0. Epidemiological Modelling at macro and micro levels: Examples of HIV and Hepatitis diseasesby
Livingstone S. Luboobi
Department of Mathematics, Makerere University
given at
Strathmore University
International Mathematics Research Meeting
Nairobi, Kenya
July 2012

1. 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

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

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

22. 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 ]
Reverse Transcription
Transcription & Translation
Uncoating
Integration
Free mutant viral Populations
Budding
Assembly
Macro-Micro Epidemic Models

23. Combined macro-micro epidemiologic dynamics of HIV/AIDS (cont1)
Micro level: Cellular level Dynamics
[Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI ]
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

24. Combined macro-micro epidemiologic dynamics of HIV/AIDS (cont2)
[Tameruet al., 2010: in Ethnicity & Disease,vol.20, pp SI ]
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

25. 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

26. 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

27. 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

28. 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

29. 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

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

31. Thank you for Listening
Macro-Micro Epidemic Models