THE ROLE OF MATHEMATICAL
MODELLING IN
EPIDEMIOLOGY WITH PARTICULAR
REFERENCE TO HIV/AIDS
Senelani Dorothy Hove-Musekwa
Department of Applied Mathematics
NUST- BYO- ZIMBABWE
Outline of Talk
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Aim and objectives
Epidemiology
Model Building
Example
Conclusion
AIM
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To bring awareness to medical
epidemiologists and pubic health providers
of how mathematical models can be used in
epidemiology
OBJECTIVES:
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To highlight the purpose of mathematical
modelling in epidemiology .
To give basic principles on epidemic
mathematical modelling
To highlight one of the mathematical
models which have been developed.
Background
Empirical Modelling-data driven
 Application of statistical extrapolation techniques
 Back calculation method
 Short term projection only
Disadvantages
 Requires reliable and substantial complete data
WHAT IS MATHEMATICAL
MODELLING?
 An
activity of translating a
real problem into
mathematics for
subsequent analysis of the
real problem
Model Development Steps
Identify the
problem
Identify existing
knowledge
Formulation of
Mathematical
model
No agreement
Comparison with
The real world
(model validation)
agreement
Report writing
Interpretation of
solution
Mathematical
Solution
What is epidemiology?
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DEFINITION:- THE STUDY OF THE
DISTRIBUTION, FREQUENCY AND
DETERMINANTS OF HEALTH PROBLEMS
AND DISEASE IN HUMAN POPULATION
PURPOSE:- TO OBTAIN, INTERPRET AND
USE HEALTH INFORMATION TO PROMOTE
HEALTH AND REDUCE DISEASE
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BASED ON TWO FUNDAMENTAL ASSUMPTIONS:
- HUMAN DISEASE DOES NOT OCCUR AT
RANDOM
- HUMAN DISEASE HAS CAUSAL AND
PREVENTIVE FACTORS THAT CAN BE IDENTIFIED
THROUGH SYSTEMATIC INVESTIGATION OF
DIFFERENT
POPULATIONS OR SUBGROUPS OF INDIVIDUALS
WITHIN A POPULATION IN DIFFERENT PLACES
OR AT DIFFERENT PLACES OR AT DIFFERENT
TIMES
KEY QUESTIONS FOR SOLVING
HEALTH PROBLEMS
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WHAT? IS THE HEALTH PROBLEM,
DISEASE OR CONDITION, ITS
MANIFESTATIONS, CHARATERISTICS
WHO? IS AFFECTED:- AGE, SEX SOCIAL
STATUS,ETHNIC GROUP
WHERE? DOES THE PROBLEM OCCUR IN
RELATION TO PLACE OF RESIDENCE,
GEOGRAPHICAL DISTRIBUTION AND
PLACE OF EXPOSURE
QUESTIONS-contd
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WHEN? DOES IT HAPPEN IN TERMS
OF DAYS, MONTHS, SEASON OR
YEARS
HOW? DOES THE HEALTH PROBLEM
DISEASE OR CONDITION OCCUR,
SOURCES OF INFECTION,
SUSCEPTIBLE GROUPS. OTHER
CONTRIBUTING FACTORS
QUESTIONS-contd
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WHY? DOES IT OCCUR IN TERMS OF THE
REASONS FOR ITS PERSISTENCE OR
OCCURANCE
SO WHAT? INTERVENTIONS HAVE BEEN
IMPLEMETED AS A RESULT OF THE
INFORMATION GAINED, THEIR
EFFECTIVENESS, ANY IMPROVEMENTS IN
HEALTH STATUS
The General Dynamic Of An
Epidemic
Individuals pass from one class to
another with the passage of time.
 Mathematical model tries to capture
this flow by using compartments
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The purpose of mathematical
modelling in epidemiology
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To develop understanding of the interplay between the
variables that determine the course of the infection
within an individual and the variables that control the
pattern of infection within the communities of people.
To provide understanding of the pathophsiology of a
disease e.g. HIV.
To estimate the incidence and prevalence of a disease
e.g.HIV infection in both current and in the past.
To identify the groups of the population that are
currently at highest risk of contracting a particular
disease e.g. HIV.
Functions of mathematical models
– understanding
 Explicit assumptions – testable predictions
 Framework for data analysis
 Projections
 Interventions:
Outcome
Impact
Perverse outcomes
Combining Interventions
Target Setting
Impact of new technologies
Advocacy
Model Example:
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A TWO-STRAIN HIV-1 MATHEMATICAL
MODEL TO ASSESS THE EFFECTS OF
CHEMOTHERAPY ON DISEASE
PARAMETERS
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Developed by Shiri, Garira and Musekwa
2005
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VARIABLE INFECTIOUSNESS OVER THE HIV INFECTION PERIOD
Source HIV Insite, University of California San Francisco, School of Medicine
http://hivinsite.ucsf.edu
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MODEL ASSUMPTIONS
Cell mediated response and no humoral immune
response
Infection is by two viral strains
An uninfected cell once infected remains infected for
life
Only CD4+ T cells are infected and upon infection
cells become productive
Treatment drugs: RTIs and Pis act only on the wildtype strain with drug efficacy of RTI and PI
respectively
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Mutant strain viral particles not susceptible to the
drug’s antiviral effects
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No pharmacological and intracellular delays when
drugs are administered
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Eight interacting species
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Mass action principle employed, i.e., rate at which T
cells are infected is proportional to the product of
abundances of T cells and viral load
• Constant mutation in viral genes would
lead to continuous production of viral
variants able to evade to some extent the
CTL defenses operating. Genetic
mutations lead to changes in the
structure of viral peptides, i.e. epitopes
and these can become invisible to the
body’s defenses. If there are too many
variants, the immune system becomes
incapable of controlling the virus.
• Constant mutation in viral genes would
lead to continuous production of viral
variants able to evade to some extent the
CTL defenses operating. Genetic
mutations lead to changes in the
structure of viral peptides, i.e. epitopes
and these can become invisible to the
body’s defenses. If there are too many
variants, the immune system becomes
incapable of controlling the virus.
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STABILITY ANALYSIS
Need to remark that the model is reasonable in the sense
that no population grows negative and no population grows
unbounded
The model predicts that within the nonnegative orthant, the
number of densities of the seven species attain two steady
state values, one with no virus, an uninfected steady state
and another with a virus, an endemically steady state
Basic reproductive ratio (R0) - the number of newly infected
cells that arise from any one cell when almost all cells are
uninfected.
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THE BASIC REPRODUCTIVE RATIO
The Ratio determines:
•Whether an infection can occur
determines whether disease will progress or not
•Growth rate of infection
speed of disease progression
•Asymptomatic Period
determines time to progress to disease
•Necessary effort to control
controlling the ratio parameters, we can control the disease
T(0)
T(1)
T(2)
R0 = 2
Transmission
No Transmission
Infectious
Susceptible
T(0)
T(1)
T(2)
R0 = 1.5
Transmission
No Transmission
Infectious
Susceptible
T(0)
T(1)
T(2)
R0 = 2
Transmission
No Transmission
Infectious
Susceptible
Immune
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The wild-type strain reproductive ratio is given by
The mutant strain’s reproductive ratio is given by
D
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epartt m
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ent of A pplied M at hem at ics
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and T
T echnology
echnology
CTL EFFECTS
1. CTLs only kill infected cells (a2 = b2 = 0 and h2 ≠ 0), ratio is given by
R021 =
sα2β2N2
μ2μT(α2 + h2C2 )
2 . CTLs reduce infection rate of T cells and viral burst size (h2 = 0, a2 ≠ 0 and b2 ≠ 0)
-b2C2
R022 =
sN2e
-a2C2
β2e
μ2μT
3 . CTLs kill infected cells and reduce viral infectivity (b2 = 0, a2 ≠ 0 and h2 ≠ 0)
-a C
sα2N2β2e 2 2
R023 =
μ2μT(α2 + h2C2 )
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Continued – CTL Effects
4. CTLs kill infected cells and reduce viral burst size (a2 = 0, b2 ≠ 0 and h2 ≠ 0)
R024 =
sα2N2e
-b2C2
β2
μ2μT(α2 + h2C2 )
Comparing the reproductive ratios
For α2>>h2C2 and if c2≈∞
•R022 < R021,
•R022 <R023
R023 < R021
and
and
R024 < R021 .
R022 < R024
•The hierarchy of the reproductive ratios for a virus with a high rate of viral induced cell
killing (high cytopathicity) relative to infected cell CTL mediated killing (α2>>h2C2) and
a2<b2 is:
R02 < R022 < R024 < R023 < R021
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The hierarchy of the reproductive ratios for a virus with a low rate of viral
induced killing (low cytopathic effect) relative to CTL mediated killing (α2<<h2C2)
and a2>b2 is:
R02 < R023 < R024 < R021 < R022
Results
•Non- lytic effects are critical in the control of virus if the virus’s cytopathic effect
is high.
•Lytic effects of CTLs are critical in controlling viral load if the virus is less
virulent
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NUMERICAL RESULTS
A
R0-no immune response
B
C
R021-h2≠0
E
R022-a2,b2≠0
R023-a2,h2≠ 0
D R024-b2, h2≠0
R02F
R02 < R022 < R024 < R023 < R021 < R0
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ENLARGEMENT OF R1
ASYMPTOMATIC PHASE
AIDS
CHRONIC PHASE
AIDS
The parameter values are s2=20,μT=0.02, r2=0.01, β2=0.005,k2=0.0025, BT=350,
α2=0.25,h2=0.001,N2=1000, a2=0.015, b2=0.05, μ2=2.5,p=0.00001,2=1.3, BV=400
(Kirschner, 1996; Ho et al. 1995; Dixit and Perelson, 2004; Joshi, 2002)
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Discussion
•no chemokines to inhibit infection, no cytokines to
reduce burst size with or without killing, there is no
clinical latency.
•Presence of HIV-1 suppressive factors produced by
CTLs control the viral load during HIV infection –
thus the presence of chronic phase.
•Non-lytic CTL effects are crucial to control viruses
with high cytopathicity effects.
•Killing of virally infected cells is critical for low
cytopathicity viruses.
• Results provide evidence to shift our
focus to immune based therapies if we
are to control the debilitating effects of
HIV.
• Therapeutic strategies would prompt the
body’s own immune system to respond
and control HIV infection – immune based
therapies should include cytokine
modulators and active
immunotherapeutics that enhance
production of effective cytokines and
chemokines by HIV specific CTLs.
• Due to the continual generation of
new HIV variants that escape CTL
killing and resist current ARVs,
these therapies should interfere
more effectively with the replication
and budding processes of the virus.
• In conclusion, immune based
therapy is the only hope if we are to
fight the epidemic.
• Constant mutation in viral genes would
lead to continuous production of viral
variants able to evade to some extent the
CTL defenses operating. Genetic
mutations lead to changes in the
structure of viral peptides, i.e. epitopes
and these can become invisible to the
body’s defenses. If there are too many
variants, the immune system becomes
incapable of controlling the virus.
•The battle between HIV and the
body’s defensive forces is a clash
between two armies. Each member of
the HIV ARMY is a GENERALIST (able
to attack ANY enemy cell it
encounters) but each member of the
IMMUNE ARMY is a SPECIALIST (able
to recognize an HIV SOLDIER if the
soldier is waving a flag of a PRECISE
COLOUR)
WHO IS GOING TO WIN THE WAR?
Recommendations
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Introduce taught courses on the transmission
dynamics of diseases, employing some
mathematical content in the training of
medical doctors and others associated with
public health
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Meanwhile corroboration between health
workers, statisticians and mathematicians
should be intensified
THANK YOU:
Partnerships for Health Research
&Development
Questions?
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Robustness of Model Results
Proposition: Any function f(C,ai) where C is a variable (CTL count) and ai is a
parameter (i=1,2), where a1 is the effectiveness of each CTL in reducing viral
infection of naïve CD4+ T cells and a2 is the effectiveness of each CTL in reducing
viral burst size, with the following properties:
1. lim f(C,ai) = 0,
C≈∞
2. lim f(C,ai) = 1,
C≈0
3. f(C,ai) is strictly decreasing function of C and
4. f(C,ai) is positive definite
can be used to model the effectiveness on non-lytic effects of CTLs in system (1)
and still gives the same hierarchy of reproductive ratios, e.g.,
1
1+aiC