# 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

 1 2 3 4 5 6  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

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  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

 1 2 3  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

 1 2  plot(resol[,"time"],resol[,"I"],sort="l",xlab="time",ylab="",col="pink") strains(resol[,"time"],resol[,"H"],col="blue")