The Study Of Epidemic And Endemic Diseases Using Mathematical Models
The Study of Epidemic and Endemic Diseasesusing Mathematical ModelsCandidate: Jummy Funke David!Supervisors: Fred Brauer Viviane Dias LimaCommittee members: Supervisors and Priscilla (Cindy) E.GreenwoodDissertation DefenseJanuary 16, 2020Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20201 / 56
Presentation Outline1Background Knowledge2Definition of some technical terms3PART A: EpidemicII4SIRP models with heterogeneous mixing and indirect transmissionCoupled PDE-ODE model with indirect transmissionPART B: EndemicIIHIV/Syphilis co-interaction model among gbMSMModified HIV/Syphilis co-interaction model among gbMSM in BCJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20202 / 56
General Background KnowledgeEpidemic:sudden occurrence of disease in a region above the level of normalexpectancymodeled with no demographic effectsby demographics, we mean birth, death and migrationEndemic:constant presence of disease within a particular regionmodeled with demographic effectsPandemic: world epidemicJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20203 / 56
Definition of technical termsBasic reproduction number (R0 ):The average number of secondary infections caused by an averageinfective.Effective reproduction number (Re ):Often used whenever we incorporate factors aimed at controlling thespread of disease into a model.The final size relation:Gives an estimate of the total number of infections and the epidemicsize for the period of the epidemic from the parameters in the model.Direct transmission of diseases:when diseases are transmitted from Host-Hoste.g. HIV, Syphilis.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20204 / 56
Definition of technical terms (CONTINUE.)Indirect transmission of diseases:when diseases are transmitted from Host-Source-Hoste.g. Chickenpox, Influenza, Measles, Smallpox, Tuberculosis.Heterogeneous mixing:mixing that exists between populations with different characteristics.gbMSM:Gay, bisexual and other men who have sex with menInterventions:1TasP: Treatment as Prevention (HIV)2PrEP: Pre-exposure prophylaxis3ART: Antiretroviral therapy (the use of HIV medicines to treatHIV infection)Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20205 / 56
PART A: Epidemic models (Project 1)SIRP models with heterogeneousmixing and indirect transmissionJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20206 / 56
Project 1: BackgroundPREVIOUS RESEARCH:Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20207 / 56
Project 1: BackgroundPREVIOUS RESEARCH:MY WORK:Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20207 / 56
Project 1: Research QuestionsWhat is the impact of varying the pathogen shedding rate andtaking the age of infection into consideration?What is the nature of the epidemic in an heterogeneous mixingenvironment?How worst is disease spread when we consider indirecttransmission pathway?What is the role of residence time on disease dynamics?Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20208 / 56
Project 1: Heterogeneous mixing and indirecttransmission for simple SIRP epidemic modelSIRP model: 0 S1 β1 S1 P ,I10 β1 S1 P αI1 ,R10 αI1 r1 I1 r2 I2 δPI20 β2 S2 P αI2 ,R20 αI2 P0 0 S2 β2 S2 P , (P Pathogen, r1 , r2 shedding rates)Basic reproduction number: R0 β1 R1 β2 R2r1 N1r2 N2R1 R2 αδαδR0 secondary infections caused indirectly through the pathogenshed by an infectious individual in I1 & I2 respectively.The final size relation:log nnSi0S1 ( ) oS2 ( ) o 2P0 βi R1 1 R2 1 , i 1, 2.Si N1N2δJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 20209 / 56
Project 1: A two-group age of infectionmodel with heterogeneous mixingA1 (τ ) & A2 (τ ) mean infectivity at age of infection τ .Γ(τ ) fraction of pathogen remaining τ time units after having beenshed by an infectious individuals.Basic reproduction number: R0 β1 R1 β2 R2Z R1 r1 N1Z A1 (τ )dτZ0 R2 r2 N2Γ(τ )dτ ,Z0 A2 (τ )dτ0Γ(τ )dτ .0R1 & R2 secondary infections caused indirectly through thepathogen shed by an infectious individual in I1 & I2 respectively.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202010 / 56
Project 1: A two-group age of infectionmodel with heterogeneous mixingA1 (τ ) & A2 (τ ) mean infectivity at age of infection τ .Γ(τ ) fraction of pathogen remaining τ time units after having beenshed by an infectious individuals.Basic reproduction number: R0 β1 R1 β2 R2Z R1 r1 N1Z A1 (τ )dτZ0 R2 r2 N2Γ(τ )dτ ,Z0 A2 (τ )dτ0Γ(τ )dτ .0R1 & R2 secondary infections caused indirectly through thepathogen shed by an infectious individual in I1 & I2 respectively.The final size relation: hhSi0S1 iS2 i βi R1 1 R2 1 Si N1N2N1 S1 & N2 S2 oftendescribedin terms of the attack S2 rates/ratios 1 SN1 and1 .N21logJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202010 / 56
Project 1: Variable pathogen shedding ratesBasic reproduction number: R0 β1 R1 β2 R2Z R1 N1Z Q1 (v)dvZ0 R2 N2Γ(c)dc,Z0 Q2 (v)dv0Γ(c)dc0Q1 (v) & Q2 (v) shedding rates.Γ(c) proportion of viruses remaining for virus already shed c timeunits earlier .The final size relation:log hhSi0S1 iS2 i βi R1 1 R2 1 .Si N1N2Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202011 / 56
Project 1: Heterogeneous mixing and indirecttransmission with residence time2-Patch SIRP model with residence time: 0S1 β1 p11 S1 (p11 P1 p21 P2 ) β2 p12 S1 (p12 P1 p22 P2 ), I 0 β p S (p P p P ) β p S (p P p P ) αI ,1 11 1 11 121 22 12 1 12 122 2110R1 αI1 , 0P1 r1 I1 δP1 , 0S β1 p21 S2 (p11 P1 p21 P2 ) β2 p22 S2 (p12 P1 p22 P2 ), 20 I2 β1 p21 S2 (p11 P1 p21 P2 ) β2 p22 S2 (p12 P1 p22 P2 ) αI2 ,R20 αI2 , 0P2 r2 I2 δP2 ,Assumptions:β2 β1 , with short term travel between the two patches.pij (i, j 1, 2) fraction of contact made by patch i residents inpatch j.Each patch has p11 p12 1, p21 p22 1.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202012 / 56
Project 1: Numerical simulationsAssumptions:Each patch has p11 p12 1, p21 p22 1.2-Patch SIRP model simulations:(a) Infected individuals (I1 ) in patch 1.(b) Infected individuals (I2 ) in patch 2.Figure: Dynamics of I1 and I2 when we vary p11 , p12 , p21 , p22 .Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202013 / 56
PART A: Epidemic models (Project 2)Coupled PDE-ODE model with indirecttransmissionJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202014 / 56
Project 2: Background & Research QuestionsAirborne disease:any disease caused by pathogens and transmitted through the air . E.g.Chickenpox, Influenza, Measles, Smallpox, Tuberculosis.Questions:Can we estimate the impact of diffusion using an ODE model?How worst is the epidemic with increase or decrease in thediffusion rate?What is the effect of the patch location on the spread of infection?Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202015 / 56
Project 2: Background & Research QuestionsSubmitted and under review:Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202016 / 56
Project 2: Model FormulationX , T ) satisfies the partialThe spatio-temporal density of pathogens P(Xdifferential equation (PDE) given by nX P 0, P DB P δ P, T 0, X Ω \ mj 1 Ωj ; TX Ω; DB nX P rj Ij , X Ωj , j 1, . . . , m,Dimensional Parameters:1DB 0 denotes the diffusion rate of pathogens in the bulk region,2δ the dimensional decay rate of pathogens,34rj 0 the dimensional shedding rate of pathogen by an infectedindividual in the j th patch, nX the outward normal derivative on the boundary of the domainΩ.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202017 / 56
Project 2: Model FormulationThe dynamics of the diffusing pathogens is coupled to the populationdynamics of the j th patch using:Integro-differential system of equations:ZdSj µj Sj(P/pc ) dSX ;dT ΩjZdIj µj Sj(P/pc ) dSX αj Ij ;dT ΩjdRj αj Ij ,j 1, . . . , m,dTJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202018 / 56
Project 2: Dimensionless coupled modelx, t) satisfiesThe dimensionless density of the pathogens P (x nx P 0, P D P P, t 0, x Ω \ mj 1 Ωεj ; tx Ω; 2πεD nx P σj Ij , x Ωεj , j 1, . . . , m,which is coupled to the dimensionless SIR dynamics of the j th patchZdSjβj Sj P dsx ;dt2πε ΩεjZdIjβj SjP dsx φj Ij ; dt2πε ΩεjdRj φj Ij ,j 1, . . . , m,dtwhere βj , σj and φj are the dimensionless transmission, shedding andrecovery rates for the j th patch, respectively.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202019 / 56
Project 2: Reduced ODE modelIn the limit D D0 /ν, ν 1/ log(ε), ε 1, where D0 O(1), thecoupled PDE-ODE model is reduced to an ODE system.Leading-order ODE model:m1 Xdp p σ j Ij ,dt Ω j 1 dSjσj Ij βj Sj p(t) ,j 1.,m,dt2πD0 σj IjdIj βj Sj p(t) φj Ij ,j 1, . . . , m ,dt2πD0dRj φj Ij ,j 1, . . . , m.dtJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202020 / 56
Project 2: one-Patch modelBasic reproduction number: R0 R? RDR? βN (0)σβN (0)σ, and RD φ Ω 2φπD0R? & RD secondary infections contributed by indirect transmissionand diffusion respectively.The final size relation: S0S S S log R? 1 RD 1 R0 1 .S NNNJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202021 / 56
Project 2: Numerical Simulations (one-patch)11D 0.1D 2D 10D 300D 0 300 (ODE)D 0 0.1D0 20.80.8D 0 10D 0 3000.60.60.40.40.20.2000.511.522.53(a) Reduced ODE000.511.52.53(b) Coupled PDE-ODE11D 0.1D 2D 10D 300D 0 300 (ODE)D 0 0.1D0 20.80.8D 0 10D 0 3000.60.60.40.40.20.2000.511.522.5(c) Reduced ODEFigure:23000.511.522.53(d) Coupled PDE-ODE(a) & (b) ODE & coupled PDE-ODE with (S(0), I(0), R(0)) (249/250, 1/250, 0) andp(0) P (0) 0. (c) & (d) ODE & coupled PDE-ODE with (S(0), I(0), R(0)) (249/250, 1/250, 0) andp(0) P (0) 1.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202022 / 56
Project 2: Numerical Simulations (one-patch)Surface plots:(a)(b)Figure: Surface plots of the basic reproduction number R0 with respect to thediffusion rate of pathogens D0 and transmission rate, and shedding rate. (a)D0 and the transmission rate β, while (b) D0 and the shedding rate σ.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202023 / 56
Project 2: Two-Patch modelBasic reproduction number: R 0 β1 R1 β2 R2In the well-mixed limit D0 1, the basic reproduction number R0for the two-patch model reduces toR 0 β1 N1 (0)σ1 β2 N2 (0)σ2 .φ1 Ω φ2 Ω N2 (0)σ2N1 (0)σ1, and R2 φ1 Ω φ2 Ω R1 & R2 secondary infections contributed by patch 1 and 2respectively.R1 Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202024 / 56
Project 2: Two-Patch modelBasic reproduction number: R 0 β1 R1 β2 R2In the well-mixed limit D0 1, the basic reproduction number R0for the two-patch model reduces toR 0 β1 N1 (0)σ1 β2 N2 (0)σ2 .φ1 Ω φ2 Ω N2 (0)σ2N1 (0)σ1, and R2 φ1 Ω φ2 Ω R1 & R2 secondary infections contributed by patch 1 and 2respectively.R1 The final size relation:loglogS10S1 S20S2 n nS2 oσ1 N1 (0) nS1 oS1 o β 1 R1 1 R2 1 ,1 N1 (0)N2 (0)2πφ1 D0N1 (0) nnS1 oS2 oσ2 N2 (0) nS2 o β 2 R1 1 R2 1 1 .N1 (0)N2 (0)2πφ2 D0N2 (0)Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202024 / 56
Project 2: Two-patch Numerical Simulation11D 0 0.10.9D 0 100.8D 0 3000.7D 0.1D 0.5D 10D 3000.9D 0 511.5(a) Patch 1 ODE00.511.5(b) Patch 1 PDE-ODE11D 0.1D 0.5D 10D 300D 0.100.9D 0.50.90.8D 0 100.80D .51(c) Patch 2 ODEFigure:1.500.511.5(d) Patch 2 PDE-ODEPatch 1 ((a) & (b)) and Patch 2 ((c) & (d)) with (S1 (0), I1 (0), R1 (0)) (299/300, 1/300, 0),(S2 (0), I2 (0), R2 (0)) (250/250, 0, 0), and p(0) 1 and P (0) 1 for the diffusing pathogens.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202025 / 56
PART A: Summary (Projects 1 & 2)1SIRP model was proposed2The model explored the impact of age of infections, varying thepathogen shedding rates and human mobility34The SIRP model in Project 1 was improved by proposing a coupledPDE-ODE models which includes diffusion of pathogensThe coupled PDE-ODE model was reduced to an ODE system,which was used to compute R0 and the final size relationJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202026 / 56
PART A: Summary (continue.)1PDE-ODE model numerically agreed with the reduced ODE2Both model predicted a decrease in the epidemic as the diffusionrate of pathogens increases3Diffusion is important when modelling airborne diseases, and somediseases may be difficult to control if overlooked4Individuals infected through indirect transmission medium in anheterogeneous mixing populations, which had been omitted in someother previous works is worth taking into account.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202027 / 56
PART A: Future work (Projects 1 & 2)Can we obtain more credible estimates of the finalepidemic sizes if weincorporate human mobility between patches?incorporate multiple class of hosts and sources in order to comparedisease spread among different populations and viruses.Explore model behaviour when vaccination and treatmentare involved:reduce contact rates?lower R0 ?decrease the final epidemic size?Can we use this novel approach to model the dynamics ofmosquito-borne diseases (malaria, dengue, .) where mosquitoes diffusein the air?Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202028 / 56
PART B: Endemic models (Project 3)HIV/Syphilis co-interaction modelgbMSMJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202029 / 56
PART B: Endemic models (Project 3)HIV/Syphilis co-interaction modelgbMSMBackground:1234The population of gay, bisexual and other men who have sex withmen (gbMSM) remain the most affected by HIV infection in BCMajority of infectious syphilis cases (over 80% of all cases) in BCwere among gbMSMCurrently, TasP, Condom use, and PrEP have been highly effectivefor HIV prevention and control in gbMSMSimilarly, Condom use, Test & Treat diagnosed cases of syphilishave also been effectiveJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202029 / 56
Project 3: BackgroundPREVIOUS RESEARCH:Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202030 / 56
Project 3: BackgroundPREVIOUS RESEARCH:MY WORK:Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202030 / 56
Project 3: Research Questions & PopulationResearch question:How does syphilis epidemic affect HIV prevalence and vice versa?What is the impact of a change in transmission rate on diseasedynamics?Can we test and treat mono-infected individuals more to reducedisease prevalences?Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202031 / 56
Project 3: Research Questions & PopulationResearch question:How does syphilis epidemic affect HIV prevalence and vice versa?What is the impact of a change in transmission rate on diseasedynamics?Can we test and treat mono-infected individuals more to reducedisease prevalences?Study population:Gay, bisexual and other men who have sex with men (gbMSM)Co-interaction of HIV and SyphilisTesting and treamentJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202031 / 56
Project 3: HIV/Syphilis Flow DiagramForce of infection associated withHIV infection:λH βH (UH κ1 AH κN2 USH κ3 ASH )Force of infection associated withsyphilis infection:λS βS (IS φ1 USH φN2 ASH φ3 TSH )Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202032 / 56
Project 3: HIV & Syphilis sub-models dS Π (µ λH )S, dt dUH λH S (µ dU H α1 )UH ,dtdAH α1 UH ν1 TH (µ dAH ρ2 )AH , dt dTH ρ2 AH (µ ν1 )TH ,dt(UH κ1 AH ),NHwith total population given as NH (t) S(t) UH (t) AH (t) TH (t). dS Π σ1 IS (µ λS )S,whereλH βHdt dIS λS S (µ σ1 )IS ,dtISλS βS,NSJummy Funke Davidwith total population given as NS (t) S(t) IS (t)Epidemic & Endemic modelsJanuary 16, 202033 / 56
Project 3: Effective reproduction NumberHIV sub-modelReH BU BA ,βH,BU (µ dU H α1 )βH α1 κ1 (µ ν1 )BA .(µ dU H α1 ) ((µ ν1 )(µ dAH ) µρ2 )Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202034 / 56
Project 3: Effective reproduction NumberHIV sub-modelReH BU BA ,βH,BU (µ dU H α1 )βH α1 κ1 (µ ν1 )BA .(µ dU H α1 ) ((µ ν1 )(µ dAH ) µρ2 )Syphilis sub-modelβS.(µ σ1 )Mathematically, the product of the transmission of syphilisinfection and the rate that an infective progresses out of syphilisclass.ReS Biologically, the number of syphilis infection produced by onesyphilis infective during the period of infectiousness whenintroduced in a totally syphilis susceptible population.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202034 / 56
Project 3: Sensitivity analysesSensitivity analysis of ReH 0.00.51.01.52.0ν1Figure: Impact of increasing testing rate α1 , treatment rate ρ2 and rate ofincrease in treatment failure ν1 on HIV reproduction number ReH .Sensitivity analysis of ReS :ReS1510500.02.55.07.510.0σ1Figure: Impact of increasing testing and treatment rate σ1 on syphilisreproduction number ReS .Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202035 / 56
Project 3: HIV-syphilis co-interaction model1Disease free equilibrium point (DFE):E0 (S0 , I0S , U0H , A0H , T0H , U0SH , A0SH , T0SH ) Πµ , 0, 0, 0, 0, 0, 0 .2Effective reproduction Number: Re max {ReS , ReH }3Endemic equilibrium point:IE1 (S1, IS1 , 0, 0, 0, 0, 0, 0), similar to HIV free equilibrium (ES ),I E2 (S2, 0, UH2 , AH2 , TH2 , 0, 0, 0), syphilis free equilibrium (EH),IE3 (S3, IS3 , UH3 , AH3 , TH3 , USH3 , ASH3 , TSH3 ), HIV-syphilisco-interaction equilibrium.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202036 / 56
Project 3: Numerical simulationsCompartmentsHIV prevalenceSyphilis prevalence2015105004080120Infected PopulationInfected Population25Compartments4000HIV prevalenceSyphilis prevalence30002000100000Time (years)4080120Time (years)(a) Disease free equilibrium.(b) Endemic equilibrium.Figure: Number of HIV infected individuals (green) and syphilis infectedindividuals (red), with different transmission rates and reproduction number:βH 0.02, βS 0.1, Re 0.139 (left); βH 0.4, βS 5.0, Re 2.780 (right)Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202037 / 56
4000300020001000004080Time (years)120(a) HIV positive individualsSyphilis infected PopulationHIV infected PopulationProject 3: Prevalence of HIV & syphilisinfections4000300020001000004080Time (years)120(b) Syphilis positive individualsFigure: Prevalence of HIV and syphilis with βH 0.4 and βS 5.0(ReH 2.780 1, ReS 1.245 1, Re 2.780 1). (a) The prevalence ofHIV with syphilis at the initial stage of the epidemic (blue dashed line) andwithout syphilis (red solid line). (b) The prevalence of syphilis infection withHIV at the initial stage of the epidemic (blue dashed line) and without HIV(red solid line)Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202038 / 56
PART B: Endemic models (Project 4)Assessing the combined impact ofinterventions on HIV and syphilisepidemics among gbMSM in BC: aco-interaction modelJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202039 / 56
PART B: Endemic models (Project 4)Assessing the combined impact ofinterventions on HIV and syphilisepidemics among gbMSM in BC: aco-interaction modelObjectives:12To assess how the combination of TasP, Condom use, PrEP, andTest & Treat syphilis can be used to prevent/eliminate HIV andsyphilis epidemics among gbMSM in BC .To assess the impact of PrEP on the HIV epidemic in BC .Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202039 / 56
Project 4: MethodsHIV/syphilis flow diagram:ν1ΠSλHµUHµ dU HλSα1µ dAHη1 λ Sσ1σ2ρ2µTHσ3η2 λ Sη3 λSσ4ν2ISµJummy Funke DavidAHUSHASHµ dU SHµ dASHγλHEpidemic & Endemic modelsρ1TSHµJanuary 16, 202040 / 56
Project 4: MethodsHIV/syphilis flow diagram:ν1ΠSλHµUHµ dU HλSα1AHµ dAHη1 λ Sσ1σ2ρ2µTHσ3η2 λ Sη3 λSσ4ν2ISµUSHASHµ dU SHµ dASHγλHρ1TSHµModeling the force of HIV/syphilis infection:1λH (UH κ1 AH κ2 USH κ3 ASH )N(IS φ1 USH φ2 ASH φ3 TSH )λS βS (1 ξ)((1 ψ) ψRP )NβH (1 ξ)((1 ψ) (1 θ)ψRP )2Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202040 / 56
Project 4: MethodsTransmission parameters fitted and calibrated on:Public Health Agency of Canada (PHAC) estimates of HIVincidence and Prevalence for gbMSM in BC ,Annual HIV diagnoses from HIV Cascade of Care in BritishColumbia Centre for Excellence in HIV/AIDS (BC-CfE), andAnnual syphilis diagnoses from British Columbia Centre forDisease ControlAssessed the impact of optimizing:1TasP2Test & Treat syphilis,3Condom use4PrEPJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202041 / 56
Project 4: Methods & Intervention scenarios:TasP:Status Quo: based on model calibrationIntervention according to Low, Medium and High:123decreasing the time to HIV diagnosisdecreasing time to antiretroviral (ART) treatmentincreasing the time retained on ARTTest & Treat syphilis:Status Quo: based on model calibrationIntervention according to Low, Medium and High:1decreasing the time from syphilis infection to treatmentPrEP:Status Quo: 4000Intervention: linearly increases to maximum PrEP uptake in 2028according to Low: 5000; Medium: 7000; High: 10,000Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202042 / 56
Project 4: Methods & Intervention scenarios:Condom use(%):Status Quo: 65Intervention: linearly increases to maximum condom use in 2028according to Low: 70; Medium: 75; High: 80Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202043 / 56
Project 4: Methods & Intervention scenarios:Condom use(%):Status Quo: 65Intervention: linearly increases to maximum condom use in 2028according to Low: 70; Medium: 75; High: 80Impact of interventions at the end of 10 years measuredon:1HIV point prevalence, HIV and syphilis incident cases2All-cause mortality cases among PLWH3WHO threshold for disease elimination as a public health concern4univariate sensitivity coefficients for HIV and syphilis incidencechanges under three PrEP uptake, TasP and Test & Treatscenarios at the end of 20285percent change in the number of cumulative HIV and syphilisincident cases with respect to the Status Quo scenario from 2019to 2028.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202043 / 56
Project 4: Model outcomesFigure: HIV incidence rate under different intervention scenarios incomparison to the WHO threshold for disease elimination as a public healthconcern at the end of 2028Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202044 / 56
Project 4: Model outcomesFigure: Syphilis incidence rate under different intervention scenarios incomparison to the WHO threshold for disease elimination as a public healthconcern at the end of 2028Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202045 / 56
Paper 4: Model outcomesReduction in HIV point prevalence, the cumulative numberof HIV incident cases, and all-cause mortality cases amongPLWH (left), and the cumulative number of syphilis incidentcases (right), among gbMSM after 10 years of TasP, PrEP,condom use, and Test & Treat (syphilis) interventionsJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202046 / 56
Project 4: Sensitivity Analysis (HIVIncidence)Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202047 / 56
Project 4: Sensitivity Analysis (SyphilisIncidence)Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202048 / 56
Project 4: Sensitivity AnalysisSensitivity analysis for the parameters with the mostuncertainty based on the available literature. Left: Percentchange in the cumulative number of HIV incident cases incomparison to the Status Quo at the end of 2028; Right:Percent change in the cumulative number of syphilis incidentcases in comparison to the Status Quo at the end of 2028Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202049 / 56
PART B: Summary (Projects 3 & 4)Analytical & numerical results:disease-free equilibra are locally and globally asymptotically stablewhenever Re (i.e ReS & ReH ) 1 .endemic equilibra are locally and globally asymptotically stablewhenever Re (i.e ReS & ReH ) 1 .Stable HIV free endemic equilibra whenever ReS 1 & ReH 1 .Stable syphilis free endemic equilibra wheneverReS 1 & ReH 1 .Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202050 / 56
PART B: Summary (Projects 3 & 4)Analytical & numerical results:disease-free equilibra are locally and globally asymptotically stablewhenever Re (i.e ReS & ReH ) 1 .endemic equilibra are locally and globally asymptotically stablewhenever Re (i.e ReS & ReH ) 1 .Stable HIV free endemic equilibra whenever ReS 1 & ReH 1 .Stable syphilis free endemic equilibra wheneverReS 1 & ReH 1 .increasing σ1 decreases ReS below unity (possiblity of eradicatingsyphilis among mono-infected individuals).increasing ρ2 , α1 , ν1 decreases ReH , but not below unity.HIV infection increases syphilis prevalence and vice versa (one ofthe possible ways).Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202050 / 56
PART B: Summary (Projects 3 & 4)1234Optimizing TasP or combining TasP with any other interventionsto at least the medium scenario significantly reduced the HIVincidence, and elimination of HIV disease was possible.Optimizing Test & Treat syphilis, and increased proportion ofcondom use with or without TasP to the high scenario reduced thesyphilis incidence, and elimination of syphilis was possible.Optimizing TasP, Test & Treat syphilis, combined with condomuse resulted in HIV & syphilis incident rate as low as 0.11 & 0.86respectively and elimination of both diseases was possible.Only TasP significantly decreased mortality while PrEP increasedsyphilis incidence by about 5%.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202051 / 56
PART B: Conclusions (Projects 3 & 4)12Optimizing TasP, through promotion of timely HIV diagnosis,treatment initiation and higher retention, or combined withimproving time from syphilis infection to treatment and thedistribution of PrEP was the most successful strategy to control theHIV epidemic.Optimizing Test & Treat syphilis, and increased condom use wasthe most successful strategy to control the syphilis epidemic.3Frequent testing for syphilis and other STIs, particularly amonggbMSM using PrEP should be prioritized to control the syphilisepidemic.4Consistent use of condoms should continue to be encouraged andpromoted to simultaneously reduce HIV and syphilis transmission.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202052 / 56
PART B: Future work (Projects 3 & 4)1Expand the model to adjust for age and risk level:II2If we have range of data and parameters, can we constructconfidence intervals for our model outcomes?II3how do individuals in each age and risk group contribute to diseasespread?what proportion of infected individuals in each age and risk groupdo we need to treat more?Bootstrap methodMonte carlo filtering methodIf we have enough information about the prior, can we use this toinform the posterior?IBayesian approachJummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202053 / 56
Summary of main contributionsDifferent infectious disease models with possible eliminationstrategies:Indirect transmission models (epidemic models):12epidemic models with heterogeneous mixing & indirecttransmission,the epidemic model designed using a coupled PDE-ODE systemDirect transmission models (endemic models)1the co-interaction model of HIV and syphilis infections,2HIV/Syphilis co-interaction model modified to assess the impact ofdifferent interventions in BC .Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202054 / 56
Conclusions:With my convincing story of innovation trends:12I consider myself an astonishing innovator.My innovative strategies to eliminating epidemic and endemicdiseases through direct and indirect transmission pathways areIcomputationally cheaper compared to other existing PDE models,Iricher and better compared to other existing ODE models.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202055 / 56
AcknowledgementsSpecial thank you to my supervisory committee members:Fred BrauerViviane Dias LimaPriscilla (Cindy) GreenwoodA big thank you to the University Examiners:Daniel CoombsPaul GustafsonMany thanks to the Chair and the External Examiner.Jummy Funke DavidEpidemic & Endemic modelsJanuary 16, 202056 / 56
Presentation Outline 1 Background Knowledge 2 De nition of some technical terms 3 PART A: Epidemic I SIRP models with heterogeneous mixing and indirect transmission I Coupled PDE-ODE model with indirect transmission 4 PART B: Endemic I HIV/Syphilis co-interaction model among gbMSM I Modi ed HIV/Syphilis co-interaction model among gbMSM in BC Jummy Funke David Epidemic & Endemic models January .
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