Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations
|
|
- Marcia Lewis
- 6 years ago
- Views:
Transcription
1 Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Published Version Accessed Citable Link Terms of Use Awerbuch-Friedlander, T., Richard Levins, and M. Predescu. 24. Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations. Discrete Dynamics in Nature and Society 24: 8. doi:.55/24/965. doi:.55/24/965 November 8, 27 :54:5 PM EST This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at (Article begins on next page)
2 Co-Dynamics of Four Variables Involved in Dengue Transmission and its Control by Community Intervention: A System of Four Difference Equations T. Awerbuch-Friedlander and R. Levins Harvard School of Public Health Department of Population and International Health 665 Huntington Avenue, Boston, MA 25, USA M. Predescu Bentley University Department of Mathematical Sciences 75 Forest Street, Waltham, MA 2452, USA Abstract In the case of Dengue transmission and control, the interaction of nature and society is captured by a system of difference equations. For the purpose of studying the dynamics of these interactions, four variables involved in a Dengue epidemic: proportion of infected people (P), number of mosquitoes involved in transmission (M), mosquito habitats (H) and population awareness (A), are linked in a system of difference equations: P n+ = ap n + ( e imn )( P n) M n+ = lm ne An + bh n( e Mn ) H n+ = chn n =,, pa n + qa n A n+ = ra n + fp n The constraints have socio-ecological meaning. The initial conditions are such that P, (M, H, A ) (,, ), the parameters l,a, c, r (, ), and the parameters f, i, b and p are positive. The paper is concerned with the analysis of solutions of the above system for p = q. We studied the global asymptotic stability of the degenerate equilibrium. We also propose extensions of the above model and some open problems. We explored the role of memory in community awareness by numerical simulations. When the memory parameter is large, the proportion of infected people decreases and stabilizes at zero. Below a critical point we observe periodic oscillations. Key words: Global stability, Boundedness, Difference equations, Dengue control, System biology. AMS Subject Classification: 39A, 39A
3 2 Awerbuch et. al. Introduction The response to an epidemic is triggered by awareness of a coming epidemic or by an existing one. The response is aimed at reducing the incidence of the actual disease. In the case of Dengue Fever, the disease is caused by a virus that is transmitted by the bite of the mosquito, usually Aedes Aegypti. The mosquitoes deposit eggs in small containers of water. These hatch to produce larvae. Some transform into pupae and then adult mosquitoes. The breeding sites may be ephemeral, such as water in an empty beer can or used tire, an animal drinking trough near a human habitation, or in-doors stored water in large containers ([4], [8]). The information about a Dengue epidemic can come from the number of reported cases of Dengue, the abundance of mosquitoes or the numbers of breeding sites for mosquitoes, or some other indicator such as rainfall that predicts breeding sites. The information triggers consciousness, and the response can be either individual and/or one at the community level. In previous work, we studied the dynamics of a discrete time system in which we modeled the awareness as a factor that is triggered by the formation of potential breeding sites and the response was aimed at eliminating them. The system was studied by a pair of two difference equations ([5]). By expanding the model to introduce an ongoing educational program, our new model predicted that high consciousness over time kept the number of breeding sites low ([3]). In a study with three difference equations, we study a system in which the information is related to the number of adult mosquitoes. The more mosquitoes, the greater the awareness of the population, and this awareness leads to action to reduce the mosquito population by controlling breeding sites ([2]). This population awareness is prompted and dissipates at a rate determined by the abundance of mosquitoes, similar to a birth and death process. The dynamics then, is that mosquitoes are produced when adult females locate breeding sites and deposit eggs which develop into adult mosquitoes, and mosquitoes die at a rate depending on their own biology and environmental conditions as a result of control measures implemented as awareness rises. Thus the pair of variables, mosquitoes and awareness, are linked in a negative feedback loop in a system of equations were decay due to control was modeled with a rational fractional term at the environmental level. With another system of three difference equations we have explored an intervention by spraying mosquitoes ([4]). The change in the spraying parameter resulted in almost periodic behavior and fluctuations in the populations of mosquitoes. Simulations show that alertness in consciousness, by keeping the memory parameter of previous week high, has an impact on the behavior of solutions and implicitly on the number of mosquitoes. When the memory parameter is high, there will be a steady decrease in the number of mosquitoes. The present study builds upon the previous models. We present a system of four difference equations, with the proportion of infected people as an additional variable that prompts consciousness: P n+ = ap n + ( e imn )( P n ) M n+ = lm n e gan + bh n ( e smn ) H n+ = ch n d + + pa n + pa n A n+ = ra n + fp n n =,,... (.)
4 A Non-Linear System of Difference Equations 3 This discrete system links the proportion of the infected people (P n ), mosquitoes (M n ), habitats (H n ), and awareness A n. The initial conditions are such that P, (M, H, A ) (,, ), the parameters l, a, c, r (, ),and the parameters f, i, b and p are positive. The current system represents a modification of the system in [2]. The first equation describes the proportion of infected people (between and ). They prompt consciousness, while the intervention is against mosquitoes and perhaps habitats. In the relationships among variables, the awareness is prompted by the proportion of sick people. The control of both adult mosquitoes by spraying, and habitats is carried out by community intervention. The parameter i is related to the behavior of infected mosquitoes, and it can be viewed as a transmission rate. An explanation of the term ( e imn ) goes as follows. If Q represents the probability that a mosquito transmits the infection, then Q is the probability that it does not transmit the infection. Therefore, ( Q) Mn will be the probability that M n mosquitoes do not transmit the infection. One can rewrite ( Q) Mn = e ln( Q)Mn = e Mn ln( Q). We denote i = ln( Q) >. One can observe that if P then P. This is true because P = ap + ( e im )( P ) ap + ( P ). It follows by induction that P n. Also, if (M, H, A ) (,, ) then (M n, H n, A n ) (,, ). Thus, we have that (P, M, H, A ) (,,, ) then (P n, M n, H n, A n ) (,,, ). By using a series of transformations, one can rescale the parameters g, s and d in (.). We use the following changes of variables, M n = s m n (in the second and first equation), A n = g a n (in the third and fourth equation) and H n = dh n. These transformations will not change the nature of parameters a, c, l and r, as these remain between and. Thus, after relabeling the variables and parameters, one can work with a simplified system of equations as below (it is this system that will get analyzed in the next sections): P n+ = ap n + ( e imn )( P n ) M n+ = lm n e An + bh n ( e Mn ) H n+ = ch n + + pa n + pa n A n+ = ra n + fp n n =,,... (.2) In the sequel, we look at boundedness properties, local and global asymptotic stability of equilibria. Numerical simulations, open problems and further directions of improvement will be mentioned.
5 4 Awerbuch et. al. 2 Boundedness of Solutions Lemma 2.. Let {P n, M n, H n, A n } n be a positive solution of system (.2). Parameters are such that < l <, < a <, < c < and < r <. Then lim sup P n ( a), lim sup M n b ( l)( c), lim sup H n c and lim sup A n f ( a)( r). Proof. First equation of system (.2) gives P n+ ap n + ( P n ) ap n +. Thus lim sup P n ( a) and then for any positive number ɛ p, there exists N sufficiently large, such that, for all n N, P n+ < a + ɛ p (2.) Making use of (2.) in the fourth equation, we get: A n+ ra n + f a + ɛ p (2.2) f Since < r < we obtain lim sup A n and then for any positive number ( a)( r) ɛ a, there exists N sufficiently large, such that, for all n N, A n+ < f ( a)( r) + ɛ a (2.3) The third equation of (.2) yields H n+ ch n + which combined with and < c < gives lim sup H n c. Thus for any positive number ɛ h, there exists N sufficiently large, such that, for all n N, H n+ < ( c) + ɛ h (2.4) Finally, (2.4) and M n+ lm n + bh n lm n + b c + ɛ h produces lim sup M n b ( l)( c). Thus for any positive number ɛ m, there exists N sufficiently large, such that, for all n N, b M n+ < ( l)( c) + ɛ m (2.5) Some notations that will be used throughout the paper are in order: lim sup P n = S P and lim inf P n = I P (2.6) lim sup H n = S H and lim inf H n = I H (2.7)
6 A Non-Linear System of Difference Equations 5 3 Equilibria Clearly, lim sup M n = S M and lim inf M n = I M. (2.8) lim sup A n = S A and lim inf A n = I A (2.9) (,,, ) (3.) c is an equilibrium point of System (.2) for all the values of the parameters. Lemma 3... Assume that b ( c)( l). Then the degenerate equilibrium (,, only equilibrium point., ) is the c 2. Assume that b > ( c)( l) then there are two equilibrium points, namely the degenerate one and a positive one denoted by ( P, M, H, Ā). The positive equilibrium can take the form ( e i M 2 a e i M, M, pf( e i M, ) c + ( r)(2 a e i M ) f( e i M ) ) ( r)(2 a e i M. ) Proof. The equilibrium solutions verify the system P = a P + ( e i M )( P ) M = l Me Ā + b H( e M ) H = c H + pā + + pā Ā = rā + f P (3.2) The fourth equation in the above system gives P = Solving for H in the third equation yields: ( r)ā. (3.3) f H = c + pā (3.4)
7 6 Awerbuch et. al. Combining equation (3.4) with the second equation of system (3.2) produces: ( le Ā) M = b( e M ) c + pā Replacing equation (3.3) in first system equation and multiplying by f to both sides: (3.5) ( a)( r)ā = ( e i M )(f ( r)ā) (3.6) Since ( c + pā) and ( le Ā) equation (3.5) can be written in the form: Equation (3.6) gives: M e M = b ( c + pā)( le Ā) Ā = Notice that (2 a e i M ) >. Set w( M) = Notice that f( e i M ) ( r)(2 a e i M ) Ā = w( M). f( e i M ) ( r)(2 a e i M ). (3.7) (3.8) where function w(m) has the property that it is an increasing function, first order derivative w f( a)ie im (M) = ( r)(2 a e im > for M (, ). Set the real valued functions ) 2 and g(a) = Φ (M) = M e M b ( c + pa)( le A ). We have that Φ (M) is an increasing function in M, Φ ( + ) = and Φ ( M) = g(ā). From the above, g(ā) = g(w( M)) = (g w)( M) where we denote Φ 2 as Φ 2 (M) = (g w)(m) = g(w(m)). Function Φ 2 is decreasing. Let M < M 2. Since function w is increasing, one has w(m ) < w(m 2 ). But g is a decreasing function and Φ 2 (M ) = (g w)(m ) = g(w(m )) > g(w(m 2 )) = Φ 2 (M 2 ). Using that w( + ) =, we have that Φ 2 ( + ) = b ( c)( l). For equation (3.7) to have a unique solution (and thus, system to have a unique solution), one must have Φ ( + ) < Φ 2 ( + b ) or equivalently < and the proof ends. ( c)( l)
8 A Non-Linear System of Difference Equations 7 4 Stability of Equilibrium Points Next we are concerned with the local and global asymptotic stability of equilibrium points. Notations for our map are as follows: P n+ = Θ(P n, M n, H n, A n ) with Θ(P, M, H, A) = ap + ( e im )( P ). (4.) M n+ = g(p n, M n, H n, A n ) with g(p, M, H, A) = lme A + bh( e M ). (4.2) H n+ = h(p n, M n, H n, A n ) with h(p, M, H, A) = ch + pa + + pa. (4.3) A n+ = Φ(P n, M n, H n, A n ) with Φ(P, M, H, A) = ra + fp. (4.4) The Jacobian evaluated at the equilibrium point ( P, M, H, Ā) has the form: a ( e i M ) ( P )ie i M J( P le Ā + b He M b be M Ā l Me, M, H, Ā) = c p( + c H) + pā ( + pā)2 f r Using the third equilibrium equation, + c H = H( p( + c H) p H + pā). Thus, = ( + pā)2 ( + pā). The characteristic equation associated with ( P, M, H, Ā) is given by the fourth order polynomial: [ ] [a ( e i M ) λ][le Ā + b He M c λ] ( + pā) λ [r λ] [ ( P )ie i M p Hb( e M ( )] ) Ā c f + l Me + pā + pā λ = One can look at the characteristic equation in the form: - λ 4 (A + A 2 + A 3 + A 4 )λ 3 + (A A 2 + A A 3 + A A 4 + A 2 A 3 + A 2 A 4 + A 3 A 4 )λ 2 (A A 2 A 3 + A A 2 A 4 + A A 3 A 4 + A 2 A 3 A 4 + A 5 A 7 )λ + A A 2 A 3 A 4 + A 5 A 6 + A 3 A 5 A 7 =. where A = a ( e i M ) A 2 = le Ā + b c A 3 = + pā A 4 = r He M A 5 = ( P i M )ife
9 8 Awerbuch et. al Mosquitoes Habitats Proportion of people infected Awareness Figure 4.: The above graph is generated with parameter values a =.5, p = 5, b =.48, c =.4, l =.5, i =.5, r =.5, f = 2 (the parameters are placed in the region where b = ( c)( l)). One can see that the solutions converge to the degenerate equilibrium point. A 6 = p Hb( e M ) + pā Ā A 7 = l Me In the region of existence of positive equilibrium point, b > ( c)( l), the values of parameters for which the roots of the fourth order polynomial are inside unit disc, generate a locally asymptotically stable equilibrium point. The positive equilibrium point is not always locally asymptotically stable in the region b > ( c)( l) (see Figure (4.3)). The following theorem about the degenerate equilibrium point Theorem 4.. Assume that b < ( c)( l). Then ( ),, c, is globally asymptotically stable. Proof. When P =, M =, H = c ( ),, c, holds: and Ā =, the jacobian becomes:
10 A Non-Linear System of Difference Equations Mosquitoes Habitats Proportion of people infected Awareness Figure 4.2: The above graph is generated with parameter values a =.5, p =.5, b = 5, c =.6, l =.5, i =.5, r =.97, f = 2 (the parameters are placed in the region where b > ( c)( l)). One can see that the solutions display convergence to the equilibrium for big values of r. a i ( ) J,, c, l + b = c cp c c p f r with the characteristic equation a polynomial that factors into: ( (a λ) l + b ) c λ (c λ)(r λ) =. Three of the roots, namely λ = a, λ 2 = c and λ 4 = r are less than and if l + b c < (or b < ( c)( l)) then the degenerate equilibrium is a sink and thus locally asymptotically stable. It remains to be shown that this equilibrium is a global attractor. We offer a proof by contradiction as in [2]. Let s suppose S P > and S M >. Then using the last equation in the system, we conclude: S A rs A + f a Using that e Mn < M n in the second equation of the reduced system yields M n lm n + bh n M n.
11 Awerbuch et. al Mosquitoes Habitats Proportion of people infected Awareness Figure 4.3: The above graph is generated with parameter values a =.5, p =.5, b = 5, c =.6, l =.5, i =.5, r =.5, f = 2 (the parameters are placed in the region where b > ( c)( l)). One can see that the solutions display oscillatory behavior for smaller values of r. Thus S M ls M + bs H S M. Dividing by S M > to both sides one obtains l b S H c which implies that ( l)( c) b (hence the contradiction). Thus S M =. First equation in the reduced system yields the inequality P n+ ap n + im n ( P n ) or further P n+ ap n + im n. Passing to the limit one has S P as P + is M = as P Dividing by ( a) > the above yields S P which in combination with S P gives S P =. Using the inequality in the third equation: I H ci H + ci H +. + ps A + ps A It follows I H c. But S H c I H and thus S H = I H = c. S A = follows easily.
12 A Non-Linear System of Difference Equations 5 Conclusions and Open Problems The global asymptotic stability of the degenerate equilibrium was investigated (but the global asymptotic stability of the positive equilibrium remains an open problem that is worth investigating mathematically). An interesting result pertains to the role that the memory plays in controlling the epidemic. We observed oscillatory behavior for marginally low memory parameter values (r =.5, Fig.4.3), meaning that the population might recover only for a short period of time, and then getting periodically infected. High awareness with ( r =.97, Fig. 4.2) leads to a complete decrease in the proportion of infected people and the solutions stabilize. Simulations done with various parameter values, seems to suggest that the memory parameter has a threshold below which there are oscillations and above which it exhibits the equilibria, leading to the extinction of the infection. This is consistent with other findings from studies specifically designed to discover thresholds (see[7]). In [7] the authors, considered the rate of contact between susceptible people and infectious vectors, a component captured in our system in the first equation by the term ( e imn )( P n ). Their study reports that they were surprised to discover that the size of the viral introduction was not seen to significantly influence the magnitude of the threshold. In our future study, we shall focus on finding the memory parameter threshold value that leads to the extinction of the infection and to test whether changing the initial conditions of the proportion of infected people P, has an impact on the threshold value or not. The average number of mosquitoes per breeding site (parameter b), was estimated to be 9.5, ranging from 3 to 3, in field studies(see [8]). We used b = 5 (see Fig. 4.2 and 4.3), a value within the range suggested by field studies in the aforementioned reference. Computer simulations on system (.) indicate that it is possible that for large values of parameter d (high pollution level such as new empty cans and tires that collect water), the memory parameter r alone may not be sufficiently strong enough to eliminate the infection from the population, and the infection might equilibrate at levels higher than zero. In future work we shall explore the relationship between environmental pollution and the memory that creates awareness in the community. In this section we also want to bring attention to some extensions and open problems related to system (.). An interesting question to be analytically investigated in a further study is the global asymptotic stability of non-degenerate equilibrium of system (.) especially in the case when the system incorporates different parameters that measure the sensitivity of surviving habitats to communal awareness and individual awareness (hence p q). Thus, in this case the third equation reads H n+ = ch n h (pa n ) + dh 2 (qa n ). Based on biological considerations, one can take h ( ) and h 2 ( ) as decreasing functions, h, h 2 C ((, ) (, ]) with properties (i) h () = and h 2 () = ; (ii) lim y h (y) = and lim y h 2(y) =. Two most used examples of such functions (used in the previous work, [5] are for instance h (y) = /( + py) and h 2 (y) = /( + qy). Thus, a open problem that we want to pose here refers to the study of the existence and global asymptotic stability of the positive equilibrium of the general system:
13 2 Awerbuch et. al. P n+ = ap n + ( e imnpn )( P n ) M n+ = lm n e gan + bh n ( e smn ) H n+ = ch n h (pa n ) + dh 2 (qa n ) A n+ = ra n + fp n n =,,... (5.) Mathematical models may serve at designing policy interventions and provide a better understanding of phenomena at study ([5]). Because, at times interventions are implemented when consciousness is prompted by an increase in the incidence of sick people, one can work with the original system in a form as such: P n+ = ap n + ( e imnpn )( P n ) M n+ = lm n e gan + bh n ( e smn ) H n+ = ch n d n =,,... (5.2) + + pa n + qa n A n+ = ra n + fp n The first equation describes the proportion of infected people in the population (between and ). Proportion of sick people is assumed to prompt consciousness, while the intervention is against mosquitoes and perhaps habitats. The control of both, adult Mosquitoes (M) and habitats (H) where mosquitoes lay their eggs is carried out by spraying and community intervention by reducing breeding sites. One may use this system (system 5.2) to compare a few control strategies, where increase in the proportion of infected people is linked to consciousness. Insecticide spraying is a common method in mosquito control despite of its many disadvantages; and new ones are continuously being developed and tested (Alimi, Qualls et al. 23 ([] ); Dantun, Zainderberg and Santana 23 ([6]); Kaufman, Mann, Butler 2 ([2])). In the long run the mosquitoes become resistant and the insecticide ineffective (Onstad 23 [3]); it poses serious risks to humans and the environment (Jeyaratnam, 99 ([9]); Davis et al. 27; Rodrguez et al. 26 ([6]); Kaufman, Mann, Butler 2 ([2]); Wassie et al 22 ([7])). In order to assess the effect of insecticide spraying without habitat management, the equations are modified so that we eliminate the rational control on H n, and keep the population control on M n. To assess the effect of habitat control only, through citizens intervention, the equation will keep its intervention parameters as such H n+ = ch n /( + pa n ) + d/( + qa n ). for example. We believe that system (5.2) is useful not only biologically but also interesting mathematically. Both systems (5.) and (5.2) posses bounded solutions. Acknowledgments: We thank Nathalie Marchand and Vadym Barda for running the simulations. We thank the reviewers for their comments that helped improve the quality of our manuscript.
14 A Non-Linear System of Difference Equations 3 References [] Alimi TO, Qualls WA, Roque DD, Naranjo DP, Samson DM, Beier JC, Xue RD., Evaluation of a new formulation of permethrin applied by water-based thermal fogger against Aedes albopictus in residential communities in St. Augustine, Florida.J Am Mosq Control Assoc. 23 Mar;29(): [2] T. Awerbuch, E. Camouzis, G.Ladas, R.Levins, E.A.Grove, M. Predescu, A Nonlinear System of Difference Equations, linking Mosquitoes, Habitats and Community Interventions, Communications on Applied Nonlinear Analysis, vol. 5, no. 2, 77-88, 28. [3] T. Awerbuch, R. Levins, M. Predescu and G. Sirbu, On the Dynamics of a Deterministic and Stochastic Model for Mosquito Control, Applied Mathematics Letters, vol 2, , 27. [4] T. Awerbuch, R. Levins and M. Predescu, Co-Dynamics of Consciousness and Populations of the Dengue Vector in a Spraying Intervention System Modeled by Difference Equations,Far East Journal of Applied Mathematics, vol. 37, number 2 (29), pages [5] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, November 27. [6] Dantur Juri MJ, Zaidenberg M, Santana M., The efficacy of a combined larvicideadulticide in ultralow volume and fumigant canister formulations in controlling the dengue vector Aedes aegypti (Diptera: Culicidae) in Northwest of Argentina. Parasitol Res., 2(3):237-46, 23. [7] Focks DA, Brenner RJ, Hayes J, Daniels E.Transmission thresholds for dengue in terms of Aedes aegypti pupae per person with discussion of their utility in source reduction efforts, Am J Trop Med Hyg. 2 Jan;62():-8. [8] Focks DA, Chadee DD Pupal survey: an epidemiologically significant surveillance method for Aedes aegypti: an example using data from Trinidad, Am J Trop Med Hyg. 997 Feb;56(2): [9] Jeyaratnam, J., Acute pesticide poisoning: a major global health problem, World health statistics quarterly. Rapport trimestriel de statistiques sanitaires mondiales 43 (3): 3944, 99. [] S.A. Juliano, Population dynamics review, J. Am. Mosq. Control Assoc., 23 (27), [] S. Kasai, T. Shono, O. Komagata, Y, Tsuda, M. Kobayashi, M, Motoki, I. Kashima, T. Tanikawa, M. Yoshida, I. Tanaka, G. Shinjo, T. Hashimoto, T. Ishikawa, Y. Higa, and T. Tomita, Insecticide resistance in potential vector mosquitoes for West Nile virus in Japan, J. Med. Entomol., 44, (27),
15 4 Awerbuch et. al. [2] Kaufman PE, Mann RS, Butler JF, Evaluation of semiochemical toxicity to Aedes aegypti, Ae. albopictus and Anopheles quadrimaculatus (Diptera: Culicidae).Pest Manag Sci. May;66(5):497-54, 2. [3] David W. Onstad, Insect Resistance Management: Biology, Economics, and Prediction, Academic Press Elesvier, 24. [4] Padmanabha H, Soto E, Mosquera M, Lord CC, Lounibos LP (2) Ecological Links Between Water Storage Behaviors and Aedes aegypti Production: Implications for Dengue Vector Control, Variable Clim Am J Trop Med Hyg. 2 Jan;62():-8. [5] M. Predescu, R. Levins, and T.E. Awerbuch-Friedlander, Analysis of a nonlinear system for community intervention in mosquito control, Discrete and Continous Dynamical Systems, Series B, 6(3), (26), [6] Rodrguez AD, Penilla RP, Rodrguez MH, Hemingway J, Trejo A, Hernndez-Avila JE., Acceptability and perceived side effects of insecticide indoor residual spraying under different resistance management strategies. Salud Publica Mex. 48(4):37-24, 26. [7] Wassie F, Spanoghe P, Tessema DA, Steurbaut W., Exposure and health risk assessment of applicators to DDT during indoor residual spraying in malaria vector control program. J Expo Sci Environ Epidemiol., Nov;22(6):549-58, 22.
Modelling the dynamics of dengue real epidemics
Modelling the dynamics of dengue real epidemics Claudia P. Ferreira Depto de Bioestatística, IB, UNESP E-mail: pio@ibb.unesp.br Suani T.R. Pinho Instituto de Física, Universidade Federal da Bahia E-mail:
More informationGeographical Information System (GIS)-based maps for monitoring of entomological risk factors affecting transmission of chikungunya in Sri Lanka
Geographical Information System (GIS)-based maps for monitoring of entomological risk factors affecting transmission of chikungunya in Sri Lanka M.D. Hapugoda 1, N.K. Gunewardena 1, P.H.D. Kusumawathie
More informationA Mathematical Model for Transmission of Dengue
Applied Mathematical Sciences, Vol. 10, 2016, no. 7, 345-355 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.510662 A Mathematical Model for Transmission of Dengue Luis Eduardo López Departamento
More informationModeling the Existence of Basic Offspring Number on Basic Reproductive Ratio of Dengue without Vertical Transmission
International Journal on Recent and Innovation Trends in Computing and Communication ISSN: 232-869 Modeling the Existence of Basic Offspring Number on Basic Reproductive Ratio of Dengue without Vertical
More informationVector Hazard Report: Malaria in Ghana Part 1: Climate, Demographics and Disease Risk Maps
Vector Hazard Report: Malaria in Ghana Part 1: Climate, Demographics and Disease Risk Maps Information gathered from products of The Walter Reed Biosystematics Unit (WRBU) VectorMap Systematic Catalogue
More informationAedes aegypti Population Model with Integrated Control
Applied Mathematical Sciences, Vol. 12, 218, no. 22, 175-183 HIKARI Ltd, www.m-hiari.com https://doi.org/1.12988/ams.218.71295 Aedes aegypti Population Model with Integrated Control Julián A. Hernández
More informationGlobal Stability of a Computer Virus Model with Cure and Vertical Transmission
International Journal of Research Studies in Computer Science and Engineering (IJRSCSE) Volume 3, Issue 1, January 016, PP 16-4 ISSN 349-4840 (Print) & ISSN 349-4859 (Online) www.arcjournals.org Global
More informationStudent Background Readings for Bug Dynamics: The Logistic Map
Student Background Readings for Bug Dynamics: The Logistic Map Figure 1 The population as a function of a growth rate parameter observed in the experiment of [Bugs]. Graphs such as this are called bifurcation
More informationGlobal Analysis of an Epidemic Model with Nonmonotone Incidence Rate
Global Analysis of an Epidemic Model with Nonmonotone Incidence Rate Dongmei Xiao Department of Mathematics, Shanghai Jiaotong University, Shanghai 00030, China E-mail: xiaodm@sjtu.edu.cn and Shigui Ruan
More informationRole of GIS in Tracking and Controlling Spread of Disease
Role of GIS in Tracking and Controlling Spread of Disease For Dr. Baqer Al-Ramadan By Syed Imran Quadri CRP 514: Introduction to GIS Introduction Problem Statement Objectives Methodology of Study Literature
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 6, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 6, 1671-1684 ISSN: 1927-5307 A MATHEMATICAL MODEL FOR THE TRANSMISSION DYNAMICS OF HIV/AIDS IN A TWO-SEX POPULATION CONSIDERING COUNSELING
More informationA threshold analysis of dengue transmission in terms of weather variables and imported dengue cases in Australia
OPEN (2013) 2, e87; doi:10.1038/emi.2013.85 ß 2013 SSCC. All rights reserved 2222-1751/13 www.nature.com/emi ORIGINAL ARTICLE A threshold analysis of dengue transmission in terms of weather variables and
More informationStability Analysis of a SIS Epidemic Model with Standard Incidence
tability Analysis of a I Epidemic Model with tandard Incidence Cruz Vargas-De-León Received 19 April 2011; Accepted 19 Octuber 2011 leoncruz82@yahoo.com.mx Abstract In this paper, we study the global properties
More informationStability of SEIR Model of Infectious Diseases with Human Immunity
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1811 1819 Research India Publications http://www.ripublication.com/gjpam.htm Stability of SEIR Model of Infectious
More informationAccepted Manuscript. Backward Bifurcations in Dengue Transmission Dynamics. S.M. Garba, A.B. Gumel, M.R. Abu Bakar
Accepted Manuscript Backward Bifurcations in Dengue Transmission Dynamics S.M. Garba, A.B. Gumel, M.R. Abu Bakar PII: S0025-5564(08)00073-4 DOI: 10.1016/j.mbs.2008.05.002 Reference: MBS 6860 To appear
More informationMathematical models on Malaria with multiple strains of pathogens
Mathematical models on Malaria with multiple strains of pathogens Yanyu Xiao Department of Mathematics University of Miami CTW: From Within Host Dynamics to the Epidemiology of Infectious Disease MBI,
More informationDynamical models of HIV-AIDS e ect on population growth
Dynamical models of HV-ADS e ect on population growth David Gurarie May 11, 2005 Abstract We review some known dynamical models of epidemics, given by coupled systems of di erential equations, and propose
More informationName Student ID. Good luck and impress us with your toolkit of ecological knowledge and concepts!
Page 1 BIOLOGY 150 Final Exam Winter Quarter 2000 Before starting be sure to put your name and student number on the top of each page. MINUS 3 POINTS IF YOU DO NOT WRITE YOUR NAME ON EACH PAGE! You have
More informationDynamics of the equation complex plane
APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Dynamics of the equation in the complex plane Sk. Sarif Hassan 1 and Esha Chatterjee * Received: 0 April 015 Accepted: 08 November 015 Published:
More informationDifferential Transform and Butcher s fifth order Runge-Kutta Methods for solving the Aedes-Aegypti model
Differential Transform and Butcher s fifth order Runge-Kutta Methods for solving the Aedes-Aegypti model R. Lavanya 1 and U. K. Shyni Department of Mathematics, Coimbatore Institute of Technology, Coimbatore
More informationThe Existence and Stability Analysis of the Equilibria in Dengue Disease Infection Model
Journal of Physics: Conference Series PAPER OPEN ACCESS The Existence and Stability Analysis of the Equilibria in Dengue Disease Infection Model Related content - Anomalous ion conduction from toroidal
More informationThursday. Threshold and Sensitivity Analysis
Thursday Threshold and Sensitivity Analysis SIR Model without Demography ds dt di dt dr dt = βsi (2.1) = βsi γi (2.2) = γi (2.3) With initial conditions S(0) > 0, I(0) > 0, and R(0) = 0. This model can
More informationQuantifying the impact of decay in bed-net efficacy on malaria transmission
Quantifying the impact of decay in bed-net efficacy on malaria transmission The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters.
More information6. Age structure. for a, t IR +, subject to the boundary condition. (6.3) p(0; t) = and to the initial condition
6. Age structure In this section we introduce a dependence of the force of infection upon the chronological age of individuals participating in the epidemic. Age has been recognized as an important factor
More informationModelling of the Hand-Foot-Mouth-Disease with the Carrier Population
Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Ruzhang Zhao, Lijun Yang Department of Mathematical Science, Tsinghua University, China. Corresponding author. Email: lyang@math.tsinghua.edu.cn,
More informationMATHEMATICAL MODELS Vol. III - Mathematical Models in Epidemiology - M. G. Roberts, J. A. P. Heesterbeek
MATHEMATICAL MODELS I EPIDEMIOLOGY M. G. Roberts Institute of Information and Mathematical Sciences, Massey University, Auckland, ew Zealand J. A. P. Heesterbeek Faculty of Veterinary Medicine, Utrecht
More informationIntroduction to SEIR Models
Department of Epidemiology and Public Health Health Systems Research and Dynamical Modelling Unit Introduction to SEIR Models Nakul Chitnis Workshop on Mathematical Models of Climate Variability, Environmental
More informationA Comparison of Mosquito Species in Developed and Undeveloped parts of the College Station/Bryan Area in the Brazos County, Texas
A Comparison of Mosquito Species in Developed and Undeveloped parts of the College Station/Bryan Area in the Brazos County, Texas Carlos Briones, Caren Gonzalez, & Kevin Henson Texas A&M University, Department
More informationA MODEL FOR THE EFFECT OF TEMPERATURE AND RAINFALL ON THE POPULATION DYNAMICS AND THE EFFICIENCY OF CONTROL OF Aedes aegypit MOSQUITO
A MODEL FOR THE EFFECT OF TEMPERATURE AND RAINFALL ON THE POPULATION DYNAMICS AND THE EFFICIENCY OF CONTROL OF Aedes aegypit MOSQUITO L. B. Barsante F. S. Cordeiro R. T. N. Cardoso A. E. Eiras J. L. Acebal
More informationA NUMERICAL STUDY ON PREDATOR PREY MODEL
International Conference Mathematical and Computational Biology 2011 International Journal of Modern Physics: Conference Series Vol. 9 (2012) 347 353 World Scientific Publishing Company DOI: 10.1142/S2010194512005417
More informationDengue Forever. Edilber Almanza-Vasquez
International Journal of Mathematical Analysis Vol. 8, 014, no. 5, 547-559 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.41097 Dengue Forever Edilber Almanza-Vasquez Faculty of Exact
More informationTowards a risk map of malaria for Sri Lanka
assessing the options for control of malaria vectors through different water management practices in a natural stream that formed part of such a tank cascade system. The studies established conclusively
More informationWhy Should Mathematical Modeling In Biology Be Even Possible?
Why Should Mathematical Modeling In Biology Be Even Possible? Winfried Just Department of Mathematics, Ohio University March 27, 2018 An example of mathematical modeling in biology. Based on Oduro, Grijalva,
More informationLaboratory evaluation of Bacillus thuringiensis H-14 against Aedes aegypti
Tropical Biomedicine 22(1): 5 10 (2005) Laboratory evaluation of Bacillus thuringiensis H-14 against Aedes aegypti Lee, Y.W. and Zairi, J. Vector Control Research Unit, School of Biological Sciences, Universiti
More informationSTABILITY IN A MODEL FOR GENETICALLY ALTERED MOSQUITOS WITH PERIODIC PARAMETERS
STABILITY IN A MODEL FOR GENETICALLY ALTERED MOSQUITOS WITH PERIODIC PARAMETERS Hubertus F. von Bremen and Robert J. Sacker Department of Mathematics and Statistics, California State Polytechnic University,
More informationTHE STABILITY AND HOPF BIFURCATION OF THE DENGUE FEVER MODEL WITH TIME DELAY 1
italian journal of pure and applied mathematics n. 37 2017 (139 156) 139 THE STABILITY AND HOPF BIFURCATION OF THE DENGUE FEVER MODEL WITH TIME DELAY 1 Jinlan Guan 2 Basic Courses Department Guangdong
More informationImpacts of Climate Change on Public Health: Bangladesh Perspective
Global Journal of Environmental Research 5 (3): 97-15, 211 ISSN 199-925X IDOSI Publications, 211 Impacts of Climate Change on Public Health: Bangladesh Perspective M. Ruhul Amin, S.M. Tareq and S.H. Rahman
More informationDetermining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model
Bulletin of Mathematical Biology (2008) 70: 1272 1296 DOI 10.1007/s11538-008-9299-0 ORIGINAL ARTICLE Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical
More informationProgram Update. Lisle Township August 2018 Status Report SEASON PERSPECTIVE
Lisle Township August 2018 Status Report SEASON PERSPECTIVE Introduction. Weather conditions critically affect the seasonal mosquito population. Excessive rainfall periods trigger hatches of floodwater
More informationCluster Analysis using SaTScan
Cluster Analysis using SaTScan Summary 1. Statistical methods for spatial epidemiology 2. Cluster Detection What is a cluster? Few issues 3. Spatial and spatio-temporal Scan Statistic Methods Probability
More informationA mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host
A mathematical model for malaria involving differential susceptibility exposedness and infectivity of human host A. DUCROT 1 B. SOME 2 S. B. SIRIMA 3 and P. ZONGO 12 May 23 2008 1 INRIA-Anubis Sud-Ouest
More informationGIS to Support West Nile Virus Program
GIS to Support West Nile Virus Program 2008 Community Excellence Awards Leadership and Innovation Regional District Regional District Okanagan Similkameen July 2008 2008 UBCM Community Excellence Awards
More information2. Overproduction: More species are produced than can possibly survive
Name: Date: What to study? Class notes Graphic organizers with group notes Review sheets What to expect on the TEST? Multiple choice Short answers Graph Reading comprehension STRATEGIES Circle key words
More informationA New Mathematical Approach for. Rabies Endemy
Applied Mathematical Sciences, Vol. 8, 2014, no. 2, 59-67 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.39525 A New Mathematical Approach for Rabies Endemy Elif Demirci Ankara University
More informationProject 1 Modeling of Epidemics
532 Chapter 7 Nonlinear Differential Equations and tability ection 7.5 Nonlinear systems, unlike linear systems, sometimes have periodic solutions, or limit cycles, that attract other nearby solutions.
More informationDevelopment and Validation of. Statistical and Deterministic Models. Used to Predict Dengue Fever in. Mexico
Development and Validation of Statistical and Deterministic Models Used to Predict Dengue Fever in Mexico A thesis presented by Aditi Hota to the Applied Mathematics Department in partial fulfillment of
More informationThe Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
Applied Mathematics, 05, 6, 665-675 Published Online September 05 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/046/am056048 The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
More informationEpidemics in Complex Networks and Phase Transitions
Master M2 Sciences de la Matière ENS de Lyon 2015-2016 Phase Transitions and Critical Phenomena Epidemics in Complex Networks and Phase Transitions Jordan Cambe January 13, 2016 Abstract Spreading phenomena
More informationResearch Article Global Stability and Oscillation of a Discrete Annual Plants Model
Abstract and Applied Analysis Volume 2010, Article ID 156725, 18 pages doi:10.1155/2010/156725 Research Article Global Stability and Oscillation of a Discrete Annual Plants Model S. H. Saker 1, 2 1 Department
More informationLocal Stability Analysis of a Mathematical Model of the Interaction of Two Populations of Differential Equations (Host-Parasitoid)
Biology Medicine & Natural Product Chemistry ISSN: 089-6514 Volume 5 Number 1 016 Pages: 9-14 DOI: 10.1441/biomedich.016.51.9-14 Local Stability Analysis of a Mathematical Model of the Interaction of Two
More informationThree Disguises of 1 x = e λx
Three Disguises of 1 x = e λx Chathuri Karunarathna Mudiyanselage Rabi K.C. Winfried Just Department of Mathematics, Ohio University Mathematical Biology and Dynamical Systems Seminar Ohio University November
More informationSimulation of Population Dynamics of Aedes aegypti using Climate Dependent Model
Simulation of Population Dynamics of Aedes aegypti using Climate Dependent Model Nuraini Yusoff, Harun Bun, and Salemah Ismail Abstract A climate dependent model is proposed to simulate the population
More informationViable Control of an Epidemiological Model
Viable Control of an Epidemiological Model arxiv:1510.01055v1 [math.oc] 5 Oct 2015 Michel De Lara Lilian Sofia Sepulveda Salcedo January 5, 2018 Abstract In mathematical epidemiology, epidemic control
More informationSupplementary Information
Supplementary Information This document shows the supplementary figures referred to in the main article. The contents are as follows: a. Malaria maps b. Dengue maps c. Yellow fever maps d. Chikungunya
More informationModels of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005
Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor August 15, 2005 1 Outline 1. Compartmental Thinking 2. Simple Epidemic (a) Epidemic Curve 1:
More informationCan multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species
Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species Xingfu Zou Department of Applied Mathematics University of Western Ontario London, Ontario, Canada (Joint work
More informationGLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 304, pp. 1 8. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY
More informationApplied Mathematics Letters
Applied athematics Letters 25 (212) 156 16 Contents lists available at SciVerse ScienceDirect Applied athematics Letters journal homepage: www.elsevier.com/locate/aml Globally stable endemicity for infectious
More informationMath 3313: Differential Equations First-order ordinary differential equations
Math 3313: Differential Equations First-order ordinary differential equations Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Math 2343: Introduction Separable
More informationRELATIONSHIP BETWEEN RAINFALL AND AEDES LARVAL POPULATION AT TWO INSULAR SITES IN PULAU KETAM, SELANGOR, MALAYSIA
Relationship Between Rainfall and Aedes Larval Population RELATIONSHIP BETWEEN RAINFALL AND AEDES LARVAL POPULATION AT TWO INSULAR SITES IN PULAU KETAM, SELANGOR, MALAYSIA Lim Kwee Wee 1,2, Sit Nam Weng
More informationUnderstanding Uncertainties in Model-Based Predictions of Aedes aegypti Population Dynamics
Understanding Uncertainties in Model-Based Predictions of Aedes aegypti Population Dynamics Chonggang Xu 1 *, Mathieu Legros 1, Fred Gould 1, Alun L. Lloyd 2 1 Department of Entomology, North Carolina
More informationAnalysis of SIR Mathematical Model for Malaria disease with the inclusion of Infected Immigrants
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 14, Issue 5 Ver. I (Sep - Oct 218), PP 1-21 www.iosrjournals.org Analysis of SIR Mathematical Model for Malaria disease
More informationTemporal and Spatial Autocorrelation Statistics of Dengue Fever
Temporal and Spatial Autocorrelation Statistics of Dengue Fever Kanchana Nakhapakorn a and Supet Jirakajohnkool b a Faculty of Environment and Resource Studies, Mahidol University, Salaya, Nakhonpathom
More informationDelay SIR Model with Nonlinear Incident Rate and Varying Total Population
Delay SIR Model with Nonlinear Incident Rate Varying Total Population Rujira Ouncharoen, Salinthip Daengkongkho, Thongchai Dumrongpokaphan, Yongwimon Lenbury Abstract Recently, models describing the behavior
More informationGlobal Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response
Global Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response Gang Huang 1,2, Wanbiao Ma 2, Yasuhiro Takeuchi 1 1,Graduate School of Science and Technology, Shizuoka University,
More informationAustralian Journal of Basic and Applied Sciences. Effect of Education Campaign on Transmission Model of Conjunctivitis
ISSN:99-878 Australian Journal of Basic and Applied Sciences Journal home page: www.ajbasweb.com ffect of ducation Campaign on Transmission Model of Conjunctivitis Suratchata Sangthongjeen, Anake Sudchumnong
More informationLesson Plan: Vectors and Venn Diagrams
Prep Time: Minimal Lesson Plan: Vectors and Venn Diagrams Age Level: Can be modified for any grade Materials Needed: Blank Venn diagrams can be printed for students to complete (included in this document),
More information1 Vocabulary. Chapter 5 Ecology. Lesson. Carnivore an organism that only eats meat or flesh. Niche an organism s role in the habitat
1 Vocabulary Carnivore an organism that only eats meat or flesh Niche an organism s role in the habitat Community all the populations in one place that interact with each other Decomposer digests the waste
More informationON THE RATIONAL RECURSIVE SEQUENCE X N+1 = γx N K + (AX N + BX N K ) / (CX N DX N K ) Communicated by Mohammad Asadzadeh. 1.
Bulletin of the Iranian Mathematical Society Vol. 36 No. 1 (2010), pp 103-115. ON THE RATIONAL RECURSIVE SEQUENCE X N+1 γx N K + (AX N + BX N K ) / (CX N DX N K ) E.M.E. ZAYED AND M.A. EL-MONEAM* Communicated
More informationAnalysis of an SEIR-SEI four-strain epidemic dengue model with primary and secondary infections
CITATION. Raúl Isea. Reista Electrónica Conocimiento Libre y Licenciamiento (CLIC). Vol. 7 (201) 3-7 ISSN: 22-723 Analysis of an SEIR-SEI four-strain epidemic dengue model with primary and secondary infections
More informationPractical Experience in the Field Release of Transgenic Mosquitos
April 2013 Practical Experience in the Field Release of Transgenic Mosquitos structure Overview including field efficacy Separating males from females Longevity Gene and insect dispersal Gene penetrance
More informationGLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS
CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model
More informationThe effects of climatic factors on the distribution and abundance of Japanese encephalitis vectors in Kurnool district of Andhra Pradesh, India
J Vector Borne Dis 47, March 2010, pp. 26 32 The effects of climatic factors on the distribution and abundance of Japanese encephalitis vectors in Kurnool district of Andhra Pradesh, India U. Suryanarayana
More informationA Preliminary Mathematical Model for the Dynamic Transmission of Dengue, Chikungunya and Zika
American Journal of Modern Physics and Application 206; 3(2): -5 http://www.openscienceonline.com/journal/ajmpa A Preliminary Mathematical Model for the Dynamic Transmission of Dengue, Chikungunya and
More informationThe Response of Environmental Capacity for Malaria Transmission in West Africa to Climate Change
AGU Fall Meeting December 9, 2011 The Response of Environmental Capacity for Malaria Transmission in West Africa to Climate Change Teresa K. Yamana & Elfatih A.B. Eltahir MIT Dept. of Civil & Environmental
More informationQualitative Analysis of a Discrete SIR Epidemic Model
ISSN (e): 2250 3005 Volume, 05 Issue, 03 March 2015 International Journal of Computational Engineering Research (IJCER) Qualitative Analysis of a Discrete SIR Epidemic Model A. George Maria Selvam 1, D.
More informationSimple Mathematical Model for Malaria Transmission
Journal of Advances in Mathematics and Computer Science 25(6): 1-24, 217; Article no.jamcs.37843 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-851) Simple
More informationCLIMATE CHANGE, ROSS RIVER VIRUS AND BIODIVERSITY
24 CLIMATE CHANGE, ROSS RIVER VIRUS AND BIODIVERSITY PHILIP WEINSTEIN AND PENG BI Abstract Infection with the Australian Ross River virus (RRV) results in rash, fever and rheumatic symptoms lasting several
More informationPredation is.. The eating of live organisms, regardless of their identity
Predation Predation Predation is.. The eating of live organisms, regardless of their identity Predation 1)Moves energy and nutrients through the ecosystem 2)Regulates populations 3)Weeds the unfit from
More informationMathematical Analysis of Epidemiological Models: Introduction
Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010 1. Introduction. The effectiveness of improved sanitation,
More informationGlobal Attractivity of a Higher-Order Nonlinear Difference Equation
International Journal of Difference Equations ISSN 0973-6069, Volume 5, Number 1, pp. 95 101 (010) http://campus.mst.edu/ijde Global Attractivity of a Higher-Order Nonlinear Difference Equation Xiu-Mei
More informationLetter to the Editor
J. theor. Biol. (1998) 195, 413 417 Article No. jt980693 Letter to the Editor A Note on Errors in Grafen s Strategic Handicap Models In two papers published in this Journal, Grafen (1990a,b) has provided
More informationEPIDEMIOLOGY FOR URBAN MALARIA MAPPING
TELE-EPIDEMIOLOGY EPIDEMIOLOGY FOR URBAN MALARIA MAPPING @IRD/M Dukhan Vanessa Machault Observatoire Midi-Pyrénées, Laboratoire d Aérologie Pleiades days 17/01/2012 The concept of Tele-epidemiology The
More information1. INTRODUCTION REGIONAL ACTIVITIES
P 1.11 DEVELOPMENT OF CLIMATE INDICES FOR MONITORING VECTORS OF WEST NILE VIRUS Michael J. Janis Southeast Regional Climate Center, Columbia, SC Kenneth E. Kunkel Illinois State Water Survey, Champaign,
More informationME 406 S-I-R Model of Epidemics Part 2 Vital Dynamics Included
ME 406 S-I-R Model of Epidemics Part 2 Vital Dynamics Included sysid Mathematica 6.0.3, DynPac 11.01, 1ê13ê9 1. Introduction Description of the Model In this notebook, we include births and deaths in the
More informationThe SEIR Dynamical Transmission Model of Dengue Disease with and Without the Vertical Transmission of the Virus
American Journal of Applied Sciences Original Research Paper The SEIR Dynamical Transmission Model of Dengue Disease with and Without the ertical Transmission of the irus 1 Pratchaya Chanprasopchai, I.
More informationMathematical Model of Tuberculosis Spread within Two Groups of Infected Population
Applied Mathematical Sciences, Vol. 10, 2016, no. 43, 2131-2140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.63130 Mathematical Model of Tuberculosis Spread within Two Groups of Infected
More informationIN VITRO STUDIES ON THE BIOCONTROL POTENTIAL OF FISH SPECIES CARASSIUS AURATUS AGAINST MOSQUITO CULEX QUINQUEFASCIATUS
IN VITRO STUDIES ON THE BIOCONTROL POTENTIAL OF FISH SPECIES CARASSIUS AURATUS AGAINST MOSQUITO CULEX QUINQUEFASCIATUS D. Mary Cynthia 1 and Dr. R, Dhivya 2 1 PG Student, PG and Research, Department of
More informationAustralian Journal of Basic and Applied Sciences
AENSI Journals Australian Journal of Basic and Applied Sciences ISSN:1991-8178 Journal home page: www.ajbasweb.com A SIR Transmission Model of Political Figure Fever 1 Benny Yong and 2 Nor Azah Samat 1
More informationGLOBAL DYNAMICS OF A TIME-DELAYED DENGUE TRANSMISSION MODEL
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 GLOBAL DYNAMICS OF A TIME-DELAYED DENGUE TRANSMISSION MODEL Dedicated to Herb Freedman on the occasion of his 7th birthday ZHEN WANG
More informationSensitivity and Stability Analysis of Hepatitis B Virus Model with Non-Cytolytic Cure Process and Logistic Hepatocyte Growth
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 3 2016), pp. 2297 2312 Research India Publications http://www.ripublication.com/gjpam.htm Sensitivity and Stability Analysis
More informationGeneralizations of Product-Free Subsets
Generalizations of Product-Free Subsets The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Kedlaya, Kiran
More informationImpact of environmental factors on mosquito dispersal in the prospect of Sterile Insect Technique control.
Impact of environmental factors on mosquito dispersal in the prospect of Sterile Insect Technique control. Claire Dufourd a, Yves Dumont b a Department of Mathematics and Applied Mathematics, University
More informationAnalysis of a Dengue Model with Vertical Transmission and Application to the 2014 Dengue Outbreak in Guangdong Province, China
Bulletin of Mathematical Biology https://doi.org/10.1007/s11538-018-0480-9 ORIGINAL ARTICLE Analysis of a Dengue Model with Vertical Transmission and Application to the 2014 Dengue Outbreak in Guangdong
More informationPredicting Malaria Epidemics in Ethiopia
Predicting Malaria Epidemics in Ethiopia Eskindir Loha, Hawassa University Torleif Markussen Lunde, UoB 5th Conference on Global Health and Vaccination Research: Environmental Change and Global Health
More informationDie Modellierung von indirekt übertragenen Erkrankungen durch Verkettung von deterministischen und stochastischen Komponenten
Die Modellierung von indirekt übertragenen Erkrankungen durch Verkettung von deterministischen und stochastischen Komponenten Modeling vector-borne diseases by combining deterministic and stochastic approaches
More informationVISCOSITY SOLUTIONS. We follow Han and Lin, Elliptic Partial Differential Equations, 5.
VISCOSITY SOLUTIONS PETER HINTZ We follow Han and Lin, Elliptic Partial Differential Equations, 5. 1. Motivation Throughout, we will assume that Ω R n is a bounded and connected domain and that a ij C(Ω)
More informationResearch Article Global Attractivity of a Higher-Order Difference Equation
Discrete Dynamics in Nature and Society Volume 2012, Article ID 930410, 11 pages doi:10.1155/2012/930410 Research Article Global Attractivity of a Higher-Order Difference Equation R. Abo-Zeid Department
More informationFinal Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger. Project I: Predator-Prey Equations
Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger Project I: Predator-Prey Equations The Lotka-Volterra Predator-Prey Model is given by: du dv = αu βuv = ρβuv
More informationSpatial mapping of temporal risk characteristics to improve environmental health risk identification: A case study of a dengue epidemic in Taiwan
Science of the Total Environment 367 (2006) 631 640 www.elsevier.com/locate/scitotenv Spatial mapping of temporal risk characteristics to improve environmental health risk identification: A case study
More information