THE ROLE OF MATHEMATICAL MODELLING IN EPIDEMIOLOGY WITH PARTICULAR REFERENCE TO HIV/AIDS Senelani...

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

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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

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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

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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