DIMACS 2007 Modeling Transmission Dynamics of Outline HIV/AIDS: Some Results & Challenges Introduction ARVs DISP ARV Model Abba B. Gumel, Two-strain Model Department of Mathematics, Male Circumcision University of Manitoba , Winnipeg, Manitoba, HIV/TB Dynamics Canada. HIV-Malaria Challenges promo Acknowledgements DIMACS-SACEMA-AIMS Meeting; Stellenbosch, South Africa, June 25, 2007.
DIMACS 2007 Outline Outline � Introduction Introduction ARVs � Modelling Control Strategies DISP ARV Model Two-strain Model ◮ ARVs Male Circumcision ◮ Male Circumcision HIV/TB Dynamics HIV-Malaria ◮ Imperfect Prophylactic Vaccine Challenges promo � Modelling HIV Co-infection Acknowledgements ◮ HIV-TB ◮ HIV-Malaria � Conclusions and Current Challenges
DIMACS 2007 HIV: Facts and Figures � Human Immunodeficiency Virus (HIV): causative Outline Introduction agent of Acquired Immune Deficiency Syndrome Typical course (AIDS). First appeared in 1980s; Control strategies ARVs � Modes of Transmission: sexual, needle-sharing, DISP ARV Model blood transfusion, vertical etc; Two-strain Model � Global Statistics: Male Circumcision HIV/TB Dynamics ◮ Accounts for ≈ 20 million deaths; HIV-Malaria ◮ 34-46 million people live with HIV; 30% unaware Challenges of infection status. promo � Inflicts severe public health & socio-economic Acknowledgements burden. ◮ economic burden due to HIV-related death or disability in 50 countries (US, Russia, 5 in Asia, 8 in Latin America, and 35 in sub-Saharan Africa) during 1992 − 2000 estimated at $25 billion (Fleck, 2004).
DIMACS 2007 Typical Course of HIV Disease Outline Introduction Typical course Control strategies ARVs DISP ARV Model Two-strain Model Male Circumcision HIV/TB Dynamics HIV-Malaria Challenges promo Acknowledgements
DIMACS 2007 Control Mechanisms Outline (i) Therapeutic: Anti-retroviral Drugs (ARVs) Introduction Typical course ◮ Drawback: resistance development (spread of Control strategies ARVs resistant HIV); ◮ Not widely accessible in some resource-poor DISP ARV Model nations with high HIV prevalence; Two-strain Model Male Circumcision (ii) Preventive: HIV/TB Dynamics ◮ Abstinence; HIV-Malaria ◮ “Be faithful”; Challenges ◮ Correct and consistent use of condoms ; promo ◮ Education and counseling about safer sex Acknowledgements practices; ◮ Voluntary testing, screening of blood products and use of sterilized needles; ◮ Use of a vaccine; ◮ Male circumcision.
DIMACS 2007 Modeling the Impact of ARVs Outline Introduction � Anti-retroviral drugs (ARVs), particularly HAART, ARVs have had dramatic impact in curtailing HIV ARV strategies burden; DISP ARV Model Two-strain Model Male Circumcision ◮ use of ARVs, over long periods of time, reduces HIV/TB Dynamics the viral loads in HIV-infected individuals to HIV-Malaria non-detectable levels Challenges promo ◮ reduce infectiousness; extends life and quality of Acknowledgements life � ARVs not widely accessible globally
DIMACS 2007 Implementation Strategies of ARVs Outline (i) Universal: Introduction ◮ ARVs administered to all infected individuals ARVs ◮ popularly used (success story in Brazil) ARV strategies DISP ARV Model ◮ could lead to emergence and transmission of Two-strain Model ARV-resistant HIV Male Circumcision (ii) Targeted (viral-load or CD4-dependent): HIV/TB Dynamics ◮ treat only those with low CD4 count ( < 200 HIV-Malaria cells/ml) (individuals with such low CD4 count Challenges are at pre-AIDS or AIDS stage; high viral loads); promo ◮ strategy justified by the results of randomized Acknowledgements controlled trials (provide strong evidence of improved survival and reduced progression) ◮ minimize probability of resistance development and ARV-related side effects and toxicity ◮ part of new control guidelines in USA, Canada, Botswana etc.
DIMACS 2007 Flow Diagram for DISP ARV Model Outline Introduction ARVs DISP ARV Model Flow Diagram Model Dynamical features Bifurcation diagram Conclusions Two-strain Model Male Circumcision HIV/TB Dynamics HIV-Malaria Challenges promo Acknowledgements
DIMACS 2007 Viral-load Dependent Treatment Model With Multiple Infection Stages ( Sharomi and Gumel, Bull. Outline Math. Biol., 2007) Introduction ARVs DISP ARV Model Flow Diagram Model ˙ S = Π − λ S − µ S , Dynamical features Bifurcation diagram ˙ L 1 = ( 1 − σ ) λ S − ( µ + α 1 + τ 1 ) L 1 , Conclusions ˙ Two-strain Model L 2 = α 1 L 1 − ( µ + α 2 + τ 1 ) L 2 , Male Circumcision ˙ H 1 = σλ S − ( µ + η 1 α 1 + τ 2 ) H 1 , HIV/TB Dynamics ˙ H 2 = η 1 α 1 H 1 − ( µ + η 2 α 2 + τ 2 ) H 2 , HIV-Malaria ˙ A = α 2 L 2 + η 2 α 2 H 2 + α 3 T − ( µ + δ + τ 3 ) A , Challenges ˙ T = τ 1 ( L 1 + L 2 ) + τ 2 ( H 1 + H 2 ) + τ 3 A − ( µ + α 3 ) T , promo Acknowledgements λ = β ( L 1 + L 2 + θ 1 H 1 + θ 2 H 2 + θ 3 A + θ 4 T ) . N
DIMACS 2007 Dynamical Features Outline Introduction ARVs Theorem DISP ARV Model The disease-free equilibrium of the DISPT model is Flow Diagram Model globally-asymptotically stable if R T < 1 . Dynamical features Bifurcation diagram Conclusions Proof based on using a Lyapunov function Two-strain Model ( p i > 0 ): Male Circumcision HIV/TB Dynamics F = p 1 L 1 + p 2 L 2 + p 3 H 1 + p 4 H 2 + p 5 A + p 6 T , HIV-Malaria Challenges Theorem promo Model has a unique locally-stable endemic equilibrium Acknowledgements whenever R T > 1
DIMACS 2007 Forward Bifurcation Diagram Outline Introduction ARVs DISP ARV Model Flow Diagram Model Dynamical features Bifurcation diagram Conclusions Two-strain Model Male Circumcision HIV/TB Dynamics HIV-Malaria Challenges promo Acknowledgements
DIMACS 2007 Simulation Results (i) Universal strategy gives highest reduction in Outline number of cases; Introduction ARVs (ii) Low viral load strategy accounts for the highest DISP ARV Model mortality; Flow Diagram Model Dynamical features (iii) For low treatment rates (low ARV supplies), high Bifurcation diagram Conclusions viral load and the AIDS-only strategies avert Two-strain Model more deaths than any of the remaining Male Circumcision strategies; HIV/TB Dynamics (iv) For high treatment rates, the universal strategy HIV-Malaria averts more deaths than any of the other Challenges promo strategies. Acknowledgements (v) In terms of reduction of new cases, the strategies are listed in descending order of significance as follows: universal, high viral load, AIDS-only and low viral load strategies;
DIMACS 2007 ARV Model with Two Strains Outline Introduction Drawbacks of ARVs: emergence and spread of ARVs ARV-resistant strains. DISP ARV Model Two-strain Model Reasons: Flow Diagram Equations � Incomplete compliance to the specified ARV Summary: Table Male Circumcision regimen; HIV/TB Dynamics HIV-Malaria � Primary infection of susceptible individuals with Challenges the resistant strain; promo Acknowledgements � Biological factors. Motivation: what is the impact of the emergence and transmission of HIV resistant strain on HIV control?
DIMACS 2007 Mathematical Model: Flow Diagram Outline Introduction ARVs DISP ARV Model Two-strain Model Flow Diagram Equations Summary: Table Male Circumcision HIV/TB Dynamics HIV-Malaria Challenges promo Acknowledgements
DIMACS 2007 The Model Outline dS dt = Π − β ( I w + η w A w + η T I T ) S − β ( I r + η r A r ) S − µ S , Introduction N N ARVs DISP ARV Model dI w dt = β ( I w + η w A w + η T I T ) S Two-strain Model − ( µ + σ w + τ w ) I w , Flow Diagram N Equations Summary: Table Male Circumcision dI r dt = β ( I r + η r A r ) S HIV/TB Dynamics − ( µ + σ r ) I r + γ wr I T , N HIV-Malaria (1) Challenges promo dA w = σ w I w − ( τ w + µ + δ w ) A w + θσ w I T , Acknowledgements dt dA r = σ r I r − ( µ + δ r ) A r , dt dI T dt = τ w I w + τ w A w − ( µ + γ wr + θσ w ) I T .
DIMACS 2007 Summary of Dynamical Features of Multi-Strain Model Outline Introduction Treatment-free model Treatment model ARVs R t w < R t r < 1 both strains die out both strains die out DISP ARV Model Two-strain Model R t w < 1 < R t resistant strain dominates resistant strain dominates r Flow Diagram Equations Summary: Table R t r < 1 < R t wild strain dominates low endemicity w Male Circumcision co-existence equilibrium HIV/TB Dynamics R t w = R t both strains die out both strains die out r = 1 HIV-Malaria Challenges R t w = R t continuum of endemic resistant strain dominates r > 1 promo equilibria Acknowledgements R t w > R t wild strain dominates high endemicity r > 1 co-existence equilibrium R t r > R t resistant strain dominates resistant strain dominates w > 1
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