Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Authors: van den Driessche and Watmough, 2005.


This paper defines the basic reproductive number as the spectral radius of a certain matrix,

$R_0 = \rho (FV^{-1})$.

The matrices $F$ and $V$ emerge in the computation of the Jacobian. To compute the Jacobian, the authors decompose the state equations into two parts, one corresponding to new infections and the other corresponding to 'everything else':

$\displaystyle{x^{\prime}_{i} = \hat{G}_i(x) - \hat{V}_i(x)}$

If the state has the form $x = (x_1, \cdots, x_m, x_{m+1}, \cdots, x_n)$, where only $x_1$ through $x_m$ admit infection, then the Jacobian breaks into blocks:

$\displaystyle{ J \hat{G} = \left( \begin{array}{cc} F + V & 0 \\ J_3 & J_4 \end{array} \right)}$

Although this decomposition is not unique, in many cases it is apparently intuitive, and yields satisfying results in the sense that the corresponding $R_0$ is actually a functional threshhold parameter (i.e. you get one kind of behavior for $R_0 >1$, another for $R_0 < 1$.)

To Do:

1. Sections 1-4 are transparent, but section 5 contains a bunch of stuff having to do with "center manifolds". This is apparently standard fare in dynamical systems--have a look at the relevant text, see if you can make head or tail of it.

2. Starting with the examples, see if you can calculate the corresponding number $R_0$ without looking at the derivations in the paper.

3. Can you get your hands on some data and try estimating this number? If so, it would be interesting to compare the estimates to those obtained from other methodologies.

The effects of averaging on the basic reproduction ratio

Authors: F. Adler (1992).

Beautiful paper written by Adler for his thesis at Cornell under the direction of Simon Levin. Ultimately it is a bit of relatively simple matrix analysis. The paper has one theorem, which basically says that, under appropriate modeling assumptions, the largest eigenvalue of one matrix is bigger than that of another.

To Do:

1. Understand the proof on purely mathematical terms (i.e. as a statement about two matrices that live in some relation to one another.)

2. Push the model in some way: allow not just group combinations, but other sorts of distortions (though see next paper.) Joe's ideas: allow differences in susceptibilities, or heterogeneities in the $\gamma$.

3. This paper answers a question about invasability. Is the concept well defined? Can the same techniques be used to answer other questions, perhaps not at a DFE?

4. When do you get equality?

5. Is it possible to get the wrong qualitative answer under aggregation? (i.e. suppose under aggregation $R_0 < 1$, while in fact $R_0 >1$.)

Optimal spatial management of renewable resources: matching policy scope to ecosystem scale

Authors: J. Sanchirico and J. Wilen, 2004.


This paper applies the formal apparatus of optimal control theory to a particular open access fisheries model. The optimization is to maximize rents, i.e. the problem is being solved from an economic viewpoint. The objective functional has the form:

$ \max_{\tau_i, \pi_i} \int_0^{\infty} e^{-\delta t} \sum_i R_i(E_i,x_i) dt$

where $R_i$ represents the rent from spatial patch $i$ with fish population $x_i$ and effort level $E_i$. The regulatory mechanisms are taxes, in this case: $\tau_i$ and $\pi_i$ represent landing and shipping taxes, respectively. The constraint equations have the form

$ x_i^{\prime} = f_i(x_i)x_i + ND_i(x_1, x_2, \cdots , x_n) - h_i(E_i,x_i)$

and

$ E_i^{\prime} = s_i R(E_i,x_i, \tau_i, \pi_i) + \sum s_{ij}[R_i(E_i, x_i, \tau_i, \pi_i) - R_j(E_j, x_j, \tau_j, \pi_j)]$,

where the rent function is

$ R(x_i, E_i, \tau_i, \pi_i) = [(p_i - \tau_i) h_i(x_i, E_i) - (c_i + \pi_i) E_i]$,

with $p_i$ the ex-vessel price and $c_i$ the per-unit effort cost. The problem is linear in the control variables, and thus leads to bang-bang control.

From here I get lost, unfortunately. There is a lot of text, a few sketches, a long discussion, but I miss the main point.

Triple benefits from spatial resource management

Authors: M. Neubert and G. Herrera, 2008.

Considers the same model as Neubert (2003), i.e.

$ u_t(t,x) = ru(t,x)(1-u(t,x)/k) + d\frac{\partial^2 u(t,x)}{\partial x^2} - qE(t,x)u(t,x) $

but instead of approaching the problem from the perspective of a single owner, i.e. maximizing yield

$\int_0^L h(x)u(x)dx$,

this paper maximizes rents, which happen to be modeled as

$ \int_0^L [pqu(x) - (w_0 + w_1E(x))]E(x) dx $.

(Note the odd cost structure. Idea: quadratic captures the fact that high effort density means more fishermen are tripping over one another's lines.)

Solutions are numeric and graphical, but show several things:

1. Effort concentrated near boundaries, where population levels are low.

2. At rent-maximizing equilibrium, stock levels and effort levels are often both higher than at open access equilibrium.

3. Conditions that support this phenomenon basically boil down to having heterogeneity and misaligned incentives (i.e. harvestors want one thing, regulators another.)


Questions/TODO

1. Are these numerics any stabler than the one for Neubert (2003)? I'll want to try to reproduce the graphics.

2. Can some of the corresponding patch or age-structured models be easily worked out? Worth taking a look.

Marine reserves and optimal harvesting

Author: M. Neubert, 2003.

Elegantly reasoned and written, this paper looks at a one-dimensional fishery whose total fish-density is given by a PDE with Dirichlet boundary conditions, diffusion and growth terms, and a harvest term. Under this model, the paper looks at equilibrium harvest strategies that maximize total revenue, i.e.

$ y(u,h) = \int h(x) u(x) dx $

Using Pontryagin's principle, finding the optimal control $h$ is reduced to solving a two-point boundary value problem, which the authors do without comment (though my own attempts to do the same resulted in resounding failure (on the level of convergence, not comment.)) The results show that there are more and more "reserves" (i.e. regions in which $h$ is 0) as the domain expands. Indeed, as it passes some critical threshhold, there is a region of "singular control" which is entered and exited by "chattering control", i.e. the control goes to 0 an infinite number of times in a finite interval.

Questons/Items of Interest:

1. The numerical problem of how to solve for this control on a computer. Apparently, a lot is known. Is there any freeware that can solve this class of control problems?

2. There is no formula for how the number of zones increases as a function of domain size. It should be something like $C/(L - x)$, I suppose, where C is a scaling constant, $L$ is the critical domain size, and $x$ is the size of some particular domain. Is such a formula even a remote hope? Perhaps easier: is it even the case that the number of reserves goes to infinity as x --> L, or is there a jump from finite to infite?

3. Is there a stochastic variant to this formulation?

Near real-time spatial management based on habitat predictions for a longline bycactch species

Authors: A.J. Hobday and K. Hartmann, 2006.

This paper is a nice example of a theoretical model backed up by data. The data in question is actually rather difficult to collect: in an attempt to figure out the temperature preferences of a certain beleaguered species of tuna, scientists tag them with little thermometer-widgets that take and archive ambient temperature profiles. After a year with the fish, the devices detach themselves and float to the surface, where they transmit a summary of their data to certain scientific satellites in polar orbit. This temperature data is combined with satellite estimates of ocean surface temperature, estimates that are in turn plugged into an oceanic model that gives a vertical temperature profile based on the surface estimates. The thrust of this paper is that if you can figure out what temperature the fish like, and if you can figure out where that temperature obtains, you can tell fishermen to avoid that area and thus minimize the odds of infelicitous by-catch.

The paper uses a fair bit of level-headed but elementary statistical hocus-pocus. For example, the oceanographic model gives temperature at each point on a 3-D grid. To figure out whether this temperature at this depth represents an attractive environment for these tuna, the authors look at all the tag data and try to figure out which percentage of observations correspond to tuna hanging out at that depth but in a cooler temperature. (It is not clear to me if the tag data can be sorted by location; I don't think so, but if not, then there is nothing location specific about this statistic: it becomes a function of depth alone. Since the authors claim that the numbers relate to location as well, I may be missing something.) They do this for all the (discretized) depths at a given local, and then assess the overall attraction of the whole water column at that location by taking a weighted sum of these figures, where the weights correspond to percentages of time spent at that depth. (I confess that I don't really understand the data structure, so I'm not sure if this technique makes sense.)

Mathematically, I think their procedure is as follows: the data consist of $K$ sets of histograms. Two of these histograms are average temperature vs. depth and percent time vs. depth; the third is percent time vs. temp. I believe the third histogram is essentially ignored. The data from the first two are ordered pairs of the form $(d_{i},t_{ki})$ and $(d_{i}, p_{ki})$, respectively, where $i=1\cdots D$ indexes the depth and $k=1\cdots K$ indexes the observation. Given a temperature $t$ and a depth $d_i$, the authors calculate a statistic

$s(t,d_i) = \frac{1}{K}\sum_{k=1}^K n(k,t,d_i)$,

where $n(k,t,d_i)$ is equal to 0 if $t_{ki} > t$ and equal to 1 otherwise. Note that calculating this statistic involves looking at the $i$th component of each histogram, and can be done at depth intervals no finer than those reported by the PAT. Now, given a water column, discretized with depths $d_i$ and corresponding temperatures $t_i,$ the authors compute the statistics $s(t_i,d_i)$ and then use these to form

$\hat{s} := \sum_i w_i s(t_i,d_i)$

where the $w_i$ are given by

$w_i := \frac{1}{K}\sum_k p_{ki}$.

Note that $w_i$ represents the average fraction of time spent at depth $d_i$. The statistic $\hat{s}$ is thus supposed to be a measure of how desirable a water column might be: if its value is high, it means that for highly-frequented depths (big $w_I$) the temperature the hot enough that fish tend to seek out cooler spaces (i.e. $s(t_i,d_i)$ is also big.)

The rest of the paper deals with ways to automate the placement of management boundaries. There are statistical tests showing that the automation algorithm does relatively well, though it is not clear to me what "truth" might be.

Questions:

1.) What is the grid point spacing on the oceanographic model?
2.) Do the tags provide any location information?
3.) This "fish predilection" model is a touch ad-hoc (as admitted by the authors.) Is there a way to improve it? Add predation, use location information, etc.?
4.) How stable are the profiles generated by the PAT data?

The economic value of ecological stability

Authors: by Armsworth and Roughgarden, 2003.

The paper considers a toy system consisting of a wooded area $A$ of which a portion $R$ is to be into a reserve and the remainder is to be clear-cut and the wood sold on the market. The question is how large to make the reserve, and the answer, predictably, depends on the variability that characterizes the population dynamics.

To put this idea on mathematical terms, the authors consider a fecundity function of the form $f(N_t, R, a)$, where $a$ is a parameter that characterizes variability. Under discounting factor $\delta$, the optimal reserve size satisfies

$\delta L = \frac{\partial E[\pi]}{\partial R}$,

where $\pi$ is the in-situ value of the birds and $E$ is the expectation operator. By the chain rule, this can be decomposed as

$\delta L = \frac{\partial E}{\partial N} \frac{dN}{dR} + \frac{\partial E}{\partial \sigma^2} \left(\frac{\partial \sigma^2}{\partial \lambda} \frac{d \lambda}{dR} + \frac{\partial \sigma^2}{\partial \nu} \frac{d \nu}{dR}\right)$

where $N$ is the equilibrium population size, $\sigma^2$ is the population variance, $\lambda$ is the stability level of the equilibrium, and $\nu$ is the sensitivity of the population to environmental conditions. This paper supposes that per capita fecundity decreases linearly with population density, i.e. $latex f$ has the form

$f = aN(1-N/(RK))$,

and that the revenue function decreases with each additional bird, i.e.

$\pi = p(1 - e^{-\alpha N})$.

Under these assumptions, the authors show that, for appropriate parameter choices, increased variability results in decreased expected in-situ revenues, but that the optimal reserve size can actually be greater than that for a constant environment. Matlab simulation code can be found here.