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

Macro-Micro Epidemic Models 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

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

Ecological Epidemiology (cont) Interactions between individuals/species Prey – Predator relationships Competition Symbiosis Obligatory cooperation Food chain Macro-Micro Epidemic Models

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

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

Models of Interactions of Multi-species 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

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

Macro-Micro Epidemic Models Stochastic models Why? Can derive details Expectation Variance Probability distributions E.g. – in the birth-death process Deterministic model indicates exponential growth or decline Macro-Micro Epidemic Models

Stochastic models (cont) In the corresponding stochastic process we can show: 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

Macro-Micro Epidemic Models 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 Compartmentalised/structured populations such as Susceptibles –Latents – Infectives – Recovered – immunes There may be other stages Macro-Micro Epidemic Models

Epidemiology 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

HIV/AIDS Macro Level Model Simple model (early stages) 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 (1-ε)λI λS+ελI βcSI/N νI γA S I A μS μI μA Macro-Micro Epidemic Models

HIV/AIDS Macro Level Model (cont1) The Equations Macro-Micro Epidemic Models

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

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

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

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

HIV/AIDS Micro Level Model (cont3) Thus: The immune system offers a natural and the most reliable defense mechanism against HIV Interactions of the Virions, CD4+ and CD8+ T-cells of the immune system Hence the terms “viral load” and “CD4 cell count” HIV also infects the liver cells: the hepatocytes Macro-Micro Epidemic Models

HIV/AIDS Micro Level Model (cont2) Compartmental diagram X = uninfected CD4 cells Y = infected CD4 cells V = free HIV virons λ βXV X Y μX aY raY V αV Macro-Micro Epidemic Models

Macro-Micro Epidemic Models 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

HIV/AIDS Micro Level Model (cont3) The equations: Macro-Micro Epidemic Models

Combined macro-micro epidemiologic dynamics of HIV/AIDS Micro level intracellular level kinetics 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] Reverse Transcription Transcription & Translation Uncoating Integration Free mutant viral Populations Budding Assembly Macro-Micro Epidemic Models

Combined macro-micro epidemiologic dynamics of HIV/AIDS (cont1) Micro level: Cellular level Dynamics [Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI-207-210] HIV infected CD4 cells + HIV Healthy CD4 cells Productively infected Replicated virus population Chronically producing Immature virions Mature virions Defectively infected Latently infected Macro-Micro Epidemic Models

Combined macro-micro epidemiologic dynamics of HIV/AIDS (cont2) [Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI-207-210] Healthy Humans HIV infected Humans Humans with AIDS Exposure Routes *Blood transfusion *Needle Sharing (by IDU) *Percutaneous needle stick *Receptive insertive anal intercourse *Receptive penile-vaginal intercourse *Receptive insertive oral intercourse Interventions *Condom/Dam use *Compliance *Partner testing *HAART for pregnant *No needle sharing *Education Interventions *HAART (Compliance) *Education *Treating Opportunistic infections Macro-Micro Epidemic Models

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

Micro level model for HBV - Diagram Compartmental diagram ρ is a multiple of δ λ T Target cells kVT I Infected cells δI μT V Virions ρI cV Macro-Micro Epidemic Models

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

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

Macro-Micro Epidemic Models 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 PhD student, Dept of Maths L.S. Luboobi Supervisor, ,, C. Obua Supervisor, Dept of Pharmacology & Therapeutics JYT Mugisha Supervisor, CoNAS Macro-Micro Epidemic Models

Best way of generating models in epidemiological research Work with: Ecologists Public Health officers Physicians Pharmacologists Hematologists Gastrosurgeons & Others Macro-Micro Epidemic Models

Thank you for Listening Macro-Micro Epidemic Models