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Epidemiological Modelling at macro and micro levels: Examples of HIV and Hepatitis diseases by Livingstone S. Luboobi Department of Mathematics, Makerere University given at Strathmore University International Mathematics Research Meeting Nairobi, Kenya 23 - 27 July 2012 Outline – EcologicalEpidemiology (macro & micro levels) – Stage/Age structured models – Stochastic models – HIV/AIDS macro/micro level models – HBV micro level model – Therapy – Best way of generating models in epidemiological research Macro-Micro Epidemic Models 1 EcologicalEpidemiology • Ecological models are very important in Epidemiology: – No disease/epidemic can progress without a population/individual! – Population dynamics in single/multi-Species communities facilitate the epidemiological studies thro: • Processes Births/reproduction Deaths Immigration Emigration Macro-Micro Epidemic Models 2 Ecological Epidemiology (cont) • Interactions between individuals/species Prey – Predator relationships Competition Symbiosis Obligatory cooperation Food chain Macro-Micro Epidemic Models 3 Modelling at Macro level • Requires a community/ecosystem – Individuals – Species • There are interactions between individuals/species • Hence ecological considerations are important Macro-Micro Epidemic Models 4 Modelling at Micro level • Concerned with what happens to/within an individual – Interplay of different systems of cells within body • • • • • Immune system Nervous system the brain Heart Liver etc – Thus an “ecosystem” within an individual/organ Macro-Micro Epidemic Models 5 Models of Interactions of Multispecies communities • Inter-interactions as well as intra-interactions – the rate of growth of i-th species sub-population an nspecies community through an eqn. such as: The form of depends on the type of interaction Macro-Micro Epidemic Models 6 Stage/Age structured Models • Human populations Immature age-group Mature age-group (i.e. the adults capable of reproduction) Even a third age-group that have stopped giving births Sex-age structured model could be closer to reality • Application to HIV/AIDS epidemic 0 -5, 5 – 12/15 years, Adults sub-populations • Method of analysis: delay DEs Macro-Micro Epidemic Models 7 Stochastic models • Why? Can derive details o Expectation o Variance o Probability distributions E.g. – in the birth-death process o Deterministic model indicates exponential growth or decline Macro-Micro Epidemic Models 8 Stochastic models (cont) o In the corresponding stochastic process can show: we There is a possibility of extinction of the population Extinction is certain when the birth rate is equal or less than the death rate Macro-Micro Epidemic Models 9 Epidemiology • At macro level – Infectious diseases cannot spread or be transmitted without a population(s) – Mode of transmission is key in study of the epidemiology of a disease Examples o Compartmentalised/structured populations such as Susceptibles –Latents – Infectives – Recovered – immunes o There may be other stages Macro-Micro Epidemic Models 10 Epidemiology at micro level • At micro level – Even for non-infectious diseases the infection is thro’ interaction of cells (drug molecules, infected cells & the immune system) - Hence the disease-specific immunological models for: malaria, HIV, Hepatitis strains, etc. Macro-Micro Epidemic Models 11 HIV/AIDS Macro Level Model • Simple model (early stages) (1-ε)λI λS+ελI βcSI/N S μS νI I μI γA A μA – S(t) = number of susceptibles (i.e. the ‘non-infected’) at time t – I(t) = number of infectives (i.e. the infected & are infectious) at time t – A(t) = nuumber of the AIDS cases (bedridden or too week to interact) at time t Macro-Micro Epidemic Models 12 HIV/AIDS Macro Level Model (cont1) • The Equations Macro-Micro Epidemic Models 13 HIV/AIDS Macro Level Model (cont2) • Quick analysis of early stages of HIV in a community: – Hence Thus if < 1 HIV/AIDS epidemic would not develop in that community! Macro-Micro Epidemic Models 14 HIV/AIDS Micro Level Model • The development of AIDS is associated with the depletion of the CD4+ helper T lymphocyte. • HIV relies on a host to assist reproduction. • Since the CD4+ cells are depleted over time, strengthening cytotoxic responses can not occur. • Initially the transformation of immune-sensitivity to resistant genotypes occurs by the generation of mutations primarily due to reverse transcriptase. Macro-Micro Epidemic Models 15 HIV/AIDS Micro Level Model (cont1) • The extreme heterogeneity and diversity of HIV makes the design of effective vaccines extremely difficult. • The understanding of the dynamics of antigenic escape from immunological response has been that a mutation may enable the virus to have a selection advantage. • Because there is an asymmetric interaction between immunological specificity and viral diversity, the antigen diversity makes it difficult for the immune system to control the different mutants simultaneously and the virus runs ahead of the immune response. Macro-Micro Epidemic Models 16 HIV/AIDS Micro Level Model (cont2) • While most productively infected cells have a relatively short life span, many cells are latently infected and are very long lived. • A simple model for the interaction between the human immune system and HIV was developed by Perelson (2002). • A stochastic model for the HIV pathogenesis under anti-viral drugs has been developed. Macro-Micro Epidemic Models 17 HIV/AIDS Micro Level Model (cont3) • Thus: The immune system offers a natural and the most reliable defense mechanism against HIV o Interactions of the Virions, CD4+ and CD8+ T-cells of the immune system o Hence the terms “viral load” and “CD4 cell count” HIV also infects the liver cells: the hepatocytes Macro-Micro Epidemic Models 18 HIV/AIDS Micro Level Model (cont2) • Compartmental diagram λ βXV X Y μX X = uninfected CD4 cells Y = infected CD4 cells V = free HIV virons aY raY V αV Macro-Micro Epidemic Models 19 The parameters are described as follows: β = CD4+ T-cell infection rate by HIV. a = the death rate of infected CD4+ T-cells. α = the rate of removal of free virus from the system. r = number of free virus particles from an infected cell as result of bursting. λ = constant rate of production of uninfected CD4+ T-cells. μ = death rate of uninfected CD4+ T-cell Macro-Micro Epidemic Models 20 HIV/AIDS Micro Level Model (cont3) • The equations: Macro-Micro Epidemic Models 21 Combined macro-micro epidemiologic dynamics of HIV/AIDS Micro level intracellular level kinetics Uncoating Reverse Transcription Free mutant viral Populations Integration Budding Transcription & Translation Assembly Intervention Strategies: *Inhibition of binding. Blocking of the gp41 conformational changes that permit viral fusion *Nucleoside/Nucleotide Reverse Transcriptase Inhibitors (NRTIs) & Non-Nucleotide Reverse Transcriptase Inhibitors *Integrase inhibitors *Antisense antivirals or transcription Inhibitors (TIs) *Protease inhibitors (PI) [Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI-207-210] Macro-Micro Epidemic Models 22 Combined macro-micro epidemiologic dynamics of HIV/AIDS (cont1) Micro level: Cellular level Dynamics Healthy CD4 cells HIV infected CD4 cells + HIV Productively infected Chronically producing Defectively infected Replicated virus population Immature virions Mature virions Latently infected [Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI-207-210] Macro-Micro Epidemic Models 23 Combined macro-micro epidemiologic dynamics of HIV/AIDS (cont2) Healthy Humans HIV infected Humans Exposure Routes Humans with AIDS Interventions *Blood transfusion *Needle Sharing (by IDU) *Percutaneous needle stick *Receptive insertive anal intercourse *Receptive penile-vaginal intercourse *Receptive insertive oral intercourse Interventions *HAART (Compliance) *Education *Treating Opportunistic infections *Condom/Dam use *Compliance *Partner testing *HAART for pregnant *No needle sharing *Education [Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI-207-210] Macro-Micro Epidemic Models 24 Epidemiological Models for HBV Background HBV (Hepatitis B Virus) Sexually transmitted Attacks the liver cells However more destructive to the hepatocytes However there is vaccine (unlike HIV) HAART (ARVs) can administered for both If HBV is treated early there is possibility of recovery Macro-Micro Epidemic Models 25 Micro level model for HBV - Diagram • Compartmental diagram λ T Target cells μT kVT I Infected cells δI V Virions ρI cV – ρ is a multiple of δ Macro-Micro Epidemic Models 26 Micro level model for HBV - Equations • The equations derived from the HBV dynamics as in the diagram: – This is a simple model for the HBV dynamics – it can be improved as in the next slide Macro-Micro Epidemic Models 27 Micro level model for HBV - modified • Logistic generation of the target cells (hepatocytes) • where is the maximum number of hepatocytes the liver can support. Macro-Micro Epidemic Models 28 Therapy • The ARVs are used as drugs to control the effects of HIV and HBV • But they are toxic to the hepatocytes – hepatoxicity • Hence an optimal therapeutic programme is the concern of the Research team: – HasifaNampala – L.S. Luboobi – C. Obua – JYT Mugisha PhD student, Dept of Maths Supervisor, ,, Supervisor, Dept of Pharmacology & Therapeutics Supervisor, CoNAS Macro-Micro Epidemic Models 29 Best way of generating models in epidemiological research Work with: Ecologists Public Health officers Physicians Pharmacologists Hematologists Gastrosurgeons & Others Macro-Micro Epidemic Models 30 Thank you for Listening Macro-Micro Epidemic Models 31