Ebola in the Hog Sector: Modeling Pandemic Emergence in Commodity Livestock

Abstract

Commodity agriculture represents an expanding sink for a growing array of zoonotic pathogens. The emergence of novel strains of Ebola by way of economically driven shifts in husbandry and horticulture appears one such transition. Following up experimental studies of Ebola transmission, the agroeconomic origins of the Zaire ebolavirus outbreak in West Africa, and reports of endemic Reston ebolavirus in commercial hog in the Philippines and China, we develop a series of stochastic models that explicitly integrate epidemiology, spatial dynamics, and economics. Our inductive modeling suggests repeated punctuated emergence and human spillover of foodborne pathogens are intrinsic to industrial systems of production. In contrast to traditional and conservation agroecologies, by its accelerated and geographically expansive production of genetically uniform seed and stock, highly capitalized agriculture appears especially vulnerable to sudden shifts in disease evolution and spread. Industrial food production strips out environmental stochasticity that can cap pathogen population growth. The mechanisms for such explosive epidemiologies appear fundamentally founded in economic policy and practice. A variant of the Black–Scholes pricing model implies that pathogen propagation in intensive agrifood production outpaces the margins the sector allocates to biocontrol and containment across large expanses of the model’s parameter space. The resulting financial gaps appear met by externalizing the epidemiological costs of industrial food production to livestock morbidity, contract producers, farmworker and consumer health, smallholder markets, local wildlife, off-site environments, and government budgets across administrative units. By way of the models’ results, we hypothesize that as the hog sector expands for export, including across areas of Africa in which Ebola has already emerged as a human infection, multiple Ebola strains will follow Reston’s trajectory, evolving novel phenotypes in livestock and repeatedly spilling over into human populations.

Mathematical Appendix

2.1.1 Epidemic Prevention Farming

The models above focus primarily on explosive epidemic outbreaks and their containment costs, incorporating as well the influence of sudden Levy jumps. In general, however, endemic levels of infection, fluctuating about some mean, would be expected, and the central question then surrounds the transition between endemic and epidemic modes.

Khasminskii’s (1966/2006, 2012, Theorem 4.1) version of Eq. (2.1) provides insight, using a linear first approximation to some complicated, multidimensional, cross-influence function expanded about a quasi-stable equilibrium point. This gives the system of stochastic equations

$$ \displaystyle\begin{array}{rcl} dx_{t}^{i} =\sum _{ j=1}^{l}b_{ i}^{j}x_{ t}^{j}dt +\sum _{ r=1}^{n}\sum _{ j=1}^{l}\sigma _{ i,r}^{j}x_{ t}^{j}dW_{ t}^{r}\quad i = 1,\ldots,l& &{}\end{array}$$

(2.21)

where dW _t ^r is white noise and the b and σ terms are constants.

Khasminskii defines two associated matrices,

$$\displaystyle{a_{i,j}(x) =\sum _{ k,s=1}^{l}\sum _{ r=1}^{n}\sigma _{ i,r}^{k}\sigma _{ j,r}^{s}x^{k}x^{s},\,\,\,\mathbf{B} = \vert \vert b_{ i}^{j}\vert \vert }$$

under the condition that, for A, the inner product condition

$$\displaystyle{ (\mathbf{A}(x)\alpha,\alpha ) \geq m\vert x\vert ^{2}\vert \alpha \vert ^{2} }$$

(2.22)

always holds.

Khasminskii invokes two new variates, λ = x∕ | x | on the unit sphere, and ρ = log[ | x | ], expanding d ρ _t using the Ito chain rule to obtain

$$\displaystyle\begin{array}{rcl} d\rho _{t}& =& \left [(\mathbf{B}\lambda _{t},\lambda _{t}) + \frac{1} {2}\sum _{i=1}^{l}a_{ i,i}(\lambda _{t}) -\sum _{i,j=1}^{l}a_{ i,j}(\lambda _{t})\lambda _{t}^{i}\lambda _{ t}^{j}\right ]dt \\ & & +\sum _{r}(\sigma (r)\lambda _{t},\lambda _{t})dW_{t}^{r} {}\end{array}$$

(2.23)

where σ(r) =  | | σ _i, r ^j | | , i, j = 1, …, l.

Define

$$\displaystyle\begin{array}{rcl} Q(\lambda )& =& (\mathbf{B}\lambda,\lambda ) + \frac{1} {2}\sum _{i=1}^{l}a_{ i,i}(\lambda ) -\sum _{i,j=1}^{l}a_{ i,j}(\lambda )\lambda ^{i}\lambda ^{j} {}\\ J& =& \int Q(\lambda )d\lambda {}\\ \end{array}$$

where the integral is taken over the unit sphere. (Khasminskii 1966/2006, 2012, Theorem 4.1) shows that, if J < 0, the complex stochastic process converges to an endemic equilibrium distribution. If J > 0, then the probability that | x _t  | → ∞ as t → ∞ is 1.

Thus, for any given cross-influence matrix B, there is a set of structures defined by the matrix A—under the condition of Eq. (2.22)—that will contain a pathogen outbreak to endemic levels. Conversely, given an endemic distribution, sufficient alteration of either the structural matrix B or of the “noise” matrix A would trigger an epidemic outbreak | x _t  | → ∞.

Extension of this result involving jump processes can be found in Khasminskii et al. (2007).

2.1.2 Large Deviations

Something similar to Eq. (2.14) can be simply derived via a standard large deviations argument.

Following Dembo and Zeitouni (1998), let X ₁, X ₂, … X _n be a sequence of independent, standard Normal, real-valued random variables and let

$$\displaystyle{ S_{n} = \frac{1} {n}\sum _{j=1}^{n}X_{ j} }$$

(2.24)

Since S _n is again a Normal random variable with zero mean and variance 1∕n, for all δ > 0

$$\displaystyle{ \lim _{n\rightarrow \infty }P(\vert S_{n}\vert \geq \delta ) = 0 }$$

(2.25)

where P is the probability that the absolute value of S _n is greater or equal to δ. Some manipulation, however, gives

$$\displaystyle{ P(\vert S_{n}\vert \geq \delta ) = 1 - \frac{1} {\sqrt{2}\pi }\int _{-\delta \sqrt{n}}^{\delta \sqrt{n}}\exp (-x^{2}/2)dx }$$

(2.26)

so that

$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\log P(\vert S_{n}\vert \geq \delta )} {n} = -\delta ^{2}/2 }$$

(2.27)

This can be rewritten for large n as

$$\displaystyle{ P(\vert S_{n}\vert \geq \delta ) \approx \exp (-n\delta ^{2}/2) }$$

(2.28)

That is, for large n, the probability of a large deviation in S _n follows something much like the asymptotic equipartition relation of the Shannon–McMillan Theorem.

This result can be generalized to more complicated probability spaces using Sanov’s Theorem, the Gartner–Ellis Theorem, and related developments (Dembo and Zeitouni 1998) to show that large deviations paths of length n all have approximately the probability

$$\displaystyle{ P(n) \propto \exp (-n\mathcal{H}[\mathbf{X}]) }$$

(2.29)

where $\mathcal{H}$ is of the form − ∑ _i P _i log(P _i ) for some probability distribution. Under the conditions of our analysis, P(n) is the probability of an excursion from the absorbing state of n = zero infections. $\mathcal{H}$ thus quantifies an information source representing the active imposition of control strategies to prevent a large-scale outbreak of infection, and the Black–Scholes cost analysis carries through.