Transcript Conference paper

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
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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
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Ecological Epidemiology (cont)
• Interactions between individuals/species
 Prey – Predator relationships
 Competition
 Symbiosis
 Obligatory cooperation
 Food chain
Macro-Micro Epidemic Models
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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
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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
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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
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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
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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
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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
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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
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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
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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
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HIV/AIDS Macro Level Model (cont1)
• The Equations
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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!
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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
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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
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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.
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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
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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
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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
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HIV/AIDS Micro Level Model (cont3)
• The equations:
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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
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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]
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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]
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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
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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
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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
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Micro level model for HBV - modified
• Logistic generation of the target cells
(hepatocytes)
• where
is the maximum number of
hepatocytes the liver can support.
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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
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Best way of generating models in
epidemiological research
Work with:
Ecologists
Public Health officers
Physicians
Pharmacologists
Hematologists
Gastrosurgeons
& Others
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Thank you for Listening
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