Instructional exercises 4: Epidemiological Mathematical Modeling, The Cases of Tuberculosis and Dengue. Numerical Modeling of Infectious Diseases: Dynamics and Control (15 Aug - 9 Oct 2005) Jointly composed by Institute for Mathematical Sciences, National University of Singapore and Regional Emerging Diseases Intervention (REDI) Center, Singapore http://www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htm Singapore, 08-23-2005 Carlos Castillo-Chavez Joaquin Bustoz Jr. Educator Arizona State University Arizona State UniversitySlide 2
A TB model with age-structure (Castillo-Chavez and Feng. Math. Biosci. , 1998) Arizona State UniversitySlide 3
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SIR Model with Age Structure s(t,a) : Density of helpless people with age an at time t . i(t,a) : Density of irresistible people with age an at time t . r(t,a) : Density of recuperated people with age an at time t . # of helpless people with ages in ( a 1 , a 2 ) at time t # of irresistible people with ages in ( a 1 , a 2 ) at time t # of recuperated people with ages in ( a 1 , a 2 ) at time t Arizona State UniversitySlide 7
Parameters : enlistment/birth rate. (a): age-particular likelihood of getting to be tainted. c(a): age-particular per-capita contact rate. (a): age-particular per-capita death rate. (a): age-particular per-capita recuperation rate. Arizona State UniversitySlide 8
Mixing p(t,a,a`): likelihood that a person of age a has contact with a person of age a` given that it has a contact with an individual from the populace . Arizona State UniversitySlide 9
Mixing Rules p(t,a,a`) ≥ 0 Proportionate blending: Arizona State UniversitySlide 10
Equations Arizona State UniversitySlide 11
Demographic Steady State n(t,a): thickness of individual with age an at time t n(t,a) fulfills the Mackendrick Equation We expect that the aggregate populace thickness has achieved this demographic enduring state. Arizona State UniversitySlide 12
Parameters : enrollment rate. (a): age-particular likelihood of getting to be contaminated. c(a): age-particular per-capita contact rate. (a); age-particular per-capita death rate. k: movement rate from contaminated to irresistible. r: treatment rate. : lessening extent due to earlier presentation to TB. : diminishment extent because of inoculation. Arizona State UniversitySlide 13
Age Structure Model with inoculation Arizona State UniversitySlide 14
Vaccinated Age-subordinate ideal immunization procedures (Feng, Castillo-Chavez, Math . Biosci. , 1998) Arizona State UniversitySlide 15
Basic regenerative Number (by cutting edge administrator) Arizona State UniversitySlide 16
Stability There exists an endemic consistent state at whatever point R 0 ()>1. The contamination free consistent state is all inclusive asymptotically stable when R 0 = R 0 (0)<1. Arizona State UniversitySlide 17
Optimal Vaccination Strategies Two enhancement issues: If the objective is to bring R 0 () to pre-appointed esteem then discover the inoculation methodology (a) that minimizes the aggregate expense connected with this objective (diminished predominance to an objective level). In the event that the financial backing is settled (cost) discover an immunization system (a) that minimizes R 0 (), that will be, that minimizes the predominance . Arizona State UniversitySlide 18
R (y) < R * Reproductive numbers Two improvement issues: Arizona State UniversitySlide 19
One-age and two-age inoculation methodologies Arizona State UniversitySlide 20
Optimal Strategies One–age system: immunize the vulnerable populace at precisely age A. Two–age technique: immunize part of the defenseless populace at precisely age A 1 and the rest of the susceptibles at a later age A 2. . Chosen ideal technique relies on upon cost capacity (information) . Arizona State UniversitySlide 21
Generalized Household Model Incorporates contact sort (close versus easygoing) and concentrate on close and delayed contacts. Summed up families turn into the fundamental epidemiological unit as opposed to people. Use epidemiological time-scales in model improvement and examination. Arizona State UniversitySlide 22
Transmission Diagram Arizona State UniversitySlide 23
Key Features Basic epidemiological unit: bunch (summed up family) Movement of kE 2 to I class conveys nkE 2 to N 1 populace, where by suspicions nkE 2 (S 2/N 2 ) go to S 1 and nkE 2 (E 2/N 2 ) go to E 1 Conversely, recuperation of �� I irresistible take n I back to N 2 populace, where n I (S 1/N 1 )= �� S 1 go to S 2 and n I (E 1/N 1 )= �� E 1 go to E 2 Arizona State UniversitySlide 24
Basic Cluster Model Arizona State UniversitySlide 25
Basic Reproductive Number Where: is the normal number of contaminations created by one irresistible individual inside his/her group. indicates the portion that makes due over the inertness time frame. Arizona State UniversitySlide 26
Diagram of Extended Cluster Model Arizona State UniversitySlide 27
(n) Both close easygoing contacts are incorporated into the developed model. The danger of disease per powerless, , is thought to be a nonlinear capacity of the normal group size n. The consistent p measures extent of time of an "individual spread over inside a group. Arizona State UniversitySlide 28
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Role of Cluster Size (General Model ) E(n) signifies the proportion of inside group to between bunch transmission. E(n) increments and achieves its most extreme worth at The group size n * is ideal as it boosts the relative effect of inside to between bunch transmission. Arizona State UniversitySlide 31
Full framework Hoppensteadt\'s Theorem (1973) Reduced framework where x R m , y R n and is a positive genuine parameter close to zero (little parameter). Five conditions must be fulfilled (not recorded here). In the event that the decreased framework has a comprehensively asymptotically stable balance, then the full framework has a g.a.s. balance at whatever point 0< <<1. Arizona State UniversitySlide 32
1 Bifurcation Diagram Global bifurcation graph when 0< <<1 where indicates the proportion between rate of movement to dynamic TB and the normal life-range of the host (roughly). Arizona State UniversitySlide 33
Numerical Simulations Arizona State UniversitySlide 34
Concluding Remarks on Cluster Models A worldwide forward bifurcation is gotten when << 1 E ( n ) measures the relative effect of close versus easygoing contacts can be characterized. It characterizes ideal bunch (size that expands transmission). Technique can be utilized to concentrate on other transmission maladies with particular time scales, for example, flu Arizona State UniversitySlide 35
TB in the US (1953-1999) Arizona State UniversitySlide 36
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TB control in the U.S. CDC\'s objective 3.5 cases for every 100,000 by 2000 One case for each million by 2010. Could CDC meet this objective? Arizona State UniversitySlide 38
Model Construction Since d has been roughly equivalent to zero in the course of recent years in the US, we just consider Hence, N can be registered freely of TB. Arizona State UniversitySlide 39
Non-self-ruling model (lasting idle class of TB presented) Arizona State UniversitySlide 40
Effect of HIV Arizona State UniversitySlide 41
Parameter estimation and reenactment setup Arizona State UniversitySlide 42
N(t) from registration information N(t) is from evaluation information and populace projection Arizona State UniversitySlide 43
Results Arizona State UniversitySlide 44
Results Left: New instance of TB and information (spots) Right: 10% blunder bound of new cases and information Arizona State UniversitySlide 45
Regression approach Arizona State University A Markov chain model backings the same resultSlide 46
CONCLUSIONS Arizona State UniversitySlide 47
Conclusions Arizona State UniversitySlide 48
CDC\'s Goal Delayed Impact of HIV. Lower bend does exclude HIV sway; Upper bend speaks to the case rate when HIV is incorporated; Both are the same before 1983. Spots speak to genuine information. Arizona State UniversitySlide 49
Our work on TB Aparicio, J., A. Capurro and C. Castillo-Chavez, " On the long haul flow and re-rise of tuberculosis ." In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction , IMA Volume 125 , 351-360, Springer-Veralg, Berlin-Heidelberg-New York. Altered via Carlos Castillo-Chavez with Pauline van nook Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002 Aparicio J., A. Capurro and C. Castillo-Chavez, " Transmission and Dynamics of Tuberculosis on Generalized Households" Journal of Theoretical Biology 206, 327-341, 2000 Aparicio, J., A. Capurro and C. Castillo-Chavez, Markers of infection development: the instance of tuberculosis, Journal of Theoretical Biology , 215: 227-237, March 2002 . Aparicio, J., A. Capurro and C. Castillo-Chavez, " Frequency Dependent Risk of Infection and the Spread of Infectious Diseases ." In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction , IMA Volume 125 , 341-350, Springer-Veralg, Berlin-Heidelberg-New York. Altered via Carlos Castillo-Chavez with Pauline van cave Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002 Berezovsky, F., G. Karev, B. Melody, and C. Castillo-Chavez , Simple Models with Surprised Dynamics , Journal of Mathematical Biosciences and Engineering, 2(1): 133-152, 2004. Castillo-Chavez, C. what\'s more, Feng, Z. (1997), To treat or not to treat: the instance of tuberculosis , J. Math. Biol. Arizona
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