Deterministic Compartmental Models of Disease

Transcription

1 Math 191T, Spring 2019

2 1 2 3 The SI Model The SIS Model The SIR Model 4 5

3 Basics Definition An infection is an invasion of one organism by a smaller organism (the infecting organism). Our focus is on microparasites: small (invisible to naked eye), found in huge numbers in host. Examples: (i) virus measles, mumps, rubella, smallpox, flu; (ii) bacteria whooping cough, TB, typhoid fever.

4 Time Periods Pre-infectious (latent) period: the time from infection to when a host is able to transmit the agent to another host. Incubation period: time from infection to onset of disease. Infectious period: time from end of pre-infectious period until the time when a host can no longer transmit the infection to others.

5 Time Periods (cont.) Serial interval: time from onset of a primary case to a secondary case. Determined largely by duration of pre-infectious and infectious periods and incubation period. These time periods determine when secondary case has clinical onset. Clinical onsets are typically reported and are the basis of infectious disease statistics. Often details of person-to-person chains of transmission lost in the large number of cases and clinical onsets over time.

6 Immune Outcomes Complete immunity: once rid of infection, can t again be infected; e.g., measles, rubella, mumps. Partial immunity: once rid of infection, still susceptible to some extent to subsequent infection; e.g., whooping couph, malaria. Little or no immunity: can remain infected and/or infectious effectively for life e.g., HIV. Variations of the above: e.g., TB individuals may be infected for many years, but only infectious later in life or be superinfected by another exposure (infected simultaneously by two strains).

7 Frequency of infections described in terms of incidence and prevalence. Incidence: risk or rate of new cases per unit time per 100 (or 1000) susceptible individuals. Prevalence: number or proportion of affected individuals in a population at any point in time. Statistics typically broken down by age, gender, and other groups.

8 Measuring Transmissibility Secondary attack rate: proportion infected among those susceptibles in contact with a primary case; available for many infections (best measure). Reproduction number: average number of successful transmissions per infectious person. Maximum when infectious person introduced into a totally susceptible population. Then called basic reproduction number, R 0. R 0 > 1 = infection persists in a population.

9 Basic Reproduction Number, R 0 Average number of secondary infectious individuals resulting from a typical infectious person following their introduction into a totally susceptible population. If the new infection confers immunity on infected individuals, the proportion immune increases with time, so number of actual transmissions will be less than R 0.

10 Net Reproduction Number, R n Net or effective reproduction number, R n, is the number of actual transmissions Simplest form: R n = R 0 s, where s is the proportion of the population that is susceptible. Notes: s = 1 R 0 = R n = 1. s < 1 R 0 s > 1 R 0 = R n < R 0, so incidence decreases. = R n > R 0, so incidence increases.

11 Herd Immunity Threshold (HIT) The critical threshold of susceptibles typically described in terms of proportion immune (= 1 s). The critical threshold immune is the herd immunity threshold. HIT given by HIT = 1 1 R 0 = R 0 1 R 0. Thoeretically, HIT is target for immunization programs if this proportion of immune is exceeded, the incidence of infection should decrease.

12 Examples Below is a table of some illnesses, along with their serial intervals, R 0, and HIT. Infection Serial interval (range) R 0 HIT (%) diphtheria 2-30 days influenza 2-4 days malaria 20 days measles 7-16 days polio 2-45 days 2-4 (good hygiene) 8-14 (poor hygiene) not well-defined smallpox 9-45 days

13 The SI Model The SIS Model The SIR Model Simple models: compartment models based on the mass action principle. Assumptions: a population is homogeneous (all people are the same); and the only difference is their disease state. Mass action principle: rate of reaction is a function of the product of the concentrations of the reagents.

14 The SI Model The SI Model The SIS Model The SIR Model Assume the population can be broken into two classes, those susceptible to infection at time t (S(t)) and those infectious at t (I(t)). Assumption: no recovery from the disease (e.g., HIV). Let β = rate at which two individuals come into effective contact per unit time. Result: a coupled system of differential equations ds dt = βsi di dt = βsi.

15 SI Model (cont.) The SI Model The SIS Model The SIR Model If total population size fixed so S + I = N, the system can be reduced to ds = βs(n S). dt Solution: S(t) = NS(0) (N S(0))e βnt, I(t) = N S(t). + S(0) Note: lim t S(t) = 0, so all individuals eventually get infected.

16 The SIS Model The SI Model The SIS Model The SIR Model Assumption: Individuals recover from the illness, but they then become susceptible. Diagram: Susceptible S(t) λ(t) γ Infectious I(t) Parameters: λ(t) = (the force of infection) = βi(t); and γ = fraction of infectious that recover, becoming susceptible

17 SIS Model (cont.) The SI Model The SIS Model The SIR Model The system of differential equations modeling this is ds = βis + γi dt di = βis γi dt Assuming constant population N = S + I, we can reduce the system to ds = βs(n S) + γ(n S). dt

18 Analysis of the SIS Model The SI Model The SIS Model The SIR Model Solution (via Maple) given an initial susceptible population S(0) = S 0 : γ(n S 0 )e (Nβ γ)t + N(βS 0 γ) β(n S 0 )e (Nβ γ)t + βs 0 γ. Note: The limit as t of S(t) depends on the relative values of γ and Nβ. If Nβ < γ, then lim t S(t) = N, so all individuals eventually are susceptible. If Nβ > γ, then lim t S(t) = γ β. If Nβ = γ, then lim t S(t) = S 0(Nβ γ) βn γ.

19 The SIR Model The SI Model The SIS Model The SIR Model Assumption: Once an individual recovers, they are immune from the illness. Diagram: Susceptible S(t) λ(t) Infectious I(t) r Recovered R(t) Parameter: r = rate at which infectious individuals recover (per unit time)

20 SIR Model (cont.) The SI Model The SIS Model The SIR Model The system of differential equations that models this is ds dt = βsi di = βsi ri dt dr = ri dt Verify this satisfies constant population size: ds dt + di dt + dr = βsi + βsi ri + ri = 0. dt

21 Analysis of SIR Model The SI Model The SIS Model The SIR Model No closed form solution. Given β, r, and initial values, we can get a graphical solution. Example: β = 10 5, r = 1, S(0) = 10000, I(0) = 10, R(0) = 0 7

22 Human Respiratory Syncytial Virus (RSV) A leading cause of hospitalization among young children with respiratory infections. RSV strains divided into two groups, A and B. thought to occur by hands into nose or eyes. Once infected, individuals can be reinfected. See [2] for more details.

23 Structure of the Model of RSV Susceptible S(t) w s Recovered R(t) λ(t) w s w r w s kλ(t) w r (primary) I s (t) (reinfected) I r (t)

24 Parameters I s (t) = infectious (first infected as a child) I r (t) = infectious (re-infected) w s = rate infectious (primary or re-infected) or recovered become susceptible w r = rate infectious (primary or re-infected) recover k: rate at which recovered individuals are reinfected differs by a factor k from the rate at which susceptible individuals become infected.

25 DE s for RSV Model The system of DE s is ds dt = λ(t)s + w si s + w s I r + w s R di s = λ(t)s w r I s w s I s dt di r = kλ(t)r w r I r w s I r dt dr = kλ(t)r + w r I s + w r I r w s R. dt Parameters in [2] determined using model-fitting. Resulting final model is stochastic.

26 Question How would we model a system in which we have a vaccination that provides at least partial protection? For this question, let s consider something like the flu and class all types of the flu under the category of flu.

27 [1] Emilia Vynnycky and Richard White, An Introduction to Infectious Disease Modelling, Oxford University Press, [2] L.J. White, J.N. Mandl, M.G. Jones, et al, Understanding the Dynamics of Respiratory Syncytial Virus Using Multiple Time Series and Nested Models, Mathematical Biosciences (2008), 209(1):