σ + ¹h) ¡ p(1 ¡ q)γ¹h Modelling Chagas Disease s µ ¹ ¹ ¶2 µ ¹ ¹ ¶2 2 Nh + Nnt 2 Nt + Nnt R0 = R + R h N¹ t N¹ s b2¯ HN¹ N¹ R = h h v h h 2 ¹v K(N¹h + N¹nt ) secondary infections in human-vector cycle s b2¯ ® N¹ N¹ R = t t t v t t 2 ¹v ¹t (N¹t + N¹nt ) 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 bh®aN¹v bh®c (1 ¡ q)γN¹v b1 = + KN¹ ¹hKN¹ pbh®aN¹v (γ + ¹h)bh®c N¹v b2 = + KN¹ ¹hKN¹ bt ®t N¹v b3 = ¹t N¹ Modelling Chagas Disease Endemic equilibrium I P1 = (¹Ia;¹Ic ;¹It ;¹Iv ) ¹hbh¯hN¹h¹Iv (1 ¡ q)γ bt ¯t N¹t¹Iv ¹Ia = ¹Ic = ¹Ia ¹It = NK¹ ¹h + K1¹Iv ¹h bt ¯t¹Iv + ¹t N¹ ¹ ¹2 ¹ p(Iv ) = a1Iv + a2Iv + a3 K1 = bh¯h(¹h + (1 ¡ q)γ) I P1 is feasible if R0 > 1 I Linear analysis: P1 is l.a.s.