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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 Aim and objectives Epidemiology Model Building Example Conclusion
.
AIM To bring awareness to medical
epidemiologists and pubic health providers of how mathematical models can be used in epidemiology
OBJECTIVES:
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 theproblem
Identify existingknowledge
Formulation ofMathematical
model
MathematicalSolution
Interpretation ofsolution
Comparison withThe real world
(model validation)
Report writing
agreement
No agreement
What is epidemiology?
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
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 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 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 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
The purpose of mathematical modelling in epidemiology
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:
A TWO-STRAIN HIV-1 MATHEMATICAL MODEL TO ASSESS THE EFFECTS OF CHEMOTHERAPY ON DISEASE PARAMETERS
Developed by Shiri, Garira and Musekwa 2005
VARIABLE INFECTIOUSNESS OVER THE HIV INFECTION PERIOD
Department of A pplied M athematics National University of Science and T echnology
Source HIV Insite, University of California San Francisco, School of Medicine
http://hivinsite.ucsf.edu
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 wild-type strain with drug efficacy of RTI and PI respectively
Department of A pplied M athematics National University of Science and T echnology
Mutant strain viral particles not susceptible to the drug’s antiviral effects
No pharmacological and intracellular delays when drugs are administered
Eight interacting species
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.
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.
Department of A pplied M athematics National University of Science and T echnology
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
Department of A pplied M athematics National University of Science and T echnology
T(0) T(1) T(2)
R0 = 2
Transmission
No Transmission
Infectious
Susceptible
T(0) T(1) T(2)
R0 = 1.5
Transmission
Infectious
Susceptible
No Transmission
T(0) T(1) T(2)
R0 = 2
Transmission
No Transmission
Infectious
Susceptible
Immune
The wild-type strain reproductive ratio is given by
Department of A pplied M athematics National University of Science and T echnology
The mutant strain’s reproductive ratio is given by
Department of A pplied M athematics National University of Science and T echnology Department of A pplied M athematics National University of Science and T echnology
CTL EFFECTS
1. CTLs only kill infected cells (a2 = b2 = 0 and h2 ≠ 0), ratio is given by
sα2β2N2
μ2μT(α2 + h2C2 )R021 =
2 . CTLs reduce infection rate of T cells and viral burst size (h2 = 0, a2 ≠ 0 and b2 ≠ 0)
R022 =sN2e
3 . CTLs kill infected cells and reduce viral infectivity (b2 = 0, a2 ≠ 0 and h2 ≠ 0)
sα2N2β2e
μ2μT(α2 + h2C2 )
β2e
μ2μT
-b2C2 -a2C2
R023 =
-a2C2
Department of A pplied M athematics National University of Science and T echnology Department of A pplied M athematics National University of Science and T echnology
Continued – CTL Effects
4. CTLs kill infected cells and reduce viral burst size (a2 = 0, b2 ≠ 0 and h2 ≠ 0)
sα2N2e β2
μ2μT(α2 + h2C2 )R024 =
-b2C2
Comparing the reproductive ratios
For α2>>h2C2 and if c2≈∞
•R022 < R021, R023 < R021 and R024 < R021 .
•R022 <R023 and 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
Department of A pplied M athematics National University of Science and T echnology
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
Department of A pplied M athematics National University of Science and T echnology
BC
A
D
E
F
R0-no immune response
R021-h2≠0
R023-a2,h2≠ 0
R024-b2, h2≠0
R022-a2,b2≠0
R02
Department of A pplied M athematics National University of Science and T echnology
NUMERICAL RESULTS
R02 < R022 < R024 < R023 < R021 < R0
Department of A pplied M athematics National University of Science and T echnology
AIDSCHRONIC PHASE
Department of A pplied M athematics National University of Science and T echnology
ENLARGEMENT OF R1
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)
AIDSASYMPTOMATIC PHASE
Department of A pplied M athematics National University of Science and T echnology
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 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
Meanwhile corroboration between health workers, statisticians and mathematicians should be intensified
THANK YOU:Partnerships for Health Research &Development
Questions?
Department of A pplied M athematics National University of Science and T echnology
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,
2. lim f(C,ai) = 1,
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
C≈∞
C≈0
1+aiC