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Study and Analysis of Cholera Transmission MALLESSA YEBOAH OLIVIA HYLTON MENTOR: DR. URMI GHOSH -DASTIDAR 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 over 7,000 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. Modeling ◦ Epidemic models attempts to understand then prevalence and distribution of a disease, together with the factors that determine incidence, spread, and persistence. ◦ They allow us to conceptualize transmission dynamics in a quantitative way, while also giving a platform where different hypotheses can be tested. ◦ The 1st differential equation models of infectious disease were the work of Daniel Bernoulli. (1766) ◦ During the 1900s the dynamical systems approaches were applied. SIR Model Stability Analysis What is stability analysis and why is it important? The Jacobian stability approach is used to prove the stability of the Equilibriums. Linearization a model's equations give the Jacobian matrix and tells us about the behavior near equilibrium. An equilibrium is determined to locally asymptotically stable (LAS) if all the eigenvalues have negative real parts. In order to show that an infection/disease is independent of the sample population size used for the model one must show that the equilibrium is globally-asymptotically stable (GAS). Disease Free Equilibrium is when no presence of the infection. The infected class will be zero (I=0) and the entire population will consist of only susceptible people. Endemic Equilibrium is a situation where there is presence of the infection and the disease invades. I≠0 Basic Reproductive Number The expected number of new infections created during its by an infected individual under the best conditions for transmission. Calculated by taking infectivity rate over the infectivity period. ◦ The below case holds try for true for the our sample SIR model: The value of the reproductive number will determine the presence of the disease. If < 1 there exist some conditions under which the disease cannot grow, therefore the disease dies out. If > 1 there exist some conditions under which the disease can grow, therefore the disease invades. Calculating the Basic Reproductive Number ◦ Next Generation Method ◦ Next Generation Matrix K= Note that F is associated with new infections and V is associated with infected population going out. ◦ represents the expected number of secondary cases in compartment i put in by an individual in compartment j ◦ the next generation operator: ◦ Gives rate at which individuals in compartment j generate new infections in compartment i multiplied by the average length of time that individuals spend in a single visit to compartment j ◦ is given by the dominant(largest) eigenvalue of Disease Free Equilibrium D.F.E Point: (1, 0, 0) Since infection is not present there is no recovery Stability Analysis of Disease Free Equilibrium Evaluation at D.F.E: (1,0,0) • The D.F.E is stable for all negative real parts of the eigenvalues. Here the system is stable when μ > 0 and β < γ+μ β / (γ+μ ) < 1 is equivalent to Endemic Equilibrium E.E Point: Stability Analysis of Endemic Equilibrium Evaluate at E.E point: SIVWR Model 𝑑𝑆 = n(S+IA+IS+V+R) + θV(t) -bWHWHS - bWLWLS- bISSIs - bIASIA 𝑑𝑡 𝑑𝐼𝐴 = p(bWHWHS - bWLWLS- bISSIs - bIASIA) - γAIA-μIA- dIA= 0 𝑑𝑡 𝑑𝐼𝑆 = (1-p)(bWHWHS - bWLWLS- bISSIs - bIASIA)- γSIS-μIS- dIS= 0 𝑑𝑡 𝑑𝑉 = ØS - θV(t)- μV= 0 𝑑𝑡 𝑑𝑊𝐻 = αAIA + αsIs - χWH= 0 𝑑𝑡 𝑑𝑊𝐿 = χWH – εWL= 0 𝑑𝑡 𝑑𝑅 = γAIA+ γSIS – μR- ωR = 0 𝑑𝑡 N = S+IA+IS+V+WH+WL+R ØS – μS +ωR = 0 [1A] [2A] [3A] [4A] [5A] [6A] [7A] Disease Free Equilibrium IA IS =0 dR dt = γAIA+ γSIS – μR- ωR = 0 R(μ+ ω) =0 R=0 WH WL= 0 n(S+0+0+V+0) + θV(t) -0 - 0- 0 - 0 - ØS – μS + 0 = 0 n(S+ V) + θV -S(𝜃 + Ø) = 0 (n- μ- Ø)S + (n+θ)V = 0 [plug in 4B] 𝑆Ø (n- μ- Ø)S + (n+θ) =0 (𝜃+𝜇) S(n- μ- Ø + (n+θ) Ø (𝜃+𝜇) )=0 [Assuming S ≠0] (n- μ- Ø)(n+θ)+ (n+θ)Ø = 0 dV = dt ØS - θV(t)- μV= 0 S= (𝜃−𝜇)𝑉 𝑆Ø V= (𝜃+𝜇) nθ - μθ- Øθ +nμ-μ2-μØ+nØ+θØ = 0 -μ2-μ (Ø+θ)+ n(Ø+μ+θ) = 0 μ (μ+Ø+θ) = n(μ+Ø+θ) Ø [4B] [1A] [μ+Ø+θ ≠0] disease free non-equilibrium exists if μ = n (natural death rate = natural birth rate) Disease Free Equilibrium Assume μ = n V= 𝑆Ø (𝜃+𝜇) N-S= [V+S=N] 𝑆Ø S(1+(𝜃+𝜇) )=N SD = S(𝜃+𝜇+Ø )=N (𝜃+𝜇) 𝑁(𝜃+𝜇) (𝜃+𝜇+Ø) 𝑁(𝜃+𝜇) V= (𝜃+𝜇+Ø)(𝜃+𝜇) Ø VD = Disease free equilibrium: 𝑆Ø (𝜃+𝜇) 𝑁Ø (𝜃+𝜇+Ø) [from 4B] (SD, VD, IAD, ISD, WH , WL , R)= ( 𝑁(𝜃+𝜇) 𝑁Ø , , (𝜃+𝜇+Ø) (𝜃+𝜇+Ø) 0, 0, 0, 0, 0) Endemic Equilibrium •substitutions were used to get solutions for each system of equations. 𝐺 𝐹 𝐺 𝐹 =( , GØIS∗ pDIS∗IS∗ 𝐺 𝐸 𝐸 , , IS* , IS* , IS* 𝐹(𝜃+𝜇) (1−𝑝) 𝐹 χ ε GØ pD 𝐺 , , 𝐹(𝜃+𝜇) (1−𝑝) 𝐹 𝐸 𝐸 , χ , , F ) IS* ε D= E= (S*, V*, IA*, IS*, WL* , WH*, R*) = ( IS* , Where: , F IS* ) γS+μ+dS γA+μ+dA αApD+αs 1−p F= G= (1−p) θ+μ+Ø n−μ S (θ+μ) IS (1−p) ( θ+μ+ n−μ S (θ+μ) − npD − (1−p)(n+nF+ωF)) New Generation Method • The matrices F and V are calculated using the next generation method. • We will need to find in order to calculate which will aid in finding the basic reproductive number. Future Research ◦ We will calculate . ◦ After finding we will find the largest eigenvalues. This will give the spectral radius which is also the basic reproductive number. ◦ We will work on an extension of this problem by taking u(t) not equal to zero. ◦ We believe u(t) is a combination of antibiotic treatment and hydration. Reference ◦ 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] ◦ Cholera Outbreak – Haiti, October 2010, MMWR, CDC ◦ "Cholera Outbreak in Haiti." - International Medical Corps. N.p., n.d. Web. 04 June 2015. ◦ Cholera Prevention and Control, CDC ◦ Cui, Jing'an, Zhanmin Wu, and Xueyong Zhou. "Mathematical Analysis of a Cholera Model with Vaccination." N.p., 13 Feb. 2014. Web. ◦ Dimitrov, Nedialko B. B., Meyers, and Laure Ancel. "Mathematical Approaches to Infectious Disease Prediction and Control." Mathematical Approaches to Infectious Disease Prediction and Control. 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. ◦ GHOSH-DASTIDAR, URMI, and SUZANNE LENHART. "MODELING THE EFFECT OF VACCINES ON CHOLERA TRANSMISSION." World Scientific Publishing Company. N.p., 29 May 2015. Web. 04 June 2015. ◦ 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 ◦ 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. ◦ 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. Acknowledgements We would like to thank Dr. Urmi Ghosh-Dastidar for her support and guidance throughout our project. Also we would like to thank Matt, Becky, and especially to the MAA for funding and DIMACS REU Program for providing us with research facilities at Rutgers University.