In cmhoove14/DDNTD: Density Dependence in Neglected Tropical Diseases

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Snail infection dynamics

Beginning with a basic schistosomiasis model with $S-E-I$ infection dynamics and logistic population growth among the intermediate host snail population we have three ODEs and one simple relation between each infection class and the total snail population, N:

r latexImg("\\frac{dS}{dt}=r\\Big(1-\\frac{N}{K}\\Big)\\Big(S+E\\Big)-(\\mu_N+\\Lambda) S")

r latexImg("\\frac{dE}{dt}=\\Lambda S-(\\mu_N+\\sigma)E")

r latexImg("\\frac{dI}{dt}=\\sigma E - \\mu_I I")

r latexImg("N=S+E+I")

Where $\Lambda$ is the man-to-snail force of infection (FOI). At equilibrium, r latexImg("\\frac{dS}{dt}=\\frac{dE}{dt}=\\frac{dI}{dt}=0") we have:

r latexImg("E^*=\\frac{\\Lambda S^*}{\\mu_N+\\sigma}")

r latexImg("I^*=\\frac{\\sigma E^*}{\\mu_I}=\\frac{\\sigma\\Lambda S^*}{\\mu_I(\\mu_N+\\sigma)}")

and therefore:

r latexImg("N^*=S^*\\Big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}+\\frac{\\sigma\\Lambda}{\\mu_I(\\mu_N+\\sigma)}\\Big)")

In addition, we can solve equation 1 representing susceptible snail dynamics for $N^*$ in terms of $\Lambda$ as:

r latexImg("N^*(\\Lambda)=K\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)")

Which gives:

r latexImg("S^*=\\frac{K\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)}{\\Big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}+\\frac{\\sigma\\Lambda}{\\mu_I(\\mu_N+\\sigma)}\\Big)}")

and

r latexImg("I^*=\\frac{K\\sigma\\Lambda\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)}{\\Big(\\mu_I(\\mu_N+\\sigma)\\Big)\\Big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}+\\frac{\\sigma\\Lambda}{\\mu_I(\\mu_N+\\sigma)}\\Big)}")

Which simplifies to:

r latexImg("I^*=\\frac{K\\sigma\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)}{\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda}+\\mu_I+\\sigma\\Big)}=\\frac{\\sigma N^*}{\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda}+\\mu_I+\\sigma\\Big)}")

Therefore with infected snail prevalence, $I_P=I^/N^$, and other snail population and infection parameters as inputs, we can estimate $\Lambda$ as:

r latexImg("I_P=\\frac{\\sigma}{\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda}+\\mu_I+\\sigma}")

r latexImg("\\Lambda=\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}")

$R_0$ derivation

We employ the next generation matrix method to develop an analytic expression of the basic reproduction number, $R_0$. We begin by linearizing the disease system about the disease-free steady state to solve for the equilibrium snail population size, $N^*$:

r latexImg("N^*=K\\Big(1-\\frac{\\mu_N}{r}\\Big)")

We next consider the linearized subsystem of equations governing infectious state dynamics (state variables $E$, $I$, and $W$) and subsititute in for $N^*$:

r latexImg("\\frac{dE}{dt}=\\Lambda N^*-(\\mu_N+\\sigma)E")

r latexImg("\\frac{dI}{dt}=\\sigma E - \\mu_I I")

r latexImg("\\frac{dW}{dt}=\\lambda I-(\\mu_W+\\mu_H)W")

We then build matrices representing the rate of generation of new infections, $T$, and transitions between infectious classes, $\Sigma$:

r latexImg("T=\\begin{bmatrix} 0 & 0 & \\Lambda N^* \\\\ 0 & 0 & 0 \\\\ 0 & \\lambda & 0 \\\\ \\end{bmatrix}")

r latexImg("\\Sigma=\\begin{bmatrix} (\\mu_N+\\sigma) & 0 & 0 \\\\ -\\sigma & \\mu_I & 0 \\\\ 0 & 0 & (\\mu_W+\\mu_H) \\\\ \\end{bmatrix}")

And then estimate the next generation matrix as $K_L=T(-\Sigma^-)$:

r latexImg("K_L=\\begin{bmatrix} 0 & 0 & \\frac{\\Lambda N^*}{(\\mu_W+\\mu_H)} \\\\ 0 & 0 & 0 \\\\ \\frac{\\sigma\\lambda}{(\\mu_N+\\sigma)(\\mu_I)} & \\frac{\\lambda}{\\mu_I} & 0 \\\\ \\end{bmatrix}")

the largest eigen value of which is $R_0$, giving:

r latexImg("R_0=\\Big(\\frac{\\Lambda N^*\\sigma\\lambda}{\\mu_I(\\mu_N+\\sigma)(\\mu_W+\\mu_H)}\\Big)^{0.5}")

We can also arrive at this estimate (without the square root) by considering the relative rates of change of each infection class:

r latexImg("R_0=\\frac{\\Lambda N^*}{(\\mu_N+\\sigma)}\\times\\frac{\\sigma}{\\mu_I}\\times\\frac{\\lambda}{(\\mu_W+\\mu_H)}=\\frac{\\Lambda N^*\\sigma\\lambda}{\\mu_I(\\mu_N+\\sigma)(\\mu_W+\\mu_H)}")

$R_{eff}$ and breakpoint derivation with single worm burden compartment

Effective reproduction number, $R_{eff}$, derivation

The effective reproduction number, $R_{eff}$, for schistosomiasis is defined as the number of mated adult female worms produced by a single adult female worm over the course of her lifetime. Unlike the basic reproduction number, $R_0$, $R_{eff}$ changes as a function of the current level of infection, here measured in terms of the mean worm burden, $W$. From Anderson and May Infectious Diseases of Humans, we can estimate the effective reproduction number from the rate of worm burden change as:

r latexImg("\\frac{dW}{dt}=\\lambda-\\mu_\\mathcal{W}W")

r latexImg("\\frac{dW}{dt}=\\mu_\\mathcal{W}W\\Big(\\frac{\\lambda}{\\mu_\\mathcal{W}W}-1\\Big)")

r latexImg("\\frac{dW}{dt}=\\mu_\\mathcal{W}W\\Big(R_{eff}(W)-1\\Big)")

r latexImg("R_{eff}(W)=\\frac{\\lambda}{\\mu_\\mathcal{W}W}")

Where $\mu_\mathcal{W}$ is the mean lifespan of an adult worm and $\lambda$ is the worm acquisition rate or snail-to-man FOI, which is a function of the exposure coefficient, $\omega$, probability of worm acquisition given exposure, $\alpha$, and cercarial concentration estimated as a function of cercarial shedding and the infected snail population size: r latexImg("C=\\theta I"). Therefore we can express r latexImg("R_{eff}") in terms of state variables $I$ and $W$ as:

r latexImg("R_{eff}(W, I)=\\frac{\\alpha\\omega\\theta I}{W\\mu_\\mathcal{W}}")

Alternatively, if we assume fast snail infection dynamics relative to human infection dynamics as is common, we can substitute for $I$ in terms of the equilibirum snail population size, $N$, and snail FOI, $\Lambda$, both in terms of $W$:

r latexImg("R_{eff}(W)=\\frac{\\alpha\\omega\\theta\\sigma N(W)}{W\\mu_\\mathcal{W}\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda(W)}+\\mu_I+\\sigma\\Big)}")

r latexImg("N(W)=K\\Big(1-\\frac{\\mu_N+\\Lambda(W)}{r\\big(1+\\frac{\\Lambda(W)}{\\mu_N+\\sigma}\\big)}\\Big)")

The particular formulation of the snail FOI, $\Lambda$, will determine how analytically tractable this estimate is. In the simplist case where snail infection is estimated simply as the infection probability, $\beta$, and miracidial concentration, $M$, we can combine the two equations to arrive at:

r latexImg("R_{eff}(W)=\\frac{K\\alpha\\omega\\theta\\sigma\\big(1+\\frac{\\beta M(W)}{\\mu_N+\\sigma}-\\frac{\\mu_I}{r}-\\frac{\\beta M(W)}{r}\\big)}{W\\mu_\\mathcal{W}\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\beta M(W)}+\\mu_I+\\sigma\\Big)\\Big(1+\\frac{\\beta M(W)}{\\mu_N+\\sigma}\\Big)}")

This solution is nice in that it is analytically tractable, however, this simple treatment is likely to overestimate snail FOI at small worm burdens where we care most about estimating $R_{eff}$. More accurate representations of the snail FOI such as r latexImg("\\Lambda=\\beta M(W)/N(W)") or the saturating version of snail FOI, r latexImg("\\Lambda=\\Lambda_0(1-e^{-\\beta M(W)/N(W)})") are less analytically tractable, but can still be used for r latexImg("R_{eff}") estimation via rootsolving.

# Silly hoop jumping to arrive at the saem thing as above

#Alternatively, we can make one more substitution based on the fact that infected snail prevalence can be estimated as a function of snail FOI:

#`r latexImg("I_P=\\frac{\\sigma}{\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda}+\\mu_I+\\sigma}")`

#therefore we can substitute in terms of infected snail prevalence, $I_P$, to get:

#r latexImg("R_{eff}(W, I_P)=\\frac{\\alpha\\omega\\theta N I_P}{W\\big(\\mu_W+\\mu_H\\big)}")`

#And then with linear FOI, `r latexImg("\\Lambda=\\beta M/N")`substitute for N in terms of $I_P$:

#`r latexImg("N=\\frac{\\beta M\\big(\\frac{\\sigma}{I_P}-\\mu_I-\\sigma\\big)}{\\mu_I(\\mu_N+\\sigma)}=\\frac{\\beta\\omega WHmv\\Phi(W)\\rho(W)U\\big(\\frac{\\sigma}{I_P}-\\mu_I-\\sigma\\big)}{2\\mu_I(\\mu_N+\\sigma)}")`

#and get:

#`r latexImg("R_{eff}(W, I_P)=\\frac{\\alpha\\omega^2\\theta\\beta\\phi(W)\\rho(W)HmvU(\\sigma-I_P\\mu_I-I_P\\sigma)}{2\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma)\\big)}")`




#This was from another parameterization of R_eff, but the one in the document is simpler as it replaces K*(1-...) with N

#`r latexImg("I^*=\\frac{K\\sigma\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}\\Big)}{\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda}+\\mu_I+\\sigma\\Big)}")`

#Therefore we can estimate $R_{eff}$ as:

#`r latexImg("R_{eff}(W)=\\frac{\\alpha\\omega\\theta K\\sigma\\Big(1-\\frac{\\mu_N+\\Lambda}{r\\big(1+\\frac{\\Lambda}{(\\mu_N+\\sigma)}\\big)}\\Big)}{W\\Big(\\mu_W+\\mu_H\\Big)\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda}+\\mu_I+\\sigma\\Big)}")`

#Which can be (slightly) simplified to:

#`r latexImg("R_{eff}(W)=\\frac{\\alpha\\omega\\theta K\\sigma\\Lambda\\Big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}-\\frac{\\mu_N}{r}-\\frac{\\Lambda}{r}\\Big)}{W\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma+\\Lambda)+\\sigma\\Lambda\\big)\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}")`

#We also note that the man-to-snail FOI, $\Lambda$, is largely a function of miracidial density, $M$, which is determined by the mean worm burden and associated density dependent mating probability, $\Phi$, and density dependent fecundity $\rho$, therefore it is more accurate to denote $\Lambda$ as a function of $W$:

#`r latexImg("R_{eff}(W)=\\frac{\\alpha\\omega\\theta K\\sigma\\Lambda(W)\\Big(1+\\frac{\\Lambda(W)}{\\mu_N+\\sigma}-\\frac{\\mu_N}{r}-\\frac{\\Lambda(W)}{r}\\Big)}{W\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma+\\Lambda(W))+\\sigma\\Lambda(W)\\big)\\big(1+\\frac{\\Lambda(W)}{\\mu_N+\\sigma}\\big)}")`

Worm burden breakpoint estimation

The breakpoint worm burden population size, r latexImg("W_{bp}") is an unstable equilibrium at which each female worm replaces herself, i.e. r latexImg("R_{eff}=1") therefore the breakpoint is one solution for W and I of:

r latexImg("R_{eff}(W, I)=1=\\frac{\\alpha\\omega\\theta I}{W\\mu_\\mathcal{W}}")

So:

r latexImg("W_{bp}=\\frac{\\alpha\\omega\\theta I_{bp}}{\\mu_\\mathcal{W}}")

Or is the smaller solution to:

r latexImg("W_{bp}=\\frac{\\alpha\\omega\\theta\\sigma N(W_{bp})}{\\mu_\\mathcal{W}\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda(W_{bp})}+\\mu_I+\\sigma\\Big)}")

r latexImg("N(W_{bp})=K\\Big(1-\\frac{\\mu_N+\\Lambda(W_{bp})}{r\\big(1+\\frac{\\Lambda(W_{bp})}{\\mu_N+\\sigma}\\big)}\\Big)")

MDA coverage necessary to reach the breakpoint

Another useful outcome is the MDA coverage necessary to reach the breakpoint. We can model MDA as a reduction in the mean worm burden at the following time step in terms of the MDA coverage, $\mathcal{C}$, and drug efficacy, $\epsilon$:

r latexImg("W_{t+1}=(1-\\epsilon)\\mathcal{C}W_t+(1-\\mathcal{C})W_t")

If our goal is to reduce the worm burden to or below the breakpoint (i.e. r latexImg("W_{t+1}\\leq W_{bp}")) with MDA, we can estimate the coverage required as:

r latexImg("\\mathcal{C}=\\frac{W_{bp}-W_t}{-\\epsilon W_t}")

We can also incorporate excentricities of MDA such as systematic noncompliance by incorporating a proportion of the worm burden that is not affected by MDA

$R_{eff}$ for stratified worm burdens

While the mean worm burden is often modeled as a single state variable, it is more accurate to further stratify the compartment to more accurately capture the skewed distribution of worms amongst the human population or to more accurately model interventions such as MDA that are often only administered to SAC or that systematically miss particular portions of the population. We therefore wish to express r latexImg("R_{eff}") as a function of the population mean worm burden, r latexImg("\\bar{W}"), derived from multiple treatment groups, $i$, and age groups, $j$, where:

r latexImg("\\bar{W}=\\sum_i\\sum_j h_{ij}W_{ij}")

The net r latexImg("R_{eff}") can be thought of similarly as some weighted average of contributions from all treatment groups:

r latexImg("R_{eff}(\\bar{W})=\\sum_i\\sum_jh_{ij}\\frac{\\alpha\\omega_i\\theta\\sigma N(\\bar{W})}{W_{ij}\\big(\\mu_W+\\mu_{H_i}\\big)\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda(\\bar{W})}+\\mu_I+\\sigma\\Big)}")

r latexImg("N(\\bar{W})=K\\Big(1-\\frac{\\mu_N+\\Lambda(\\bar{W})}{r\\big(1+\\frac{\\Lambda(\\bar{W})}{\\mu_N+\\sigma}\\big)}\\Big)")

Which we can express as:

r latexImg("R_{eff}(\\bar{W})=\\frac{\\alpha\\theta\\sigma N(\\bar{W})}{\\bar{W}\\Big(\\frac{\\mu_I(\\mu_N+\\sigma)}{\\Lambda(\\bar{W})}+\\mu_I+\\sigma\\Big)}\\sum_i h_{i}\\frac{\\omega_i}{\\big(\\mu_W+\\mu_{H_i}\\big)}")

r latexImg("N(\\bar{W})=K\\Big(1-\\frac{\\mu_N+\\Lambda(\\bar{W})}{r\\big(1+\\frac{\\Lambda(\\bar{W})}{\\mu_N+\\sigma}\\big)}\\Big)")

Worm burden breakpoint estimation

The breakpoint worm burden population size, r latexImg("W_{bp}") is an unstable equilibrium at which each female worm replaces herself, i.e. r latexImg("R_{eff}=1") therefore:

r latexImg("R_{eff}(W_{bp}, I_P)=1=\\frac{\\alpha\\omega^2\\theta\\beta\\phi(W_{bp})\\rho(W_{bp})HmvU I_P}{2\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma)\\big)}")

From this, it's clear that interplay between the density dependence functions is key in determining r latexImg("W_{bp}"), particularly:

r latexImg("\\phi(W_{bp})\\rho(W_{bp})=\\frac{2\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma)\\big)}{\\alpha\\omega^2\\theta\\beta HmvU I_P}")

MDA coverage necessary to reach the breakpoint

Another useful outcome is the MDA coverage necessary to reach the breakpoint. We can model MDA as a reduction in the mean worm burden at the following time step in terms of the MDA coverage and drug efficacy, $\epsilon$:

r latexImg("W_{t+1}=(1-\\epsilon)cvrgW_t+(1-cvrg)W_t")

If our goal is to reduce the worm burden to or below the breakpoint (i.e. r latexImg("W_{t+1}\\leq W_{bp}")) with MDA, we can estimate the coverage required as:

r latexImg("cvrg=\\frac{{W_{bp}-W_t}{-\\epsilon W_t}")

Fitting parameters from equilibirum states

Contamination fraction, $\Omega$, and snail FOI parameters, ($\beta$ or $\Lambda_0$), as function of equilibrium worm burden in children and adults

We first assume that differences in child and adult infection rates arise predominately from variation in behavior that affects both exposure and contribution to infection in the same manner. Parameter $\Omega$ therefore represents the relative exposure/contamination of children to adults, $\Omega=\omega_C/\omega_A$, which can be estimated from the equilibrium infection ratio $\Omega=W^_C/W^_A$.

With this parameter, we can estimate the equilibirum miracidial density $M$ as:

r latexImg("M=0.5\\mathbf{H}m\\omega_A(W_C^* h_C\\Phi(W_C^*)\\rho(W_C^*)U_C\\Omega+W_A^* h_A\\Phi(W_A^*)\\rho(W_A^*)U_A)")

Saturating man-to-snail FOI

Now, since $\Lambda$ has been estimated above, $N^*$ can be estimated as a function of $\Lambda$, and we have an equilibrium estimate of miracidial density, $M$, we can estimate the baseline miracidial invasion rate, $\Lambda_0$, of the saturating form of the man-to-snail FOI:

r latexImg("\\Lambda_0=\\frac{\\Lambda}{(1-e^{-M/N^*(\\Lambda)})}")

Except in rare circumstances, $\Lambda\approx\Lambda_0$ because of the high $M/N$ ratio in transmission settings that have not been intervened upon (i.e. equilibrium settings).

Linear man-to-snail FOI

As an alternative, we can use the more common (but perhaps less accurate) linear form of the man-to-snail FOI:

r latexImg("\\Lambda=\\frac{\\beta M}{N^*(\\Lambda)}")

to estimate the probability of snail infection given miracidial density, $\beta$ as:

r latexImg("\\beta=\\frac{\\Lambda N^*(\\Lambda)}{M}")

Worm establishment rate, $\alpha$, as function of equilibrium worm burden

Assuming the observed mean worm burden in each population fraction prior to intervention is approximately at equilibrium, the worm acquisition rate, $\lambda_{ij}$, for each group can be estimated as a function of the exposure fractions, $\omega_a$ and $\omega_c$, the infected snail density, $I^=I_PN^$, the snail cercarial shedding rate, $\theta$, and the probability of cercarial establishment per exposure, $\alpha$, with $\alpha$ being the only unknown and estimated as:

r latexImg("\\alpha=\\frac{W_i^*(\\mu_W+\\mu_H_i)}{\\omega_i I^*\\theta}")

NEEDS UPDATING

r latexImg("W_{bp}=\\frac{\\alpha\\omega\\theta K\\sigma\\Lambda\\Big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}-\\frac{\\mu_N}{r}-\\frac{\\Lambda}{r}\\Big)}{\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma+\\Lambda)+\\sigma\\Lambda\\big)\\big(1+\\frac{\\Lambda}{\\mu_N+\\sigma}\\big)}")

Obviously the man-to-snail FOI, $\Lambda$ is a major determinant of the breakpoint. As noted previously, we can estimate it given infected snail prevalence,

r latexImg("\\Lambda=\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}")

And express the breakpoint in terms of infected snail prevalence as:

r latexImg("W_{bp}(I_P)=\\frac{\\alpha\\omega\\theta K\\sigma\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}\\Big(1+\\frac{\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}}{\\mu_N+\\sigma}-\\frac{\\mu_N}{r}-\\frac{\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}}{r}\\Big)}{\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma+\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma})+\\sigma\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}\\big)\\big(1+\\frac{\\frac{\\mu_I(\\mu_N+\\sigma)}{\\frac{\\sigma}{I_P}-\\mu_I-\\sigma}}{\\mu_N+\\sigma}\\big)}")

Which simplifies (somewhat) to:

r latexImg("W_{bp}(I_P)=\\frac{\\alpha\\omega\\theta KI_P\\mu_I(\\mu_N+\\sigma)\\Big(1+\\frac{I_P\\mu_I}{\\sigma-I_P(\\mu_I+\\sigma)}-\\frac{\\mu_N}{r}-\\frac{I_P\\mu_I(\\mu_N+\\sigma)}{r(\\sigma-I_P(\\mu_I+\\sigma))}\\Big)}{\\big(1-\\frac{I_P\\mu_I}{\\sigma}-I_P\\big)\\big(\\mu_W+\\mu_H\\big)\\big(\\mu_I(\\mu_N+\\sigma+\\frac{I_P\\mu_I(\\mu_N+\\sigma)}{\\sigma-I_P(\\mu_I+\\sigma)})+\\frac{I_P\\mu_I(\\mu_N+\\sigma)}{1-\\frac{I_P\\mu_I}{\\sigma}-I_P}\\big)\\big(1+\\frac{I_P\\mu_I}{\\sigma-I_P(\\mu_I+\\sigma)}\\big)}")

cmhoove14/DDNTD documentation built on Nov. 23, 2019, 7:04 p.m.