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Study and Analysis of Cholera Transmission MALLESSA YEBOAH OLIVIA HYLTON Cholera Diarrhoeal disease caused by an intestinal bacteria (Vibrio cholerae). ◦ Thrives on environmental conditions such as salinity and temperature ◦ Can persist for extended periods of time in water sources under appropriate conditions ◦ Found in the feces of infected individuals ◦ Spreads when infected individuals shed pathogens into drinking water or contaminates food ◦ Contact may occur from person-to-person ◦ Can be life-threatening Outbreak in Haiti Following the earthquake of January 2010 in Haiti the first cholera outbreak in almost a century was announced in October of that year. To date, over 470,000 cases of cholera have been reported in Haiti with 6,631 attributable deaths. This marks the worst cholera outbreak in recent history, as well as the best documented cholera outbreak in modern public health. Since the beginning of the outbreak the CDC (Center of Disease Control) has worked closely with the Haitian MSPP (Ministry of Public Health and Population) to combat the cholera epidemic and reduce the impact of the disease. Outbreak in Haiti It is uncommon for developed countries to have large outbreaks of this disease. ◦ Advanced water and sanitation systems prevent the occurrences of such waterborne diseases Controlling an outbreak remains a major challenge in both developing and underdeveloped countries. Mathematical Modeling • A tool that complements statistical analysis and field epidemiology. • Study the progression of infectious diseases is a significant aspect of mathematical research. • Allow us to use information about a specific disease to show the projected outcome of an epidemic, control the spread of disease, and reduce disease related costs. Modeling Cholera Outbreak The SIWR model was developed by Tien and Earn. S-Susceptible I-Infectious W-Waterborne Pathogen Concentration R-Recovered • Described dynamics of a disease transmission that occurred through contaminated water and by personal contact. • No intervention strategies are included. The model is modified by Tuite, Tien, Eisenberg, Earn, Ma, and Fishman to include some interventions to identify optimal control interventions. • This model predicted the sequence and timing of regional cholera epidemics in Haiti and explored the potential effects of disease-control strategies. • Based on SIR model Miller, Schaefer, Gaff, Fister, and Lenhart modeled disease dynamics with added features. • Less infectious, hyper infectious, symptomatic, asymptomatic, disease waning immunity. • Considered several intervention strategies such as rehydration and antibiotic treatment, vaccination, sanitation. • Suggested optimal intervention strategies to regulate cholera transmission during an outbreak. SIR Diagram • Black arrows: Susceptible people become infected, recover, then become immune. • Blue arrows: Infectious people contaminate the water supply with bacteria and the bacteria decay. • Red arrow: Susceptible people are exposed to contaminated water and may become infected. • Gray arrows: People are born into the susceptible population; they may die as a result of cholera infection or other reasons. SVIR-B Model Modified SIWR model ◦ Vaccination is added ◦ Disease waning immunity term is added ◦ Disease related death term is added dS n ( S I R ) bWWS bI SI V (t ) S S R dt dI bWWS bI SI I I dI , dI disease related death dt dW I W dt dR I V (t ) S R R dt Ghosh-Dastidar and Lenhart, JBS, June 2015 Parameters Definition units Introduced Values here are given per day n µ bW natural birth rate natural death rate transmission rate for water-toperson day-1 day-1 ml cells-1 day-1 0.044/365 0.033/365 (2.14*10-5)/365 A Amplitude of seasonality of bW(t) Average value of bW(t) ml cells-1 day-1 0.88 Endemic Values here are given per day 0.044/365 0.033/365 (3.9*10-5)/365 (A+1)*B=(0.88+1)*(2.14*10 ^(-5))/365 0.5 ml cells-1 day-1 2.14 * 10-5 3.9*10-5 Individuals-1 day-1 year-1 (3.65 x 10-4)/365 (9.12*10-4)/365 transmission rate for personto-person immunity waning rate -1 recovery rate 0.4/365 (Low population density) 0.7/365 (High population density) 3 d disease related death rate 0.033 0.033 α rate of pathogen shedding into reservoir mean pathogen lifetime in water reservoir disease duration Individuals1day-1 Individuals1day-1 Cells ml-1 day-1 individuals-1 weeks 0.4/365 (Low population density) 0.7/365 (High population density) 3 3650/365=10 3650/365=10 7 14 B bI -1 T days 300-600 Reproductive Number Equilibria, and Stability Mathematical results show that the control reproduction number satisfies a threshold property with threshold value 1. When Rv < 1, disease free equilibrium E0 is globally asymptotically stable under sufficient conditions. When Rv > 1, disease free equilibrium E0 is globally asymptotically stable. Research Objectives New model: We combined the model proposed by Tien and Earn, (SIWR: Susceptible S – Infectious I -Waterborne Pathogen Concentration W – Recovered), model proposed by Neilan et. al (introducing the symptomatic and asymptomatic classes), the model developed by Cui et. al. (introducing a vaccination class), and the model proposed by Qiu et. al. Mainly we added: ◦ ◦ ◦ ◦ Vaccination, Hyperinfectious and less infectious class in waterborne pathogens Symptomatic and Asymptomatic Introduce treatment such as rehydration and antibiotics Our model: Parameters I A : asymptomatic state I S : symptomatic state p : proportion of inf ected who are asymptomatic 1 p : proportion of inf ected who are symptomatic bWH : transmission coefficient from waterborne hyper inf ectious pathogens to humans bWL : transmission coefficient from waterborne less inf ectious pathogens to humans bIH : transmission coefficient from humans hyper inf ectious pathogens to humans bLH : transmission coefficient from humans less inf ectious pathogens to humans A : disease re cov ery rate of asymptomatic (inf ected who do not have symptoms ) H : disease re cov ery rate of symptomatic who has not received the treatment Hu : disease re cov ery rate of symptomatic who received the treatment ( rehydration and antibiotic ) Assume A Hu H d A : disease related death of asymptomatic people : natural death rate u (t ) : proportion of the symptomatic population who received the treatment 1 u (t ) : proportion of the symptomatic population who did not receive the treatment d H : disease related death of hyper inf ectious population who did not receive the treatment d Hu : disease related death of hyper inf ectious population who received the treatment d A d Hu d H : proportion of the susceptible population who were vaccinated : rate of waning immunity of vaccination A : pathogens shedding rate by asymptomatic individuals S : pathogens shedding rate by symptomatic individuals : rate of hyper inf ectious pathogens moving to less inf ectious state : decay rate of less inf ectious pathogen rate : disease ralated waning immunity rate Our Modified model dI A p(bWH WH S bWLWL S bIH SI H bIL SI A ) A I I A d A I A dt dI H (1 p)(bWH WH S bWLWL S bIH SI H bIL SI A ) (1 u (t )) H I H u (t ) Hu I H I H (1 u (t ))d H I H u (t )d Hu I H dt dV S V V dt dWH A I A S I H WH dt dWL WH WL dt dR A I (1 u (t )) H I H u (t ) Hu I H R R dt Research Objectives 1. Are the solutions (S(t), I(t), W(t), R(t)) non-negeative for all t > 0 with the initial conditions? 2. Are all solutions (S(t), I(t), W(t), R(t)) bounded? (3) Does the system have a unique endemic equilibrium when Rv > 1? 3. What happens when Rv < 1? 4. Does any disease free equilibrium exist and if it does, then under what conditions is it locally asymptotically stable? 5. Under what conditions the endemic equilibrium is locally and globally stable? References (include the papers that I gave you if it is not included) Cholera Prevention and Control, CDC Cholera Outbreak – Haiti, October 2010, MMWR, CDC "Cholera Outbreak in Haiti." - International Medical Corps. N.p., n.d. Web. 04 June 2015. "Flooding Intensifies Cholera Outbreak in Haiti - New America Media."Flooding Intensifies Cholera Outbreak in Haiti - New America Media. N.p., n.d. Web. 04 June 2015. Fung, Isaac Chun-Hai. "Cholera Transmission Dynamic Models for Public Health Practitioners." Emerging Themes in Epidemiology. BioMed Central, n.d. Web. 04 June 2015. Suzanne Lenhart and John T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC, pp. 1-56. New York, 2007. Albert MJ, Neira M, Motarjemi Y., The Role of Food in the Epidemiology of Cholera, World Health Stat Q., 1997; vol. 50: 111-8. [PMID:9282393] Rachael L. Miller Neilan, Elsa Schaefer, Holly Gaff, K. Renee Fister, and Suzanne Lenhart, Modeling Optimal Intervention Strategies for Cholera, Bulletin of Mathematical Biology, vol. 72, 2004 – 2018, 2010. S. M. Moghadas, Modeling the Effect of Imperfect Vaccines on Disease Epidemiology, Discrete and Continuous Dynamical Systems – Series B, vol. 4, no. 4, pp. 999-1012, Nov. 2004. Joseph H. Tien and David J.D. Earn, Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model, Bulletin of Mathematical Biology, vol. 72, 1506-1533, 2010 Update: Outbreak of Cholera --- Haiti, 2010, MMWR, CDC Tien et al., Herald Waves of Cholera in Nineteenth Century London, Journal of the Royal Society, 2010. Tuite et al., Cholera Epidemic in Haiti, 2010: Using a Transmission Model to Explain Spatial Spread of Disease and Identify Optimal Control Interventions, Annals of Internal Medicine, vol. 154, pp. 593-601, May 2011.