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Research
Model Parameters and Outbreak
Control for SARS
Gerardo Chowell,*†
Carlos Castillo-Chavez,‡ Paul W. Fenimore,* Christopher M. Kribs-Zaleta,§
Leon Arriola,* and James M. Hyman*
*Los Alamos National Laboratory, Los Alamos, New Mexico, USA; †Cornell
University, Ithaca, New York, USA; ‡Arizona State University, Tempe, Arizona,
USA; and §University of Texas at Arlington, Arlington, Texas, USA
Appendix
Local Sensitivity Analysis of the Basic Reproductive Number
The sensitivity analysis approach through an exhaustive sampling of the
parameter space provides a global measure of the sensitivity of model
parameters. Another approach is to compute the sensitivity indices of
the model parameters through local derivatives (1).
This approach only provides a local measure as the sensitivity indices
can change when the parameter values change. Here, we use local sensitivity
analysis to corroborate our global sensitivity analysis results and discuss
how this approach can be applied in the analysis of cost as part of a
policy of outbreak control.
Let λ represent any of the 10 nonnegative parameters, β,
ρ, p, q, k, γ1, γ 2, δ, α,
and l, that define the basic reproductive number of our model (2)
(1)
If a "small" perturbation δλ is made to the parameter
λ, a corresponding change will occur in R0 as
δR0, where
The normalized sensitivity index Ψλ is the ratio
of the corresponding normalized changes and is defined as
(2)
An approximation of the perturbed value of R0, in terms
of the sensitivity index is
where the 10 normalized sensitivity indexes are
with η = p(1–ρ)+ρ and γ2 = a γ1/(
α –γ1). For the values of the parameters used in
this model, the sensitivity indices Ψβ, Ψρ,
Ψp, Ψq, and Ψl are
positive, Ψk = –Ψq, and the remaining indexes
are negative. Furthermore, since all of the indexes (except Ψβ)
are functions of the parameters, the sensitivity indexes will change as
the parameter values change.
For our specific case where β = 0.25, q = 0.1, p = 1/3, k = 0.15707,
α = 0.2061, γ1 = 0.035285, γ2
= 0.0426, δ = 0.0279, and ρ = 0.77 and Toronto (l = 0.1)
or Hong Kong (l = 0.43), the normalized sensitivity indices are
computed. The sensitivity indices and the associated percentage changes
needed to affect a 1% decrease in R0 are given in Tables
1 and 2. Since the effective rate of patient
isolation and the average rate of diagnosis are two feasible intervention
strategies, we examine how changes to the parameters l and α
affect R0.
Let us first consider the outbreak in Hong Kong. The value α =
0.2061 means that the mean time to diagnose an infected person's illness
is approximately 4.85 days. The sensitivity index Ψα
= –0.1933 means that a 5.2% increase in α, which in turn requires
a decrease of 5.7 hours of mean time to diagnosis, would result in a decrease
of approximately 1% in R0. Similarly, the sensitivity
index Ψl = 0.5183 suggests that a 1.9% decrease
in the value of l, that is going from 0.43 to 0.42 isolation effectiveness,1
results in a 1% decrease in R0. In other words, individually
a 5.2% increase in a or a 1.9% decrease in l both result in approximately
a 1% decrease in R0. For the particular values of the
parameters chosen for Hong Kong, the most effective way to reduce R0
is to decrease the transmission rate β and the parameter l
(improve the effective isolation rate). In the case of Toronto, Ψα
= –0.4758 means that a 2.1% increase in α, results in a 1% decrease
in R0, whereas Ψl = 0.2001 means
a 5% decrease in l also results in a 1% decrease in R0.
As can be seen from these two examples, the importance or ranking of
the sensitivity indices can change as the values of the parameters change.
Specifically, the sensitivity indices Ψl and Ψα
satisfy the relationship
(3)
For the particular values of the parameters given above, the Appendix
Figure shows the level curve for the pair (l,α), where
l(α+2γ1+2δ) = δ+γ2.
The particular parameter values are either for Toronto (l,α)
= (0.1, 0.2064) or for Hong Kong (l,α) = (0.43, 0.2064).
Choosing the parameter values (l,α) below the level
curve means that ||Ψl|| < ||Ψα||and
the converse is true if (l,α) is chosen above the
curve. Along the level curve, the magnitude of the sensitivities is equal.
Notice that the level curve divides the parameter space into two regions,
each of area Aabove and Abelow, respectively. Since
Aabove >> Abelow, Ψl
will be the dominant sensitivity index for randomly chosen (l,α).
One aspect of implementing an efficient intervention policy is the fact
that limited resources are available. If one assumes, for example, that
the strategies of isolation and diagnosis have associated 1% incremental
costs in implementation of δCI and δCD,
respectively, then a mixed strategy should be formulated that maximizes
the effectiveness of a combined intervention. Specifically, if x
denotes the magnitude of percentage decrease in l, and y
denotes the % increase in a and assuming that there is a maximum amount
of total additional resources available (δCT), then the
total additional cost of a new mixed isolation and diagnosis intervention
policy must satisfy the inequality δCIx+δCDy
< δCT. Since the objective is to maximize the
decrease in R0, this means we want to maximize the objective
function P = ||Ψl||x+||Ψα||y
under the appropriate constraints. In a more general setting, additional
nonlinear constraints could be involved, which case would require one
to solve a nonlinear optimization problem. The situation in which the
cost of diagnosis of infected persons may be much greater than the cost
of isolation or vice versa is certainly of interest.
References
- Caswell H. Matrix population models. 2nd ed. Sunderland
(MA): Sinauer Associates, Inc. Publishers; 2001.
- Chowell G, Fenimore PW, Castillo-Garsow MA, Castillo-Chavez C. SARS
outbreaks in Ontario, Hong Kong, and Singapore: the role of diagnosis
and isolation as a control mechanism. J Theor Biol. 2003;224:1–8.
| Appendix
Table 1. Sensitivity indices for Toronto with l = 0.1 |
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Positive sensitivity indices
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Negative sensitivity indices
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Ψβ = 1
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–1%
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Ψα = –0.4758
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2.10%
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Ψr = 0.6063
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–1.65%
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Ψδ = –0.1707
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5.86%
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Ψl = 0.2001
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–4.99%
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Ψγ2
= –0.1208
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8.28%
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Ψγ = 0.1172
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–8.53%
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Ψk = –0.1172
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8.53%
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Ψp = 0.0906
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–11.04%
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Ψγ1
= –0.1156
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8.65%
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| Appendix
Table 2. Sensitivity indices for Hong Kong with l = 0.43 |
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Positive sensitivity indices
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Negative sensitivity indices
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Ψβ = 1
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–1%
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Ψγ2
= –0.3129
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3.19%
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Ψρ = 0.6063
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–1.65%
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Ψδ = –0.3016
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3.32%
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Ψl = 0.5183
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–1.93%
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Ψα = –0.1933
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5.17%
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Ψp = 0.0906
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–11.04%
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Ψγ1
= –0.1216
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8.22%
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Ψq = 0.0706
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–14.16%
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Ψk = –0.0706
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14.16%
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1Recall that l = 0 corresponds to
complete isolation, whereas l = 1 means no effective isolation
occurs. Hence, a decrease in l means an increase in the effective
isolation of the infected persons.
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