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Population models of infection transmission D . Gurarie Spread of parasite in host population is similar to colonization/survival in fragmented...

""I am using the book Manual Mathematical Biology, it is a similar question to the one in the book

I am supposed to set up ‘discrete-time’ or ‘continuous’ equations for the system, and computing their BRN.

Set up SEIR diagrams, explain the meaning of all variables and parameters for the following infection systems smallpox-type pathogen (with life-long immunity) spreading in a homogeneous community with natural mortality - u (mu) , disease-specific mortality uD (mu subscript D), where fraction 0<f<1 of total population is vaccinated (fully protected). Sketch and discuss the resulting dynamic patterns of diseases spread. Explain what happens in the long run (t-> infinity), is there and ‘endemic’ state? Explain the effect of vaccination (f=0 vs. f>0) on disease
patterns.

Latency period 10 days
Infectious period 7 days
Natural Resistance level 0.9
Contact rate 2.7/day

"That sounds fine. I wanted to share with you what I have completed on the problem, currently I am stuck at this point

C=Contact Rate=2.7/day
q- Probability to Stay Latent- 10-1/10=.90 q=.90
r-Probability to Stay Infected- 7-1/7= r=.86 r=.90
s-Probability to Stay Immune-Infinity/All Life s=1
p=Probability to Survive Infectious Contact (1-p)=Susceptibility 1-p=.33 p=.66
Probability of Infection/Infectious contact 0.9/2.7=.33 Susceptibility=.33
Recovery rate= 1-r= 1-.9=.10

Attached is the handout with similar questions

Population models of infection transmission D. Gurarie Spread of parasite in host population is similar to colonization/survival in fragmented environment (meta-population view). But here host population (environment) plays active dynamic role. Questions to answer: Will infection spread What fraction of host population is affected Prevalence of endemic disease Control and eradication The effect of age structure Outline Basic SIR methodology of infection transmission a. BRN: its meaning and implications b. Control strategies: treatment, vaccination/culling, quarantine c. Extension to multiple-host transmission: zoonotics and vector-born diseases? d. Age structured populations Simple SIR and SEIR Basic host strata: S susceptible E exposed (non-infectious) I -infectious R recovered
Total population: S I R N   , or S E I R N   Compartmental diagrams with arrows indicating transitions from one group to another Figure 1: SIR and SEIR diagrams Parameters: Table 1 - Force of infection q - 1/”latency period” r - recovery tare (= 1/”mean duration”) - Mortality (natural and/or disease associated) - Immune loss Continuous DE models               (SIR) (SEIR) dS SB dS dt SB dE dt S q E dI dt S r I dI dt qE r I dR dt rI dR dt rI R R R R dt         (1) B – source of susceptible. Force of infection depends on specifics of transmission. Example : I S N  - with transmission coefficient b  (“mean contact rate ” x “probability of infection/contact”), / IN - infective fraction of contacts. The key dimensionless parameter is BRN: S I R r q S E I R r
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