Neuraminidase inhibitors — is it time to call it a day?

Stockpiling neuraminidase inhibitors (NAIs) such as oseltamivir and zanamivir is part of a global effort to be prepared for an influenza pandemic. However, the contribution of NAIs for treatment and prevention of influenza and its complications is largely debatable. Here, we developed a transparent mathematical modelling setting to analyse the impact of NAIs on influenza disease at within-host and population level. Analytical and simulation results indicate that even assuming unrealistically high efficacies for NAIs, drug intake starting on the onset of symptoms has a negligible effect on an individual's viral load and symptoms score. Increasing NAIs doses does not provide a better outcome as is generally believed. Considering Tamiflu's pandemic regimen for prophylaxis, different multiscale simulation scenarios reveal modest reductions in epidemic size despite high investments in stockpiling. Our results question the use of NAIs in general to treat influenza as well as the respective stockpiling by regulatory authorities.


Introduction
reduce the risk of illness in close contacts of infected persons (Nguyen- Van-Tam et al., 2014). 48 From the available evidence, it can be seen that NAIs require a strict and narrow time window for 49 small treatment effects to be achieved and, in order to have prophylactic effects, healthy individuals 50 need to take the medicine daily for a long period. This is undoubtedly debatable. The review of 51 Jefferson et al. (2014) has suggested that no clinical trials provide concrete evidence for patients, 52 clinicians or policy-makers to use NAIs in annual and pandemic influenza. Furthermore, prophy-53 lactic use was also questionable because virus culture was not performed on all trial participants. 54 Therefore, it is not clear whether this is because participants were not infected or because they had 55 an asymptomatic infection Jefferson and Doshi (2014). 56 Here, we attempt to clarify these claims both mathematically and computationally. Using a 57 within-host infection model of influenza infection, we evaluate the effectiveness of NAIs in reducing 58 viral load and symptom severity as a function of the initiation time of post-infection treatment. 59 Furthermore, using a contact network model of epidemics, we assess the prophylactic effects of 60 NAIs in a population, and discuss treatment strategies with a focus on the cost and availability of 61 the drugs. Our numerical analysis employs oseltamivir as a case study; however, our results and 62 their implications are applicable to NAIs in general. 63

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NAIs are unlikely to attain high efficacies even in a best-case scenario 65 Assuming an idealistic case scenario of instantaneous absorption of NAIs by a treated host as described in Materials and Methods-Eq. (12), the quasi-steady states of the drug concentration are given by (details in Appendix 1) (2) In other words, the drug concentration stabilises to well-defined values after a few doses: an upper bound and a lower bound , respectively (Appendix 1). For a given drug and a given treatment regimen, the value of represents a best-case scenario for the therapy. That is, the simplified system defined by Eqs. (7)-(11), supplemented with This relationship is illustrated in Fig. 1 A clearer picture may be obtained by expressing the equation above as a relationship between the drug parameters themselves, fixing the form of the therapy instead. This removes the dependence of the landscape on the half-maximal concentration of a particular drug, and helps shed light on the behaviour of different NAIs. This relationship reads 50 = 1 − * * 0 1 − − .
(6) Figure 2 shows the landscape of peak efficacies for a given drug, assuming that the therapy follows 77 either the curative or the pandemic regimen of oseltamivir.
Here, for the pandemic regimen, we find that in order to achieve a peak efficacy of at least * = 0.95, a drug with, e.g., a unit elimination 79 rate ( = 1 days −1 ) would need to have a half-maximal concentration no larger than ca. 20 mg. In 80 the curative regimen, in turn, this upper limit goes down to about 10 mg. If we take again the range 81 of reported 50 -values for oseltamivir as an example in this hypothetical case of a drug that is 82 slowly cleared, we see that the great majority of these concentrations fall below a peak efficacy of 83 * = 0.95-over 90% if we assume they are uniformly distributed.

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Even with high efficacy, NAIs effects are negligible in practical settings 85 The top panels of Figure 3 show the fraction of reduction in the viral load and symptoms AUC in the best-case scenario ( ( ) = * ), which we denote by , with where AUC T corresponds to the case with treatment and AUC 0 to the base case without treatment. 86 Results are shown for different starting times of the therapy, 0 , in the curative regimen of oseltamivir. 87 The bottom panels of  In order to avoid artificial growth of the virus late after infection, simulations with a viral load below a prescribed tolerance-in this case, 10 −3 TCID 50 mL −1 -one day post infection are assumed stop growing, i.e., the right hand side of Eq. (10) becomes − from day 1 onwards for these cases. In all cases, the treatment corresponds to the curative regimen for oseltamivir: 0 = 75 mg, = 0.5 days. 89 Simulations show that even for very large values of * , there exists an optimal starting time for 90 the therapy, and we can appreciate that a late start of the treatment has little to no effect on the 91 dynamics of the infection. In a real-life scenario, it is highly unlikely that the therapy will start during 92 the optimal time window: treatment would not start without symptoms, and seasonal influenza  The densities are calculated with non-parametric density estimation using the Sheather-Jones method (Sheather and Jones, 1991). Note that the epidemic size distribution in scenario 8 is bimodal. 105 The right panel of Fig. 4 shows that that there were reductions in epidemic size, but in many 106 cases the reduction was small given the investment. In scenario 7, for example, 30 million dollars 107 were spent over 6 weeks but the epidemic size reduction was lower than 20% compared to the 108 case without intervention (scenario 1). Tripling the allocated resources (scenario 16) brought the 109 epidemic size further down only by ca. 25%. 110 In addition, the right panel of Fig. 4 also shows that there was a trade-off between coverage 111 and duration. Generally, prolonging the duration while keeping a low coverage was not efficient 112 (scenarios 5-7, 10-12, and 14-16). However, a very high coverage with a short duration (scenario best-case scenario for the therapy. 128 Our analytical results suggest that, even if we assume that the drug reaches its peak concentra-129 tion at the time of initiation of the treatment, and remains at this constant value from that moment 130 onwards, it is unlikely that an extremely large reduction in viral replication rate will be achieved 131 under realistic conditions. In order for this to occur for typical dosages and intake frequencies, the affected countries, selection of prevention strategies need be well-informed for cost-effectiveness. 156 We found that there was a trade-off between duration and coverage and it seemed that prolonging 157 coverage over three weeks is not cost-effective. A best-case scenario corresponds to providing the 158 drug for more than two weeks with as high a coverage as possible. However, here we observed some in this case can be very difficult to asses. 169 It is important to remark that, by turning to a description based on the peak efficacy for a given 170 treatment regimen, we have rendered the analysis essentially independent of the drug parameters 171 in the sense that only the actual value of the peak efficacy from Eq. (6) will depend on them, but 172 their functional relationship and the landscape represented in Fig. 2 will not. Therefore, while we 173 have carried out our analysis using oseltamivir as a case study, we expect our results to hold for 174 NAIs in general. We also stress that all of the analysis above has considered the best possible case, 175 and that the impact of the therapy in a practical scenario will be lower than observed here due Here, we extend the model further to take into account the effects of treatment with NAIs. The full system of equations is given bẏ The system considers a population of target (epithelial) cells, divided into susceptible ( ), infected  (Canini et al., 2014). 203 Here, we essentially equate the concentration of the latter with . Since we are interested in 204 finding an upper bound for constant-concentration efficacy, this simplified model is sufficient for 205 our analysis. 206 The parameter values for Eqs. (7)-(11) are taken directly from Lukens et al. (2014). We start 207 by focusing on the effects of therapy with oseltamivir, and therefore we consider an elimination 208 rate = 3.26 days −1 (Wattanagoon et al., 2009) (Canini et al., 2014). All parameters are listed in Table 1. The Python code for this 215 section can be visited at systemsmedicine/neuraminidase-inhibitors. Table 1. Parameter values of the within-host model, Eqs. (7)-(12) (Lukens et al., 2014). The initial conditions correspond to a completely susceptible target cell population, i.e., 0 = 1 and an inoculum size 0 = 7.5 × 10 −6 TCID 50 mL −1 .    We can find the quasi-steady states illustrated in Appendix 1- Fig. 1 by noting that, when we start the therapy, ( − 0 ) = 0, ( + 0 ) = 0 . After a time , the initial value of will have decayed by an amount equal to − , so that the drug concentration just before and just after the second administration will be given by where we have used that the interval between intakes is constant and equal to . Repeating this reasoning a few more times we find that, when we reach time , the drug concentration has the form The summations in the expressions above have a closed form when → ∞, provided that > 0 (Abramowitz and Stegun, 1965), which is naturally the case since both parameters are positive. Therefore, we can obtain the quasi-steady state values as A similar approach can be used to obtain the values to which the peaks and valleys of the drug concentration converge after a long time has passed for the more complicated model