key: cord-0967301-y4wv5e5d authors: Bussell, Elliott H.; Cunniffe, Nik J. title: Optimal strategies to protect a sub-population at risk due to an established epidemic date: 2021-09-11 journal: bioRxiv DOI: 10.1101/2021.09.10.459742 sha: 79fb14646c87033de0a08efc1e9954fae764982b doc_id: 967301 cord_uid: y4wv5e5d Epidemics can particularly threaten certain sub-populations. For example, for SARS-CoV-2, the elderly are often preferentially protected. For diseases of plants and animals, certain sub-populations can drive mitigation because they are intrinsically more valuable for ecological, economic, socio-cultural or political reasons. Here we use optimal control theory to identify strategies to optimally protect a “high value” sub-population when there is a limited budget and epidemiological uncertainty. We use protection of the Redwood National Park in California in the face of the large ongoing state-wide epidemic of sudden oak death (caused by Phytophthora ramorum) as a case study. We concentrate on whether control should be focused entirely within the National Park itself, or whether treatment of the growing epidemic in the surrounding “buffer region” can instead be more profitable. We find that, depending on rates of infection and the size of the ongoing epidemic, focusing control on the high value region is often optimal. However, priority should sometimes switch from the buffer region to the high value region only as the local outbreak grows. We characterise how the timing of any switch depends on epidemiological and logistic parameters, and test robustness to systematic misspecification of these factors due to imperfect prior knowledge. Management of emerging infectious disease is most likely to be successful when it starts 28 as soon as possible. Smaller epidemics are easier and less expensive to control than the 29 larger epidemics which would result if management were to be delayed (Althaus et al., 30 on the agenda. In 2001 a spatially-distinct epidemic was identified in Curry County, Ore-83 gon, and since then over $20 million USD has been spent on identifying and treating 84 that localised epidemic (Grünwald et al., 2019) . There are also other areas ahead of the 85 main epidemic front under active management, most notably a pair of relatively large 86 outbreaks in Humboldt County, in California ( Figure 1a ; the southern border of boldt County is at a latitude of around 40 N). Although further south than the outbreak 88 in Oregon, the outbreaks in Humboldt county are some distance "ahead" of the main 89 bulk of the epidemic (Alexander and Lee, 2010). 90 Disease management in Humboldt County exemplifies the challenges now posed by 91 the control of sudden oak death. The goal is to design a management scheme that can 92 effectively achieve a smaller, more local, objective than complete elimination or eradica-93 tion. This could be, for example, slowing local rates of disease spread or protection of 94 valuable resources (Cobb et al., 2013) . This must be done when there is a limited budget 95 available for control (Cunniffe et al., 2016) . We focus here on strategies which reduce 96 impacts of disease on a particular "high value" region, within which it is important to mit-97 igate the effects of disease for ecological, economic, socio-cultural or political reasons. 98 The particular example we select to motivate our analysis is based upon the Redwood we deliberately phrase our mathematical model to omit many system specific details, al-129 lowing our results to illustrate epidemiological principles important whenever a subset 130 of host individuals must be protected in the face of a large, ongoing epidemic. In particular we use optimal control theory to understand how time-dependent dis-132 ease management can be optimised in order to protect a high value region at risk from 133 a growing, spreading epidemic. We account for economic and logistical limitations in Epidemiological model 148 We split the host landscape into three disjoint regions: a generally infested area in which 149 the disease is already well-established, a buffer region in which the disease might per-150 haps be present, but has not yet become established, and a high value region that is 151 currently entirely uninfected and that must be protected. In the context of Redwood Creek and sudden oak death, the generally infested area would correspond to the large region. This simplification is valid when considering new, relatively isolated outbreaks of 160 disease: a common situation for diseases such as sudden oak death that are spread over 161 long distances through rare long-distance dispersal events (Meentemeyer et al., 2011) . 162 A test of the robustness of our results to this assumption is in Appendix 1. The buffer and high value regions are modelled as well-mixed patches, meaning the 164 only spatial component in our model is between-patch coupling. We use a ( )usceptible- rogued, is assumed to be . This gives the following optimal control problem: characterising the broad features of optimal controls, and how these features are con-235 ditioned on parameter values. We therefore identify a plausible, although arbitrary, pa-236 rameterisation of our model (Table 1) , and use it to drive our analysis. 237 We follow previous modelling work targetting the sudden oak death system (Meen- For our default parameter set (Table 1) , the optimal strategy prioritises control of infec- value region from within itself becomes greater than that from outside. This is exempli-292 fied by Figure 3 (e), which shows that even for this parameterisation, the force of infection 293 upon the high value region from itself remains smaller than that from the buffer region 294 throughout the epidemic, despite the switch in focus of the optimal control at ≈ 6 units 295 of time. Further switches, i.e. multiple changes in focus between the two regions, can occur 297 whilst budgets are not limiting. In these cases since the budget is not limiting, control can 298 be maximal in both regions and so the switch has no effect on the control performed. Figure 3 . An alternative optimal strategy initially focuses management upon the buffer region, but switches to concentrate roguing within the high value region as the epidemic takes hold. The optimal strategy as found for a model parameterisation slightly altered from the default in Table 1 , with a smaller infection rate ( = 0.0028 host −1 t −1 ) and smaller maximum budget ( = 5 hosts). For the range of parameters we considered here, we do not see strategies with multiple 300 switches which impact the control performed. The optimal strategy depends on epidemiological and logistic param- Table 1 , although with only a single initially infected host in the buffer region ( (0) = 1) and using a longer timescale of interest ( = 20) to allow patterns in switching times to be more clearly seen. strategies with additional switches are found; these are the grey regions in Figure 5b . As 346 described already, these switches occur whilst the budget is not limiting, and so have no 347 effect on the control realised. To illustrate the full range of possibilities for the optimal control strategy, we focus 349 on three particular cases taken from Figure 5b . Case A finds a switch but as the budget 350 is not limiting, it has no effect. In case B the budget is limiting so the switch has an effect. In case C there is an additional switch early in the epidemic, but since the budget is not 352 limiting at that point it has no effect. in (b) ). (c) shows the wasted control resources using an infection rate overestimated by 20 %. Here too much control is allocated to the high value region that cannot be spent, and so is wasted. Similarly, not enough resources are allocated when the infection rate is underestimated, also leading to wasted control resources. Landscape scale control of sudden oak death in California has not been possible for some 370 years (Cunniffe et al., 2016) . However highly valuable sub-populations at risk due to the 371 epidemic -such as commercially valuable timber stocks, or areas important for tourism 372 or cultural reasons, or for ecological reasons to conserve biodiversity -might yet be pro-373 tected. Slowing spread to these regions is itself valuable (Cobb et al., 2013) . An obvious 374 question is then how this goal can best be achieved. This requires partitioning limited (Craig et al., 2018) , but must nevertheless work together (Laranjeira et al., 2020) . 380 We used the outbreak of sudden oak death in Humboldt County, California to mo-381 tivate our work. After developing a simple mathematical model representing disease 382 within the "high value" region of the Redwood National Park and in a "buffer" region 383 surrounding it, we used optimal control theory to find the most effective time-varying 384 allocation of a limited budget for disease management between these two regions. To 385 minimise the final amount of infection in the National Park, we found it is very often bet-386 ter to prioritise exclusively that area for treatment ( Figure 2 ). However, while it is clearly 387 very intuitive, we have identified that this strategy is not always optimal. It can instead 388 be better to start by prioritising disease control in the buffer region, and only to switch 389 to prioritising the high value region later in the epidemic (Figure 3) . We have found that 390 such a "switching" strategy is most likely to be optimal for intermediate values of the 391 infection rate (Figure 3) . For the simple model we considered here, the optimal strategies could in fact proba-410 bly have been identified by an exhaustive scan over switching times. However for more 411 complex models with more complex switching strategies this will not generally be true. Whilst the setting was highly simplified, we can already begin to see some of the poten- for example by adding a term penalising rapidly changing controls (Clarke et al., 2013) . 418 Direct use of optimal control theory also facilitated the extensive scans over alter-419 native parameterisations of our model (Figures 3 and 5) , allowing our relatively intuitive 420 characterisation of a number of outcomes based on which region is prioritised. Although 421 some work in optimal control of plant disease does translate strategies for practical ap-422 plication, and consider the impact of parameters (e.g. (Hamelin et al., 2021) ), more often 426 We have also shown how imprecision in knowledge of parameters controlling disease 427 spread can lead to less effective disease management ( Figure 6 ). Indeed in such cases 428 it may even be better to use the simplest possible strategy rather than use optimal con-429 trol theory at all. to suffer much worse outcomes following infection (Palmer et al., 2021) . 451 We reiterate this paper is a purely illustrative analysis, intended to highlight the epi- (Rowthorn et al., 2009) Controlling established invaders: integrating economics and 555 spread dynamics to determine optimal management Controlling established invaders: integrating economics and 558 spread dynamics to determine optimal management Optimising reactive disease management using spatially explicit 561 models at the landscape scale Optimizing the Control of Disease Infestations at the Landscape 564 Scale Factors that make an infectious disease outbreak 567 controllable The Life, Death, and Rebirth of a Perfect Tree SODmap and SODmap Mobile: Two Tools to Monitor the 571 Spread of Sudden Oak Death Citrus canker and citrus huanglongbing, two exotic bacterial diseases threatening 573 the citrus industries of the Western Hemisphere. 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Science, policy and Dutch Elm Disease in the UK Novel approaches to SOD manage-693 ment in California wildlands: a case study of "eradication" and collaboration in Redwood Valley Proceedings of the Sudden Oak Death Fifth Science Symposium Direct and indirect methods for trajectory optimization On the implementation of an interior-point filter line-search algorithm 698 for large-scale nonlinear programming Poppy 701 GM, Waage J. Infectious diseases of animals and plants: an interdisciplinary approach Under control prioritising the high value region, and using a smaller infection We conclude that slowly growing epidemics in the generally infested region, or at least growth that has little impact on the force of infection in the buffer region, is likely to have little impact on our results Effect of a growing epidemic in the generally infested region 706 Our model assumes a constant external force of infection on the buffer region (the parameter  in Equation 1 of the main text). This can be generalised to instead represent a growing epidemic in the generally infested region via an exponentially increasing formin which 0 corresponds to the initial size of the external epidemic. Figure 2 . When the external epidemic is more significant, the approximation is slightly less precise. We consider a case in which there is a smaller within-region infection rate ( =0.003 host −1 t −1 rather than =0.005 host −1 t −1 ) and when control is being done in the high value region. Since the dynamics of the external forcing are then more important in relative terms, there are visible differences between the results of the two models. However the differences in fact remain relatively small over the full time course of the epidemic.