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For an infectious disease such as Covid-19, we present a new four-stage vaccination model (un-vaccinated, dose 1+2, booster, repeated boosters), which examines the impact of vaccination coverage, vaccination rate, generation interval, control reproduction number, vaccine efficacies, and rates of waning immunity, upon the dynamics of infection. We derive a single equation that allows computation of equilibrium prevalence and incidence of infection, given knowledge about these parameter and variable values. Based upon a 20 compartment model, we develop a numerical simulation of the associated differential equations. The model is not a forecasting or even predictive one, given the uncertainty about several biological parameter values. Rather, it is intended to aid qualitative understanding of how equilibrium levels of infection may be impacted upon, by the parameters of the system. We examine one at a time sensitivity analysis around a base case scenario. The key finding which should be of interest to policy makers, is that while factors such as improved vaccine efficacy, increased vaccination rates, lower waning rates, and more stringent non-pharmaceutical interventions might be thought to improve equilibrium levels of infection, this might only be done to good effect, if vaccination coverage on a recurrent basis, is sufficiently high.

In this paper, for an infectious disease such as Covid-19, we present a SIR model which examines the impact of waning immunity, vaccination rates, vaccine efficacy, and the proportion of the susceptible population who aspire to be vaccinated. Under an assumed constant control reproduction number, we provide simple conditions for the disease to be eliminated, and conversely for it to exhibit the more likely endemic behaviour. With regard to Covid-19, it is shown that if the control reproduction number is set to the basic reproduction number (say 6) of the dominant delta (B1.617.2) variant, vaccination alone, even under the most optimistic of assumptions about vaccine efficacy and high vaccine coverage, is very unlikely to lead to elimination of the disease. The model is not intended to be predictive but more an aid to understanding the relative importance of various biological and control parameters. For example, from a long-term perspective, it may be found that in the UK, through changes in societal behaviour (such as mask use, ventilation, and level of homeworking), without formal government interventions such as on-off lockdowns, the control reproduction number can still be maintained at a level significantly below the basic reproduction number. Even so, our simulations show that endemic behaviour ensues. The model obtains equilibrium values of the state variables such as the infection prevalence and mortality rate under various scenarios.

This paper relates to data from the Wellcome Sanger Institute, UK, regarding Covid-19 genomic surveillance. We use a simple model to give point estimates of the effective reproduction numbers of the B.1.617.2 and B.1.1.7 lineages in England, from sequenced data as at 15 May 2021. Comparison with the estimated reproduction number of B.1.1.7 enables an estimate of the increased transmissibility of B.1.617.2. We conclude that it is almost certain that there is increased transmissibility that will rapidly lead to B.1.617.2 becoming the prevailing variant in the UK. The derived estimates of increased transmissibility have uncertainty relating to the actual distribution of the generation interval, but they do point, under present conditions of vaccination coverage and NPIs, to exponential growth of positive cases.

This paper considers SEPIR, an extension of the well-known SEIR continuous simulation compartment model. Both models can be fitted to real data as they include parameters that can be estimated from the data. SEPIR deploys an additional pre-symptomatic (also called asymptomatic) infectious stage not in SEIR but known to exist in COVID-19. This stage can also be fitted to data. We focus on how to fit SEPIR to a first wave. Both SEPIR and the existing SEIR model assume a homogeneous mixing population with parameters fixed. Moreover neither includes dynamically varying control strategies deployed against the virus. If either model is to represent more than just a single wave of the epidemic, then the parameters of the model would have to be time dependent. In view of this we also consider how reproduction numbers can be calculated to investigate the longer term overall result of an epidemic. © 2021 SW 2021. All rights reserved.

In this paper I extend the work of Bernhardt and Donnelly (2019) dealing with modern explicit tontines, as a way of providing income under a specified bequest motive, from a defined contribution pension pot. A key feature of the present paper is that it relaxes the assumption of fixed proportions invested in tontine and bequest accounts. In making the bequest proportion an additional control function I obtain, hitherto unavailable, closed-form solutions for the fractional consumption rate, wealth, bequest amount, and bequest proportion under a constant relative risk averse utility. I show that the optimal bequest proportion is the product of the optimum fractional consumption rate and an exponentiated bequest parameter. I show that under certain circumstances, such as a very high bequest motive, a life-cycle utility maximisation strategy will necessitate negative mortality credits analogous to a member paying life insurance premiums. Typical scenarios are explored using UK Office of National Statistics life tables., 21 pages, 6 figures

We suggest corrections to the paper by Kanabar and Simmons and provide analyses for deriving the internal rate of return arising from a UK state pension deferral decision and for an actuarially fair deferral scheme.

In this paper I examine the sensitivity of total UK Covid-19 deaths and the demand for intensive care and ward beds, to the timing and duration of suppression periods during a 500- day period. This is achieved via a SEIR model. Using an expected latent period of 4.5 days and infectious period of 3.8 days, R 0 was first estimated as 3.18 using observed death rates under unmitigated spread and then under the effects of the total lockdown ( R 0 =0.60) beginning 23 March. The case fatality rate given infection is taken as 1%. Parameter values for mean length of stay and conditional probability of death for ICU and non-ICU hospital admissions are guided by Ferguson et al. (2020). Under unmitigated spread the model predicts around 600,000 deaths in the UK. Starting with one exposed person at time zero and a suppression consistent with an R 0 of 0.60 on day 72, the model predicts around 39,000 deaths for a first wave, but this reduces to around 11,000 if the intervention takes place one week earlier. If the initial suppression were in place until day 200 and then relaxed to an R 0 of 1.5 between days 200 and 300, to be followed by a return to an R 0 of 0.60, the model predicts around 43,000 deaths. This would increase to around 64,000 if the release from the first suppression takes place 20 days earlier. The results indicate the extreme sensitivity to timing and the consequences of even small delays to suppression and premature relaxation of such measures.

Persons who are eligible for a defined benefit social security pension may defer their pension and receive, through accruals, an extra pension or possibly a lump sum, on termination of deferral. In certain cases, partners of the deferrer may inherit such benefits. For such a scheme, the concept of actuarial fairness to a category of pensioners is defined. A scheme that is actuarially fair will not be cost neutral to the pension provider unless the discount rate is the same for both parties. In addition to this asymmetry, adverse selection will impact upon both actuarial fairness and cost to the provider. Expressions are derived for the cost penalty to the provider for attempting to achieve actuarial fairness both with and without an acknowledgement of adverse selection. Similarly, when the objective is to achieve cost neutrality for the provider, expressions for the cost to the deferrer are obtained. Some numerical examples are given in the case of the UK state pension scheme. Under present rules it is shown that there are significant departures from actuarial fairness, particularly for those who achieved state pension age before 6 April 2016 and those with partners eligible to inherit benefits. Even when a pension provider attempts to achieve either actuarial fairness or cost neutrality, if adverse selection is ignored, then significant departures from both are quite possible.

Good things are meant to come to those who wait. But does the old adage apply to UK residents who defer their state pension? John Dagpunar weighs the pros and cons ahead of a rule change next year Is it worthwhile? John Dagpunar weighs the pros and cons ahead of a UK rule change next year

John Dagpunar introduces a probability distribution to model real-valued positive measurements, such as time spent queuing for a taxi or waiting for a hospital bed

The high cost of maintenance in the processing industry implies the need for optimal planning of maintenance strategy. In order to achieve this there is a need to understand the underlying failure processes, which are often very complex. In this paper, a new semi-parametric approach, combining Cox regression with density kernal smoothing, is introduced to estimate the underlying performance. The approach has been applied to several processes and it allowed insight into each process, which would not have been achieved if traditional approaches had been used. Particularly, the refurbishment of processes had a significant impact on the rate failure. This paper concludes by assessing this impact of refurbishment on the maintenance programme.

In developing any successful maintenance strategy it is important to be able to estimate the rate of occurrence of maintenance events. In a previous paper the authors have argued that there are advantages to using a non-parametric or semi-parametric approach to such analysis. In this paper the authors explore the use of the approach to data from the water industry using a case study. The approach provides insight into the context revealing the impact of maintenance on the process. The effect of the maintenance strategy can be evaluated. The paper explores the practical issues of missing values and implementation of an appropriate maintenance strategy.

Changes to the financial framework regulating housing associations require them to create a sinking fund that will accrue the necessary cash for future maintenance work to their stock. This paper presents an alternative to the conventional methods by using a Linear Programming model to obtain sinking fund strategies. The relevance of such modelling is discussed with reference to the evolving Sinking Fund system in the UK and European Housing Association movements. The methodology is applied to data from a Housing Association in Scotland. The paper concludes that the LP model offers a degree of flexibility not found in existing practice. The work has relevance to the long-term maintenance management of any built asset.

We consider a general repair process where the virtual age V i after the ith repair is given by V i = O(V i-1 + X i ), O(.) is a specified repair functional, and X i is the time between the (i - 1)th and ith repair. Some monotonicity and dominance properties are derived, and an equilibrium process is considered. A computational method for evaluating the expected number/density of repairs is described together with an approximation method for obtaining some parameters of the equilibrium process.

We consider a component which undergoes instantaneous general repairs on failure. Suppose that V i = Φ(V i-l + Y i ), where V i is the virtual age immediately after the ith failure, Y i is the time between the (i - 1)th and ith failures, Φ(.) is a specified repair functional, and V 0 =s. We derive integral equations for the repair density, and also for the joint density of repairs with respect to chronological age and virtual age on failure. In the asymptotic case, approximations are obtained for the mean and variance of virtual age on failure, virtual age after repair, and time between failures. Two policies are then considered. The first, an upgraded repair strategy employs minimal repairs until the component reaches a specified age, thereafter repairs restore the virtual age to that prescribed level. The second policy considered is the Kijima Type 2 model which is not amenable to the usual g-renewal analysis that is used in the Type 1 model. The repair density is obtained by numerical solution of the relevant integral equation. Finally, approximations for the asymptotic moments are found to be in close agreement with simulated results.

Since the Housing Act 1988, UK housing associations (HAs) have been responsible, with the removal of government grants, for funding all maintenance necessary for the upkeep of new stock over its lifespan. Sinking funds (SFs) have to be created, out of rental income, that are adequate to finance the long term projected maintenance demand. This paper reviews the use of SFs for building asset management with particular reference to the requirements being made of HAs. Current assumptions of what constitutes an adequate level of SF investment by their funding agencies are questioned with an analysis of case study data. Using linear-programming, the benefits of modelling an SF over merely calculating it are demonstrated. The technique is used to formulate SF strategies, some of which admit the possibility of going into deficit in the future, and these are compared with results from conventional SF projections. Finally, the effect that varying element and component lifespans can have on SF projections are invest...

A component has time to failure X with density f(·). Opportunities arise as a Poisson process of rate λ. At an opportunity the component may at choice, be replaced at cost c2. At failure the components can either be minimally repaired at cost c1 or replaced at cost c4. Finally the component may be replaced at any time at an ‘interrupt’ cost of c3. We derive expressions for the long-run expected cost rate, and give examples of numerical optimisation with respect to control limits on age at an opportunity, and at an interrupt replacement.

A producer sells equipment with a product warranty of duration W. We determine the optimal number of imperfect preventative maintenances to be made, the timing of these, and the age reduction at each such event, assuming that minimal repairs are made at any failures between preventative maintenances.

This paper determines the optimal number of preventive maintenance actions to make over a warranty period under a free maintenance warranty. Each preventive maintenance action reduces the system age by a fixed amount and all system failures are minimally repaired. A numerical example is given for Weibull failure times to show the cost improvement obtained using the policy discussed here instead of a previously published periodic policy.

In this paper we consider an age replacement strategy, where downtimes are non-zero. Although this model is well known, the literature gives no necessary and sufficient conditions for age replacement to be preferred to replacement on failure. In this paper we derive such conditions in terms of the minimum of the mean residual life function. When age replacement is indicated, we derive sufficient conditions for the existence of a global minimum to the asymptotic expected cost rate function. These results are illustrated for the Weibull and Gamma distributions.

Availability measures are given for a repairable system under minimal repair with constant repair times. A new policy and an existing replacement policy for this type of system are discussed. Each involves replacement at the first failure after time T, with T representing total operating time in the existing model and total elapsed time (i.e. operating time + repair time) in the new model. Optimal values of T are found for both policies over a wide range of parameter values. These results indicate that the new and administratively easier policy produces only marginally smaller optimal availability values than the existing policy.

An algorithm is described for generating von Mises variates using Forsythe's method. The performance is analysed with respect to speed, and compactness. Conclusions are drawn as to when it is preferred to Best and Fisher's envelope rejection method.

In this paper we discuss the use of a new method of importance sampling in the pricing of European options on a basket of assets and on average price (Asian) options. In standard Monte Carlo estimation the expectation is taken with respect to a vector which is distributed as n variate normal, where n is the number of assets in the basket, or the number of points in the averaging computation in the case of an Asian option. A change of measure which results in sampling the ratio of arithmetic to geometric average payoffs is central to the importance sampling method employed. Since this ratio varies little, considerable variance reduction can be achieved, particularly when the call option is 'in the money'. Experimental results are given.

This paper examines the problem of lot-sizing in a production facility in the face of machine breakdowns. Failures are dealt with by minimal repairs, until the required lot has been produced, at which point a new cycle is set-up and the machine restored to “as-new” condition. Given the expected duration of a repair, and the duration of a set-up, optimal lot sizes are obtained for Weibull distributed times to failure. Empirical evidence suggests that for an exponential time to failure, the failure process can usually be safely ignored, whereas for machines with a sharply increasingly failure rate, this is not the case.

The traditional method of obtaining optimal claim limits for vehicle insurance is to discretise the state space and use successive approximations. In this paper we show how the stochastic dynamic programming equations reduce to a set of differential equations, in which these are easily solved to provide exact continuous time solutions. The resulting model can be used for evaluating alternative levels of excess.

Using the ‘ratio’ method an easily implemented algorithm is derived for the generalised inverse Gaussian distribution. Computer timings and efficiency calculations show that the procedure is fast over a wide range of distribution parameter values.

An algorithm is developed for the efficient generation of random variates from the tail of a t-distribution. This is specialised to the case of a Normal distribution. Theoretical measures of efficiency are derived. Computer timings for the algorithms are given, and, in the Normal case, compared with existing tail generation procedures.