Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission

doi:10.1016/j.jtbi.2009.11.001

Journal of Theoretical Biology

Volume 262, Issue 4, 21 February 2010, Pages 757-763

https://doi.org/10.1016/j.jtbi.2009.11.001 Get rights and content

Abstract

There is growing interest in incorporating economic factors into epidemiological models in order to identify optimal strategies for disease control when resources are limited. In this paper we consider how to optimize the control of a pathogen that is capable of infecting multiple hosts with different rates of transmission within and between species. Our objective is to find control strategies that maximize the discounted number of healthy individuals. We consider two classes of host–pathogen system, comprising two host species and a common pathogen, one with asymmetrical and the other with symmetrical transmission rates, applicable to a wide range of SI (susceptible–infected) epidemics of plant and animal pathogens. We motivate the analyses with an example of sudden oak death in California coastal forests, caused by Phytophthora ramorum, in communities dominated by bay laurel (Umbellularia californica) and tanoak (Lithocarpus densiflorus). We show for the asymmetric case that it is optimal to give priority in treating disease to the more infectious species, and to treat the other species only when there are resources left over. For the symmetric case, we show that although a switching strategy is an optimum, in which preference is first given to the species with the lower level of susceptibles and then to the species with the higher level of susceptibles, a simpler strategy that favors treatment of infected hosts for the more susceptible species is a robust alternative for practical application when the optimal switching time is unknown. Finally, since transmission rates are notoriously difficult to estimate, we analyze the robustness of the strategies when the true state with respect to symmetry or otherwise is unknown but one or other is assumed.

Introduction

Many plant and animal pathogens can infect more than one host species. Amongst diseases of contemporary interest that can spread in this way, are sudden oak death in plant communities (Rizzo et al., 2002, Rizzo and Garbelotto, 2003), foot and mouth disease (Ferguson et al., 2001, Keeling et al., 2001), blue-tongue virus in livestock (Bethan et al., 2005), and bird influenza that can spread between wild and domestic birds with a risk to humans (Alexander, 2000). The spread of disease in each case can be viewed as a series of coupled epidemics on sub-populations of each species, with occasional, or sometimes frequent, transmission of infection between species. It follows that targeting control of infection (and hence disease) on one of the host species influences the infection pressure and disease risk for the other species. Host species may differ in susceptibility to the pathogen, in amenability and cost of control, or in intrinsic value. The question naturally arises of how best to deploy resources for disease control, especially when resources are limited. Here, we address the problem using a combination of economic control theory (Goldman and Lightwood, 2002, Rowthorn, 2006, Forster and Gilligan, 2007) in combination with a metapopulation framework for disease dynamics (Hanski, 1998, Park et al., 2003). The metapopulation framework is a convenient devise to separate each host species into a sub-population with coupled epidemics between sub-populations. We motivate the problem for the control of sudden oak death caused by the Oomycete, Phytophthora ramorum, a fungal-like organism, that is mainly transmitted by rain splash. The pathogen has a wide host range and is currently spreading rapidly through coastal regions of California (Rizzo et al., 2002, Rizzo et al., 2005). Here we focus initially on spread through communities dominated by two major host species with asymmetrical transmission between bay laurel (Umbellularia californica) and tanaok (Lithocarpus densiflorus) (Maloney et al., 2005).

Rowthorn et al. (2009) recently analyzed optimal strategies for the deployment of disease control on a single host species comprising two or more spatially separated sub-populations in which infected individuals recover and can be reinfected. This form of SIS (Susceptible–Infected–Susceptible) model is commonly used to describe some sexually transmitted diseases such as gonorrhea. Rowthorn et al. (2009) revealed a counter-intuitive result for the SIS type of disease in which giving preference to the sub-population with the higher level of infection is the worst strategy for diseases of the SIS form. The optimal strategy instead involved giving preference to the species with the lower level of infected individuals and hence the higher level of susceptibles. What should happen in an SI (Susceptible–Infected) system typical of sudden oak death and many other plant and animal diseases hosts in which hosts do not recover but new susceptible hosts arise by reproduction and transmission rates within species differ? To answer this, we consider whether or not preference should be given to treating the species with the higher transmission rate, and show that it is possible to identify an analytical solution for the optimal strategy. Many epidemics, including those of sudden oak death, involve cryptic spread of infection from hosts that are infected prior to spread. Additional realism is conferred to the analysis by the introduction of a threshold for detection of visible symptoms of disease.

The differential transmission rates for P. ramorum for bay laurel and tanoak, and several other species (Meentemeyer et al., 2004), can be inferred from simple pathology experiments. The relative transmission rates may not be known in advance for some emerging epidemics prior to the implementation of control strategies. Accordingly, we generalize the methods for asymmetric transmission to consider a two-species model for SI epidemics with births of susceptibles, in which there is symmetrical transmission of infection between host species. The additional analysis serves two purposes. Firstly, it provides a lower bound to the (zero) difference between transmission rates from which it is possible to test the consistency of the results for asymmetrical and symmetrical transmission. Secondly, it allows a test of the robustness of the optimal control strategies when transmission rates are erroneously assumed to be asymmetric when in fact they are not, and vice versa.

Section snippets

The model

We consider a community comprising two susceptible species with a pathogen that can infect both hosts. The model is motivated and parameterized for the spread of sudden oak death through mixed species stands of bay laurel and tanoak, in which there is asymmetrical transmission of infection between species. Parameterization (Table 1) was derived from Meentemeyer et al. (2004). The model is then generalized to consider what happens when there is symmetrical infection between the two host species.

Asymmetric case

When there are more detected individuals than can be culled, a systematic analysis shows that there are only three candidates for optimality:

1.
‘High $β$ strategy’: Give priority to the species with the higher transmission rate.
2.
‘Low $β$ strategy’: Give priority to the species with the lower transmission rate.
3.
‘Switch high to low $β$ strategy’: A single switch from giving priority to species with the higher transmission rate to giving priority to the species with the lower transmission rate.

Numerical

Conclusions and discussion

We have used a simple SI two-species model with vital host dynamics to describe an epidemic spreading through a mixed two-species stand of host plants that are susceptible to a common pathogen. For simplicity, the vital dynamics of the species are described by a simple monomolecular-type function to limit the total population size. Our results, however, hold for more complex functions such as a logistic function and incorporating Lotka-Volterra competition to describe the intrinsic host

Acknowledgments

This work was supported by a Gates Cambridge Scholarship (MN-M) and a BBSRC (Biotechnology and Biological Research Council) Professorial Fellowship (CAG) which we gratefully acknowledge.

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View full text

Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission

Abstract

Introduction

Section snippets

The model

Asymmetric case

Conclusions and discussion

Acknowledgments

Veterinary Microbiology

Mathematical Biosciences

Forest Ecology and Management

Preventive Veterinary Medicine

Mathematical Biosciences

Population biology of infectious disease

Nature

Optimal control of deterministic epidemics

Optimal Control Application and Methods

Climate change and the recent emergence of bluetongue in Europe

Nature Reviews Microbiology

Pathogen-induced reversal of native dominance in a grassland community

Proceedings of the National Academy of Science USA

Investment Under Uncertainty

An economic interpretation of optimal control theory

The American Economic Review

The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions

Science

Optimizing the control of disease infestations at the landscape scale

Proceedings of the National Academy of Science USA

Cost optimization in the SIS model of infectious disease with treatment

Topics in Economic Analysis and Policy