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

1. THE ROLE OF MATHEMATICAL MODELLING IN EPIDEMIOLOGY WITH PARTICULAR REFERENCE TO HIV/AIDS Senelani Dorothy Hove-Musekwa Department of Applied Mathematics NUST- BYO- ZIMBABWE

2. Outline of Talk • Aim and objectives • Epidemiology • Model Building • Example • Conclusion

3. AIM • To bring awareness to medical epidemiologists and pubic health providers of how mathematical models can be used in epidemiology .

4. 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.

5. Background Empirical Modelling-data driven • Application of statistical extrapolation techniques • Back calculation method • Short term projection only Disadvantages • Requires reliable and substantial complete data

6. WHAT IS MATHEMATICAL MODELLING? • An activity of translating a real problem into mathematics for subsequent analysis of the real problem

7. Model Development Steps Identifythe problem Identify existing knowledge Formulation of Mathematical model No agreement Comparison with The real world (model validation) Interpretation of solution Mathematical Solution agreement Report writing

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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.

16. 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

17. 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

21. VARIABLE INFECTIOUSNESS OVER THE HIV INFECTION PERIOD Source HIV Insite, University of California San Francisco, School of Medicine http://hivinsite.ucsf.edu

23. 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 RTIand PI respectively

24. 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

26. 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.

28. 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.

32. 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.

33. 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

34. T(0) T(1) T(2) R0 = 2 Transmission No Transmission Infectious Susceptible

36. T(0) T(1) T(2) R0 = 1.5 Transmission No Transmission Infectious Susceptible

38. T(0) T(1) T(2) R0 = 2 Transmission No Transmission Infectious Susceptible Immune

40. The wild-type strain reproductive ratio is given by • The mutant strain’s reproductive ratio is given by

41. -b2C2 -a2C2 β2e μ2μT -a2C2 R023 = CTL EFFECTS • CTLs only kill infected cells (a2 = b2 = 0 and h2≠ 0), ratio is given by sα2β2N2 R021 = μ2μT(α2 + h2C2 ) 2 . CTLs reduce infection rate of T cells and viral burst size (h2 = 0, a2≠ 0 and b2 ≠ 0) sN2e R022 = 3 . CTLs kill infected cells and reduce viral infectivity (b2 = 0, a2≠ 0 and h2 ≠ 0) sα2N2β2e μ2μT(α2 + h2C2 )

42. Continued – CTL Effects 4. CTLs kill infected cells and reduce viral burst size(a2 = 0, b2≠ 0 and h2 ≠ 0) -b2C2 sα2N2e β2 R024 = μ2μT(α2 + h2C2 ) • 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

43. 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

44. NUMERICAL RESULTS A R0-no immune response B R022-a2,b2≠0 R021-h2≠0 E C R023-a2,h2≠ 0 R024-b2, h2≠0 D R02 F R02 < R022 < R024 < R023 < R021 < R0

45. 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)

46. 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.

47. 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.

48. Due to the continual generation of new HIV variants that escape CTL killing and resist current ARVs, these therapies should interferemore 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.

49. 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.