Modeling Pandemics (3)

In Statistical Inference in a Stochastic Epidemic SEIR Mannequin with Management Intervention, a extra complicated mannequin than the one we’ve seen yesterday was thought-about (and is named the SEIR mannequin). Take into account a inhabitants of measurement N, and assume that S is the variety of prone, E the variety of uncovered, I the variety of infectious, and R for the quantity recovered (or immune) people, displaystyle{start{aligned}{frac {dS}{dt}}&=-beta {frac {I}{N}}S[8pt]{frac {dE}{dt}}&=beta {frac {I}{N}}S-aE[8pt]{frac {dI}{dt}}&=aE-b I[8pt]{frac {dR}{dt}}&=b Iend{aligned}}Between S and I, the transition price is beta I, the place beta is the typical variety of contacts per particular person per time, multiplied by the chance of illness transmission in a contact between a prone and an infectious topic. Between I and R, the transition price is b (merely the speed of recovered or lifeless, that’s, variety of recovered or lifeless throughout a time frame divided by the whole variety of contaminated on that very same time frame). And at last, the incubation interval is a random variable with exponential distribution with parameter a, in order that the typical incubation interval is a^{-1}.

In all probability extra fascinating, Understanding the dynamics of ebola epidemics prompt a extra complicated mannequin, with prone folks S, uncovered E, Infectious, however both in neighborhood I, or in hospitals H, some individuals who died F and eventually those that both get well or are buried and due to this fact are not prone R.

Thus, the next dynamic mannequin is taken into accountdisplaystyle{start{aligned}{frac {dS}{dt}}&=-(beta_II+beta_HH+beta_FF)frac{S}{N}[8pt]frac {dE}{dt}&=(beta_II+beta_HH+beta_FF)frac{S}{N}-alpha E[8pt]frac {dI}{dt}&=alpha E+thetagamma_H I-(1-theta)(1-delta)gamma_RI-(1-theta)deltagamma_FI[8pt]frac {dH}{dt}&=thetagamma_HI-deltalambda_FH-(1-delta)lambda_RH[8pt]frac {dF}{dt}&=(1-theta)(1-delta)gamma_RI+deltalambda_FH-nu F[8pt]frac {dR}{dt}&=(1-theta)(1-delta)gamma_RI+(1-delta)lambda_FH+nu Fend{aligned}}In that mannequin, parameters are alpha^{-1} is the (common) incubation interval (7 days), gamma_H^{-1} the onset to hospitalization (5 days), gamma_F^{-1} the onset to loss of life (9 days), gamma_R^{-1} the onset to “restoration” (10 days), lambda_F^{-1} the hospitalisation to loss of life (Four days) whereas lambda_R^{-1} is the hospitalisation to restoration (5 days), eta^{-1} is the loss of life to burial (2 days). Right here, numbers are from Understanding the dynamics of ebola epidemics (within the context of ebola). The opposite parameters are beta_I the transmission price in neighborhood (0.588), beta_H the transmission price in hospital (0.794) and beta_F the transmission price at funeral (7.653). Thus

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epsilon = 0.001 Z = c(S = 1-epsilon, E = epsilon, I=0,H=0,F=0,R=0)
p=c(alpha=1/7*7, theta=0.81, delta=0.81, betai=0.588, betah=0.794, blambdaf=7.653,N=1, gammah=1/5*7, gammaf=1/9.6*7, gammar=1/10*7, lambdaf=1/4.6*7, lambdar=1/5*7, nu=1/2*7)

If boldsymbol{Z}=(S,E,I,H,F,R), if we write frac{partial boldsymbol{Z}}{partial t} = SEIHFR(boldsymbol{Z})the place SEIHFR is

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SEIHFR = operate(t,Z,p){ S=Z[1]; E=Z[2]; I=Z[3]; H=Z[4]; F=Z[5]; R=Z[6] alpha=p["alpha"]; theta=p["theta"]; delta=p["delta"] betai=p["betai"]; betah=p["betah"]; gammah=p["gammah"] gammaf=p["gammaf"]; gammar=p["gammar"]; lambdaf=p["lambdaf"] lambdar=p["lambdar"]; nu=p["nu"]; blambdaf=p["blambdaf"] N=S+E+I+H+F+R dS=-(betai*I+betah*H+blambdaf*F)*S/N dE=(betai*I+betah*H+blambdaf*F)*S/N-alpha*E dI=alpha*E-theta*gammah*I-(1-theta)*(1-delta)*gammar*I-(1-theta)*delta*gammaf*I dH=theta*gammah*I-delta*lambdaf*H-(1-delta)*lambdaf*H dF=(1-theta)*(1-delta)*gammar*I+delta*lambdaf*H-nu*F dR=(1-theta)*(1-delta)*gammar*I+(1-delta)*lambdar*H+nu*F dZ=c(dS,dE,dI,dH,dF,dR) checklist(dZ)}

We are able to clear up it, or a minimum of research the dynamics from some beginning values

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library(deSolve)
instances = seq(0, 50, by = .1)
resol = ode(y=Z, instances=instances, func=SEIHFR, parms=p)

As an illustration, the proportion of individuals contaminated is the next

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plot(resol[,"time"],resol[,"I"],sort="l",xlab="time",ylab="",col="pink")
strains(resol[,"time"],resol[,"H"],col="blue")



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