Infectious Diseases of Humans Seminar The basic model

1. Infectious Diseases of Humans SeminarThe basic model Zhuobin Li

2. Outline • Framework of the basic model • Basic reproduction number, R0 • Average age of infection • Transmission parameter,

3. Frame work of the basic model • Assumption • Previous model Age-specific host death rate, per capita Age-specific recovery rate Age-specific disease-induced death rate ‘force of infection’ at time t

4. Frame work of the basic model • Assumption • X(a),Y(a),and Z(a) are time independent • Birth rate and death rate are exactly balanced • Ignore mortality associated with the infection is assumed to be zero

5. Frame work of the basic model • Model

6. Frame work of the basic model • Boundary condition

7. Frame work of the basic model • Solve the model where Survivorship function

8. Frame work of the basic model

9. Frame work of the basic model

10. Basic reproductive number R0 • R0 for a microparasite • The basic reproductive number R0, is essentially the average number of successful offspring that a parasite is intrinsically capable of producing. • For a microparasite, R0 is defined as the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is suscetible. • R 0 >1 is the condition of being capable of invading and establishing itself within a host population

11. Basic reproductive rate R0 • R0 for a microparasite • At the equilibrium, the rate of the susceptible being infected is balanced against a rate of newly susceptible individuals appearing. • At the equilibrium, each infection will be on average produce exactly one secondary infection; R=R0 x*=1 where x* is the fraction of the host population that is susceptible at equilibrium

12. Basic reproductive rate R0 • R0 for a this model • It seems to more plausible that the net rate of acquisition of new infectious is proportional to than to . “doubling the number of susceptibles in a school is arguably more likely to double the incidence of infection than is doubling the number of infectious individual” • Basic reproductive number

13. Basic reproductive number R0 • Two types of mortality Type I: Type II: where L is life expectancy.

14. Basic reproductive number R0 • Two types of mortality Type I: Type II: Survivor ship function:

15. Basic reproductive number R0 • Two types of mortality Type I: Type II:

16. Basic reproductive number R0 • Two types of mortality Type I: Type II: • Approximation of L “The difference between L and G can be significant, especially in some developing country. For an example, life expectancy L of India is around 40 years, while births are around 40 per 1000 per annum which corresponds to G~25 years”

17. Average age at infection • The average age at which individuals acquire infection, A. • Type I • Type II

18. Average age at infection • The average age at which individuals acquire infection based on the proportions susceptible. • If λ is age-dependent, then

19. Average age at infection • Come back to basic reproductive number Type I:

20. Transmission parameter, • Transmission parameter  • Combines a multitude of epidemiolagical, environmental, and social factors that affect transmission rates •  is “ force of infection”, the per capita rate of acquisition of infection. If  is time dependent, (t)t represents the probability that a given susceptible host will become infected in the small time interval t.

21. Transmission parameter, • Type I: • Type II: