North American Actuarial Journal
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Actuarial Applications of Epidemiological Models Runhuan Feng Ph.D., ASA & Jose Garrido Ph.D., ASA
To cite this article: Runhuan Feng Ph.D., ASA & Jose Garrido Ph.D., ASA (2011) Actuarial Applications of Epidemiological Models, North American Actuarial Journal, 15:1, 112-136, DOI: 10.1080/10920277.2011.10597612
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ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS
Runhuan Feng* and Jose Garrido†
ABSTRACT
The risk of a global avian flu or influenza A (H1N1) pandemic and the emergence of the worldwide SARS epidemic in 2002–2003 have led to a revived interest in the study of infectious diseases. Mathematical models have become important tools in analyzing the transmission dynamics and in measuring the effectiveness of controlling strategies. Research on infectious diseases in the actuarial literature goes only so far in setting up epidemiological models that better reflect the transmission dynamics. This paper attempts to build a bridge between epidemiological and ac- tuarial modeling and set up an actuarial model that provides financial arrangements to cover the expenses resulting from the medical treatments of infectious diseases.
Based on classical epidemiological compartment models, the first part of this paper proposes insurance policies and models to quantify the risk of infection and formulates financial arrange- ments, between an insurer and insureds, using actuarial methodology. For practical purposes, the second part employs a variety of numerical methods to calculate premiums and reserves. The last part illustrates the methods by designing insurance products for two well-known epidemics: the Great Plague in England and the SARS epidemic in Hong Kong.
1. INTRODUCTION
The Severe Acute Respiratory Syndrome (SARS) epidemic in 2002–2003 drew tremendous attention to the treatment and prevention of infectious diseases and their impact on our society’s welfare. The adverse economic impact caused by SARS in East Asia has often been compared with that of the 1998 financial market crisis in that area.
From a social point of view, an effective protection against diseases depends not only on the devel- opment of medical technology to identify viruses and to treat infected patients, but also on a well- designed health care system. The latter can reduce the financial impact of a sudden pandemic outbreak, such as surging costs of medications, hospital infrastructures and medical equipment, and prevention measures such as vaccination and quarantine. Broader insurance programs can even cover financial losses resulting from the interruption in regular business operations. As a profession with the reputa- tion of applying mathematical techniques to model and quantify financial risk, actuaries are certainly well placed to expand their expertise and deal with epidemics within health care systems.
Due to their front-line experience with the SARS epidemic, many health insurers in Asia provided coverage to compensate for medical costs of SARS treatment by listing the disease as an extended liability on regular health insurance policies. Still, many problems arose. Traditional actuarial models for human mortality lack the flexibility required to model infectious diseases, which in many respects are significantly different from natural causes of death.
* Runhuan Feng, Ph.D., ASA, is an Assistant Professor in the Department of Mathematical Sciences at the University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin, fengr@uwm.edu.
† Jose Garrido, Ph.D., ASA, is Professor in the Department of Mathematics and Statistics at Concordia University, 1455 de Maisonneuve W., Montreal, Quebec, Montreal, Quebec, Canada, garrido@mathstat.concordia.ca.
112
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 113
One of these remarkable differences is that in a population exposed to an epidemic outbreak several mutually dependent groups are involved, with different levels of vulnerability to the disease. This con- trasts with mortality rates that are often assumed to be constant among homogeneous age-specific groups.
How fast an infectious disease spreads within a population relies on the number of susceptible in- dividuals, the number of infectious individuals, and the social structure between these two groups. To be more specific, in the context of a health insurance for an initially complete susceptible group, the number of insureds bearing premiums would actually decrease in time, whereas the number of insureds claiming benefits due to infection increases as the epidemic breaks out. Applying traditional life table methods overlooks epidemiological dynamics and dependence between insurance payers and benefici- aries. It consequently violates the fair premium principle.
Other than the traditional life table methods, there have been developments in the recent actuarial literature on alternative multistate Markov models (MSMMs). These models do not center around a partition of the population into compartments and the evolution of these group sizes during the epi- demic. Instead, MSMMs put the focus on the insured individuals and their transitions between different states, for instance, being alive, disabled, or dead. MSMMs are well suited for traditional insurance products in which premium and benefit payments may differ over a long period according to the par- ticular status of the insured. For instance, in life insurance death benefits are payable upon the death of the insured, whereas annuities are payable as long as the insured stays alive. For more detailed accounts of MSMMs, interested readers can consult Waters (1984), Hoem (1988), or Jones (1994).
Although multistate models have their own merits in traditional insurance mathematics, we propose here an alternative approach in which actuarial calculations are based on epidemiological models. We believe that this approach presents advantages in modeling insurance coverage against infectious dis- eases over a short term:
1. Compartment models in the epidemiology are constructed in accordance with the law of mass action widely used in many areas of biology, chemistry, and physics. As alluded to earlier, the unique feature of the infectious disease, with its dynamics of transmission depending on the sizes and interactions of two or more subgroups of a population, can be reflected only in the utilization of the law of mass action.
2. There have been extensive studies and substantial empirical data analysis on the validity and param- eter estimations of compartment models. Actuaries have also been involved and gained expertise in fitting of these models to data, as seen in Jia and Tsui (2005). There is also a vast array of stochastic compartment models in the literature, such as Gathy and Lef`evre (2009), Lef`evre and Utev (1999), and Picard and Lef`evre (1990), which are extensions of deterministic compartment models and may provide additional tools to account for randomness in modeling epidemiological dynamics.
3. Epidemiological models can be used in sensitivity tests for prevention and intervention measures and hence can be used to analyze the impact on the financial obligations of insurance products for infectious diseases.
Therefore, we present in this paper a new approach that uses epidemiological models as building
blocks, and we develop a framework for actuarial calculations, with applications to insurance coverages targeting infectious diseases.
To make the paper self-contained, Section 2 is devoted to a brief review of a simple model from the mathematics of epidemiology, the three-compartment model. An insurance model is developed to make arrangement between an insurer and insurance policyholders in order to protect the insured from the potential financial burden resulting from infection by a disease.
To apply the ideas in an insurance context, Section 3 formulates the epidemiological model in stan- dard actuarial notation and analyzes the quantitative relations among some insurance concepts, namely, the actuarial present value of continuous payments for hospital and medical services, death benefits, and premium income.
114 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
In Section 4, several ratemaking methods are presented for various infectious disease insurance policies. Using an algorithm that calculates premiums under the fair premium principle, we then look at the solvency of these insurance plans.
Because benefit reserves generally reflect a policy’s cash value that is refundable to the policyholders, it is expected that the benefit reserves remain positive throughout the life of the policy. However, as we shall see, level net premiums lead to negative reserves, because of the distinct nature of infectious diseases. Therefore Section 5 analyzes the reasons behind the negative reserves and proposes a nu- merical method developed to determine adjusted premiums that keep benefit reserves from falling below a minimum tolerance level.
Based on models calibrated in the epidemiological literature, we analyze in Section 6 the dynamics of the Great Plague in Eyam, England, and that of the SARS epidemic in Hong Kong. This leads to insurance policy designs to cover the resulting financial losses. The analytical procedures could easily be adapted to enable an analysis of a wide range of scenarios.
2. EPIDEMIOLOGICAL COMPARTMENT MODEL
Over the last century, many contributions to the mathematical modeling of infectious diseases have been made by a great number of public health physicians, epidemiological mathematicians, and stat- isticians. Their brilliant work ranges from empirical data analysis to the theory of differential equations. Many have been applied successfully in clinical data analysis to make effective predictions. For a com- plete review of a variety of mathematical and statistical models, see Hethcote (2000) and Mollison et al. (1994). Building on the work of such pioneers, actuaries can add economical considerations to epidemiological models and design insurance policies that can provide financial means to protect the general public against the adverse economic impact of epidemics. For an account of existing cooper- ative opportunities for actuaries and epidemiologists, readers are referred to a report by Cornall et al. (2003). More recently epidemiological models have been used in applications to Entreprise Risk Man- agement (ERM; see, for instance, Chen and Cox 2009), and pandemic risks are frequently discussed in actuarial circles; see Stracke and Heinen (2006) for influenza, M ̈akinen (2009) for H1N1, and CIA (2009) for pandemic scenarios.
To illustrate the possible actuarial applications, we first look at a simple deterministic epidemiolog- ical model, which could lead to a straightforward actuarial analysis. Although most infectious diseases, such as SARS, are far too complex to fit such a three-compartment model, the generalization to mul- tidimensional models follows similar procedures. A detailed account of compartment models can be found in Anderson and May (1991), Brauer and Castillo-Ch ́avez (2001), Hethcote et al. (1981), and Lef`evre (2005).
In epidemiological models, the whole population is usually separated into compartments for different individuals. They are often labeled by acronyms, such as S, I, and R, in different patterns according to the transmission dynamics of the studied disease. Generally speaking, class S denotes the group of healthy individuals without immunity, or in other words, those who are susceptible to a certain disease or virus. In an environment exposed to the disease, some of these individuals come into contact with the virus. The individuals who are infected and able to transmit the disease are classified in class I. Through medical treatment, individuals, removed from the epidemic due to either death or recovery, are all counted in class R. The upper part of Figure 1 gives the transferring dynamics among the three compartments.
Another merit of this partition, from an actuarial perspective, is that the three compartments play significantly different roles in an insurance model. As shown in the lower part of Figure 1, susceptibles facing the risk of being infected in an epidemic form a market that could contribute premiums to an insurance fund, in return for the coverage for medical expenses incurred if infected. During the out- break of an epidemic, the infected policyholders would benefit from the claim payments provided by the insurance fund. Furthermore, following an insured individual’s death, a death benefit for funeral
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 115
Figure 1
Transmission and Insurance Dynamics among Compartments S, I, and R
and burial expenses may also be paid to the beneficiaries designated by the insured person. Once the insurance fund is set up, interest accrues at a certain rate on the unpaid reserves.
We first look at a simple mathematical (SIR) model that characterizes the interaction among the three compartments. Let S(t) denote the number of susceptible individuals at time t, whereas I(t) is the number of infected and R(t) the number of removed individuals from class I. According to the law of mass action, which is commonly used in chemistry and biological studies, the rate of change of reactions is proportional to the concentration of participants. In the context of epidemiology, the rate of increase in the number of the infected is hence proportional to the number of susceptible individuals and the number of individuals previously infected. Translated into mathematical language, the rate of change in the size of compartments can be interpreted as respective derivatives. Therefore, the evo- lution of compartment sizes is driven by the following system of ordinary differential equations (ODEs) known as the SIR model:
S(t) S(t)I(t)/N, t 0, I(t) S(t)I(t)/N I(t),
(2.1) (2.2) (2.3)
t 0,
with given initial values S(0) S0, I(0) I0, S0 I0 N, and constant rates [0,1], 0. The
R(t) I(t), t 0, model is based on the following assumptions:
1. The total number of individuals remains constant, N S(t) I(t) R(t), representing the total population size.
2. An average susceptible makes an average number of adequate contacts (i.e., contacts sufficient to transmit infection) with others per unit time.
3. At any time a fraction of the infected leave class I due to death, and is considered to be the fatality rate of the specific disease.
4. There is no entry into or departure from the population, except possibly through death from the disease. For our purpose of setting up an insurance model, the demographic factors such as natural births and deaths are negligible, as the timescale of an epidemic is generally shorter than the demographic timescale.
Because the probability of a random contact by an infected person with a susceptible individual is
S/N, then the instantaneous increase of new infected individuals is (S/N)I SI/N. The third as- sumption implies that the instantaneous rate of death in the number of infected individuals is propor- tional to the current size of compartment R.
116 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
3. ACTUARIAL ANALYSIS
The idea of insurance coverage against the financial impact due to infectious diseases is akin to that of coverage for other contingencies, such as accidental death and destruction of properties. Yet it is distinctive in nature from property and casualty insurance, because the number of policyholders bearing the premiums and the number of policyholders eligible for compensations vary over time throughout the epidemic.
Because mortality analysis is based on ratios instead of absolute counts, we now introduce the de- terministic functions s(t), i(t), and r(t), interpreted, respectively, as the fractions of the population in each of class S, I, and R. Dividing equations (2.1)–(2.3) by the constant total population size N yields
s(t) i(t)s(t), t 0,
i(t) i(t)s(t) i(t), t 0,
r(t)1s(t)i(t), t0,
(3.1) (3.2) (3.3)
wheres(0)s0 andi(0)i0,giventhats0 i0 1.
One can interpret the ratio functions s(t), i(t), and r(t) as the probability of an individual being
susceptible, infected, or removed from infected class, respectively, at the time t. However, it should be noted that because of the law of mass action, movements between the compartments depend on the relative sizes of one another. Thus these probabilities correspond to mutually dependent risks for the SIR model, as opposed to the usual independent hazards in multiple-decrement life insurance models. With these probability density functions s(t), i(t), and r(t), we now incorporate actuarial methods to formulate the quantities of interest for an infectious disease insurance.
3.1 Annuity Premium and Annuity Benefit
We assume that an infectious disease insurance plan collects premiums in the form of continuous annuities from the susceptibles. In other words, policyholders are committed to paying premiums con- tinuously as long as they remain healthy and susceptible. Meanwhile, medical expenses are continuously reimbursed for each infected policyholder during the whole period of treatment. Once the individual dies from the disease, the plan terminates immediately.
Following the principles of International Actuarial Notation, we denote the actuarial present value (APV) of premium payments from an insured person for a t-year period by a ̄s with the superscript
t
indicating payments from class S. The APV of benefits paid by the insurer to the infected at the rate
of one monetary unit per time unit is denoted by a ̄i with the superscript indicating payments to t
class I.
We shall use the current payment technique to evaluate the annuities. In other words, we identify
the present value of payments due at time t, which is the discounted value of one monetary unit for a basic annuity, multiply by the probability of making such a payment, and then integrate these actuarial present values for all payment times t. A detailed account of evaluations of annuities can be found in Bowers et al. (1997).
Hence, on the insurance liability side, the total discounted value of a t-year annuity of benefit pay- ments is given by
t
a ̄i exi(x)dx, (3.4)
t
0
where 0 is the discounting force of interest. On the revenue side, the total discounted value of a t-year annuity premium of payments is
t
a ̄s exs(x)dx. (3.5)
t
0
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 117
Our study is based on the fundamental notion of the Equivalence Principle for the determination of level premiums, which requires that
[present value of benefit outgo] [present value of premium income]. Therefore, the level premium for a unit annuity claim payment plan is determined by
As in life insurance, where the force of mortality is defined as the additive inverse of the ratio of the derivative of the survival function to the survival function itself, we define here the force of infection (leaving class S) as
a ̄i P ̄(a ̄i) t.
(3.6)
t a ̄s t
s s(t)
t s(t), t0,
and the force of removal (leaving class I) as
i i(t)
Consequently from (3.1)–(3.2), we see that s i(t) and i s(t) . tt
Note that the above definitions imply that
and
s(t) exp s
i(t) exp i
t i(t), t0.
t t
dx exp x
i(x)dx , t0,
(3.7)
(3.8)
00
t t
dx exp x
s(x)dxt , t0.
00
For mathematical convenience, we shall first analyze the policy with an infinite term. When the policy term is relatively long, the premium based on an infinite term may serve as a rough estimation of the cost of the insurance.
Proposition 3.1
In the SIR model in (3.1)–(3.2),
1a ̄i a ̄s 1. (3.9)
Note that the right-hand side of (3.9) represents the present value of a unit perpetual annuity. The
intuitive interpretation of the left-hand side is that, if each insured in the whole insured population is
provided with a unit perpetual annuity, the APV of payments to class S is given by a ̄s , and the APV of
payments to class I is given by a ̄s . Recall that at any time a fraction of the infected moves to class
R. To be fair, each one in class R is also entitled to a perpetuity worth a value of 1/ at the time of transition.HencetheAPVofpaymentstothegroupleavingclassIisgivenby(/)a ̄i .Itisreasonable
that all three types of annuity payments should add up to the present value of a unit perpetual annuity
1/ paid regardless of which compartment a policyholder lies in.
From relation (3.9), we easily find the net level premium in (3.6) for a policy of an infinite term
with both premium and claim annuity payments given by the formula
a ̄i a ̄i ̄i
P(a ̄)a ̄s 1()a ̄i . (3.10)
118 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
3.2 Annuity Premium and Lump-Sum Benefit
An insurance plan that pays a lump-sum compensation would be analogous to a whole life insurance
in actuarial mathematics. When an insured person is diagnosed with the infectious disease and im-
mediately hospitalized, the medical expenses are to be paid immediately in a lump sum, and the in-
surance plan terminates as its obligation is fulfilled. Then the APV of benefit payments, denoted by
̄i
A, is given by
e s(t)i(t) dt, since the probability of being newly infected at time t is s(t)i(t).
Proposition 3.2
In the SIR model in (3.1)–(3.2),
and
1 ̄is1
A a ̄ s0,
11 ̄iii i0 A a ̄ a ̄.
̄i t
A 0
(3.11)
(3.12)
(3.13)
Equation (3.12) also provides insight into the breakdown of expenses in each class. Assuming that
every susceptible individual who initially enters the policy claims a unit perpetual annuity, then the
APV of the total cost is s0/. From a different perspective, it is equivalent to give every one a unit of
annuity as long as the insured remains healthy in the group and then grant each a unit perpetuity
immediately as they become infected. The APV of these two types of payments is exactly given by
̄is (1/)A a ̄.
If one thinks of class I as a transit stage, then we can count the costs of payments on both the
incoming and outgoing sources. Assume that every one currently or previously in class I receives a unit
of perpetuity. From the incoming sources, the left-hand side of (3.13) gives the expenses for initial
̄i
members i0 / and the expenses for those who just entered the class (1/)A. For the outgoing sources,
the costs for individuals who continue to stay in the class is given by a ̄i , and those deceased
are compensated with a perpetuity worth 1/. Thus the right-hand side sums up to (/)a ̄i a ̄i . ̄ ̄i
Therefore the net level premium P(A) for the plan of an infinite term insurance with lump-sum compensation and annuity premium payments is given by the equivalence principle:
3.3 Death Benefit
Note that in the epidemiological literature the class R is composed of all individuals removed chron- ologically from a previous compartment, who either recover with immunity or die from the disease. A more refined model would have separate compartments for deaths and recovered individuals. For our purpose of investigating actuarial implications of the epidemiological model, we keep the simple as- sumption of only one R compartment exclusively for deaths caused by the disease.
Health insurance plans often have death benefits that differ in value from health care benefits. In this infectious disease plan, we assume a death benefit of a monetary unit paid immediately at the
̄d
moment of death. Thus, the APV of a lump-sum death benefit payment, denoted by A, is given by
̄ii
A ()a ̄i0
̄ ̄i
P(A)a ̄s 1()a ̄i .
̄d t i
e i ( t ) d t a ̄ . A 0
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 119
Therefore, the net level premium for the plan of an infinite term with both a unit lump-sum death benefit and health care claim is obtained by
i ̄di
̄i ̄d P(a ̄ A)
a ̄ A a ̄s
(1 )a ̄ 1()a ̄i .
Finally, the net level premium for a plan with both a lump-sum medical care benefit and a lump-sum
death benefit is given by
4. RATEMAKING
̄i ̄di
A A P(A A) a ̄s
()a ̄ i0 1()a ̄i .
̄ ̄i ̄d
So far net premiums have only been expressed in terms of a ̄i , which is a Laplace transform of i(t). An
implicit integral solution to the SIR model in (3.1)–(3.2) is as follows:
N 0 exp N 0 s(x)dxu du , t0,
s(t)1expt u
i(t) 1 exp t u
N 0 exp N 0 i(x)dx udu , t0.
No general explicit solution is available for s(t) and i(t). Therefore we propose numerical methods and
approximations that can provide satisfactory solutions for insurance applications. The estimation of i(t)
enables us to compute a ̄i , which in turn gives a ̄s via the relation between a ̄i and a ̄s .
In addition the proposed techniques are extended to the more realistic finite term policy. These numerical methods generally apply to the calculations of both the infinite and finite term policy.
4.1 Infection Table–Based Approximation
In practice it is difficult to keep record of susceptible individuals, partly because of their large numbers in a population and partly because of the difficulty in distinguishing a person susceptible to a certain disease from one with immunity. But we can keep track of infected people using public data from government health agencies and hospitals. Hence we rely on the function i(t), instead of s(t), for all premium-rating calculations.
A natural analogy here is with the life table in life insurance mathematics, which virtually describes an empirical survival distribution of an average person’s lifetime. Similarly, an infection table can be generated to keep record of the number of infected cases reported during each sampling period (e.g., every day for SARS). Table 2 (see below) is a simple example of an infection table dated back to the seventeenth century.
Now from the infection table, we have a piecewise constant empirical approximation of the contin-
uous function i(t) given by
̃i(t)ik, k1tk, 0, otherwise
where ik is the rate of infection in the kth period on the infection table. Using this function in place of i(t) in (3.4) gives an approximation to a ̄i :
t
t t ̃ n e(k1)ek
a ̄i exi(x) dx exi(x) dx i , t0 0 k1k
where n [t] is the integer part of t, and n is large enough. To compute a perpetuity a ̄i , one needs ̄i
to choose a sufficiently large term n and hence find P(a ̄) by (3.10) as an approximate asymptotic values.
120 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
Following the same logic as in Proposition 3.1, the following relation between a ̄s and a ̄i is easily t t
obtained:
Therefore, the premium for the policy with annuity claims and payments can be calculated by
n
1
4.2 Power Series Solutions
a ̄s 1(1et)et t i
t 0i(r)dr1a ̄t.
a ̄i (1/)
t k
The power series method is one of the oldest techniques used to solve linear differential equations. This method can be adapted well to our SIR model.
Because every point in the system is an ordinary point, in particular, t 0, we look for solutions of the form
(4.1) (4.2)
s(t) i(t)
Therefore, differentiating term by term yields
n
at, t0,
s(t) i(t)
Multiplying (4.1) by (4.2) gives
where
From (3.1) and (3.2), we obtain
n1 nat
n (n1)a t, t0,
n1
n1
n0
n0
n n0
n
n1
n1 nbt
n (n1)b t, t0.
n
n1
s(t)i(t)
n
ct, t0,
n n0
n n0
(k1) k k [e e]i
.
ta ̄s n n
t (1/)(1 et) (/)et k 2 (k1) k k
i (/ 1/)
k1 k1
[e e]i
n
bt, t0.
cn a0bn a1bn1 an1b1 anb0. nn
(n1)a t
ct 0, n1 n
n0 n0 nnn
(n1)b t
ct
n1 n n
bt 0. n0 n0 n0
To satisfy these equations for all t, it is necessary that the coefficient of each power of t be zero. Hence we obtain a0 s0, b0 i0 and the following recursion:
an1 (a0bn a1bn1 an1b1 anb0), n1
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 121
bn1 an1 bn. n1
(n 1)(n 1, t), (n 1)(n 1, t),
1 t
(n,t)(n) 0 xn1ex dx, n0,t0,
which is readily available numerically in most mathematical or statistical software.
Hence the premium for the policy with annuity claims and payments can be calculated by
Therefore,
be xdx
t b x n n
a ̄i
t n0 0 n n0n1
ae xdx
t a a ̄s x n n
t n0 0 n n0n1 where (n, t) is the incomplete gamma function
(b /n1)(n 1)(n 1, t) a ̄i n0 n
t n0n Interestingly, when t → , these formulas simplify to
t . t a ̄s (a /n1)(n 1)(n 1, t)
a ̄i
(n!)bn and a ̄s (n!)an ,
n0 n1
n0 n1
which implies that
4.3 Insurance-Related Quantities and Runge-Kutta Method
Among many numerical methods for solving ODE’s, the Runge-Kutta method is the most popular. It can be adapted for any order of accuracy. For applications in insurance, the fourth-order Runge-Kutta method (RK-4), given by the following recursion formulas, represents a good compromise between simplicity and accuracy:
yi1 yi 1(k1i 2k2i 3k3i k4i), i1,2,…,n, 6
k1i hf(ti,yi), k2i hfti h,yi 1k1i, 22
k3i hfti h,yi 1k2i, k4i hf(ti h,yi k3i), 22
where yi is given by the ODE:
dy f(t, y), dt
evaluatedattti,andwherehti tt1 isthetimestep,fori1,2,…,n.TheRunge-Kutta method is discussed in detail by Boyce and DiPrima (1986).
̄
P ( a ̄ i ) n 0
.
n
n1 (n!)b /
n1 (n!)a /
n n0
122 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
Figure 2
Benefit Reserve Function V(t) of Great Plague Plan with 0.86
Actuaries will be interested particularly in the properties of insurance-related quantities, such as the total discounted benefits, the total discounted premiums, and the premium reserves. Based on the RK-4 method, we need to express these quantities into a system of differential equations. Let P(t) denote the accumulated value of premiums collected up to time t and B(t) the corresponding accu- mulated value of benefits paid up to time t. Using a retrospective approach, we consider V(t), the accumulated benefit reserve at time t, as the difference between the accumulated value of premiums and the accumulated value of claims.
Assume that the infectious disease plan with annuity premium payments provides one monetary unit of compensation per time unit for infected policyholders. Then the connections among these insurance- related and epidemiological quantities could be described by the following ODE system:
P(t) s(t) P(t), B(t) i(t) B(t), V(t) P(t) B(t),
t 0, t 0,
t 0,
(4.3) (4.4) (4.5)
where P(0) s0, B(0) i0, and is a testing premium rate. The rationale behind the ODE is as follows. The instantaneous change in the accumulated value of total premiums P(t) is given by the sum of the instantaneous rate of premium income s(t) and the instantaneous rate of interest return on the current total premiums P(t). The instantaneous change in the accumulated value of total claims
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 123
Figure 3
Benefit Reserve Function V(t) of Great Plague Plan with 0.25
is the sum of the instantaneous rate of claims i(t) and the instantaneous rate of interest return on the current total claims B(t).
These ODE systems can be readily solved in most mathematical software such as Maple. Information
about programming with the ODE tool kits in Maple can be found in Barnes and Fulford (2008) and
Coombes et al. (1997).
̄i
We may use P(a ̄) as a starting point to test the behavior of the reserve function. Then we can
gradually increase the premium rate to produce an acceptable reserve schedule. 5. PREMIUM ADJUSTMENTS
We shall first investigate the demographic changes in the insured group over time. Throughout the section, we assume , which is usually the case in reality or otherwise the infectious disease dies down immediately.
Proposition 5.1
For the SIR model in (3.1)–(3.3), s(t) is monotonically decreasing in t, and r(t) is monotonically in- creasing. If s0 /, then i(t) is monotonically decreasing, while if s0 /, i(t) increases up to the time t*, at which point s(t*) /, and then decreases afterwards.
In actuarial mathematics mortality rates mostly rise with age. If premiums are held constant, the insurer’s future liability exceeds the future premium revenue. To cover future liabilities, life insurers
124 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
set aside benefit reserves. Unlike the ‘‘U’’ shape of mortality curves, a unique feature of epidemics is that the infection rates increase rapidly at the beginning but then drop after reaching a peak, as proved in Proposition 5.1. Figure 4 illustrates a typical path of a benefit reserve function obtained for the policy in (4.3)–(4.5), where the benefit premium is determined by (3.6). As one can see, because infection rates increase rapidly in the early stage and drop significantly at the later stage, an insurer’s liability is larger at the beginning but decreases over the time. Level premiums over a long term imply negative benefit reserves, as claims exceed premiums in the early stages of the epidemic, and then fall below premiums at later stages. Because of a likelihood of policy withdrawal after the peak of infection, an insurer may not be able to collect enough premiums. Hence the need to increase the premium to a level that guarantees positive benefit reserve even if this means that the insurer will have to pay a cash value to policyholders at the term of the policy.
We shall now investigate how the premium rate affects the shape of benefit reserve function using the example of a plan with both annuity premiums and claim payments.
It follows immediately from (4.3)–(4.5) that
V(t) s(t) i(t) V(t), t 0.
For simplicity, consider the case where the force of interest 0, to better grasp the connection between and the shape of the benefit reserve. We leave the more complicated case 0 for the numerical analysis at the end of this section.
Since the sign of the instantaneous change in V(t) depends on two competing forces, the monoton- ically decreasing s(t) and the decreasing or increasing-then-decreasing i(t). There are four possible shapes of the graph of the benefit reserve V(t), namely, strictly increasing concave, strictly increasing concave-then-convex, nonmonotonic concave-then-convex, and nonmonotonic convex, as shown in Fig- ures 2–5, respectively. The following propositions provide conditions under which the four scenarios appear (see the Appendix for the proofs).
Proposition 5.2
(Convexity) In (4.3)–(4.5) with 0, the shape of benefit reserve V(t) is determined by the premium rate as follows:
1. V(t) is concave, if
where the constant s limt→ s(t).
2. V(t) changes from concave to convex, if
1, s
(5.1)
(5.2)
(5.3)
(5.4)
1 1. s0 s
The point of inflection tf is given by
3. V(t) is convex, if
s(tf)
(1 )
1. s0
.
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 125
Proposition 5.3
(Monotonicity) In (4.3)–(4.5), the reserve V(t) is strictly increasing, if exp c 1 1,
where the constant c 1 ln(s0)/.
Because s(t) is a decreasing function, it is easy to see that
1expc11 1. s s0
(5.5)
Table 1 summarizes the four possible shapes of reserve functions.
When is small relative to i(r), for r [0, t), the shape of the reserve function does not change
significantly. Hence we can use the relation between and V(t), when 0, as our starting approxi- mation to search for an accurate premium in the case when 0.
As we can see from Figures 4 and 5, premium rates are quite low, but there might be undesirable negative reserves during the policy term. On the other hand, the insurer may not favor a policy design with the strictly increasing liability shown in Figure 3, as premiums are relatively high, increasing moral hazard. It means that healthy policyholders are more likely to shop for lower premium rates, while unhealthy ones keep the policy, ultimately increasing insurance costs. Hence the need for a design with more marketable premiums that produce a bell-shaped reserve, that is, a concave-then-convex shape, with the relatively low final cash values shown in Figure 7.
Figure 4
Benefit Reserve Function V(t) of Great Plague Plan with 0.10
126 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
Figure 5
Benefit Reserve Function V(t) of Great Plague Plan with 0.03
The above analysis shows the importance of determining whether the reserve function is concave at the term of the policy. If so, premiums can gradually increase so that the reserve reaches zero at the near the end of the policy, producing a bell-shaped reserve. Otherwise, a premium that produces a bell- shaped reserve might not exist, hence the need to settle down for a concave-then-convex reserve func- tion, producing a positive cash value paid out at the end of the policy term.
The following algorithm calculates a premium rate for a t-year policy with nonnegative cash values over the whole policy term. The main idea of the algorithm is summarized as follows: It starts with an initial premium rate that ensures that V(t) has a concave-then-convex shape. Because a small will not affect the shape of V(t) in a significant way, this step gives a tractable initial premium that can be used for other values. As Step 1 is part of a loop, it is later used to check whether V(t) is in a concave phase with the newly adjusted premium rate. If so, proceed to Step 2.1, which adjusts premium rates
Table 1
Possible Shapes of V(t) When 0
Shape of V(t)
Interval for Values of
Increasing concave
Increasing concave-then-convex Nonmonotonic concave-then-convex Nonmonotonic convex
[/(s0) 1, )
[/ exp{c/ 1} 1, /(s0) 1) [/(s) 1, / exp{c/ 1} 1) [0, /(s) 1)
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 127
according to the sign of V(t). If a premium with zero final cash value is feasible, it will be obtained in Step 2.1. Step 2.2 deals with the case when V(t) is in a convex phase.
• Step 0: Set n : 1 and (1) is set to be relatively small such that /(s(t)) 1 (1) /(s) 1.
• Step 1: If (n) [i(t)s(t) i(t) i(t) 2V(t)]/[s(t) s(t)i(t)], proceed to Step 2.1. Other- wise, proceed to Step 2.2.
•Step2.1:IfV(t)0,then(n1) :(n) 0.01,setn:n1andgotoStep1.IfV(t)0,then (n1) :(n) 0.01,setn:n1andgotoStep1.IfV(t)0,stopand(n) isthepremiumrate with zero final cash value.
•Step2.2:IfV(t)0,testwhetherV(s)0forall0st.Whenthetestfails,(n1) (n) 0.01, set n n 1 and go to Step 2.2. When the test is past, (n) is the premium rate with positive final cash value.
If the algorithm produces a positive value of V(t), it implies that the reserve function reaches its lowest point, a nonnegative value, prior to the end of the policy. By the Equivalence Principle, the cash value V(t) should be paid back to all surviving policyholders at the policy term.
6. NUMERICAL EXAMPLES
The epidemiological model in our first numerical example of the Great Plague in Eyam was originally studied by Raggett (1982). It is included as a classical case study in many textbooks because predictions from the model are remarkably close to actual data. The second example of a six-compartment model comes from Chowell et al. (2003), where parameters are primarily used for measuring the mean num- bers of secondary cases a single infected will cause in a population with no immunity, in the absence of interventions to control the infection. However, our focus is on the insurance arrangement to alle- viate the financial burden on the population, rather than model fitting and selection. These two datasets and compartment models are assumed to give an accurate depiction of the dynamics of corresponding diseases upon which actuarial models are built.
6.1 Great Plague in Eyam
The village of Eyam near Sheffield, England, suffered a severe outbreak of bubonic plague in 1665– 1666. Only 83 of an initial population of 350 villagers survived the plague. Detailed records were preserved as shown in Table 2. In Raggett (1982), the SIR model is fitted to the Eyam data, over the period from mid-May to mid-October 1666. Time is measured in months, with an initial population of 7 infectives and 254 susceptibles and a final population of 83. Since the disease was fatal at the time, all infected individuals eventually died from the disease.
Table 2
Eyam Plague Susceptible and Infective Populations in 1666
Date
Susceptibles
Infectives
Initial
July 3–4
July 19
August 3–4 August 19 September 3–4 September 19 October 4–5 October 20
254 235 201 153.5 121 108
97 Unknown 83
7 14.5
22 29 21
8
8 Unknown 0
Source: Raggett (1982), Table II.
128 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
To set up an actuarial model based on the arrangement proposed in Figure 1, we shall base the actuarial analysis on an accurately calibrated SIR model.
According to (A.5),
i0 s0 lns0 i s lns,
from which we obtain an expression for / in terms of measurable quantities s0 and s, ln(s0/s) .
1 s
(6.1)
It is generally difficult to estimate the contact rate because of its dependency on social and behavioral factors. Hence it is important that we can estimate from (6.1) with an estimation of the fatality rate .
Because s0 254/261 0.97318, i0 7/261 0.02682, and s 83/261 0.31801, (6.1) gives / 1.64004. The parameter is determined by its reciprocal, which has the clinical interpretation of the average infectious period. From clinical observations, an infected person stays infectious for an average of 11 days or 0.3667 months before death, so that 1/0.3667 2.73 and 4.4773. The resulting graphs of s(t) and i(t) are given in Figure 6.
Insurance coverage will not directly reduce the transmission of the disease if an epidemic similar to that described in table were to occur today. But a well-designed insurance program could provide financial incentives for prevention measures and compensations for hospitalization and other medical costs and services. To develop this insurance model, we assume that everyone in the village participates in a mutual insurance fund set up at the beginning of the epidemic. The insurance fund earns interest at the force of interest of 0.2%. The insurance term lasts five months, which matches the duration of the epidemic.
The insurance plan provides infection benefits continuously at the rate of $1,000 per month until death, for every infected individual, for the whole period when he or she has been infected and hospi- talized. The insurance liability is terminated after death. It is purchased by susceptible individuals, through continuous premium payments.
Using the algorithm provided in Section 5, we start with an initial premium rate (1) 1000[(/)exp(c/) 1] 188.27. The initial premium is quite high because the algorithm requires the reserve to stay in its concave phase at the start, and then gradually reduce the premium to a desired level. After many rounds of premium reduction, we reach the final premium rate of 114.58. Such a premium rate avoids the reserve from ever being negative and keeps the cash value at policy end at a reasonable level of V(5) 49.44, which means each survivor receives a reward of $49.44 at the end of the epidemic to clear off the insurance fund. The reserve function of such a policy is given in Figure 7.
6.2 SARS Epidemic in Hong Kong
In the classical SIR model, the implicit assumption that the mixing of members from different com- partments is geographically homogeneous is probably unrealistic. The susceptible people in geograph- ical neighborhoods of an infectious virus carrier are more likely to be infected than those who are remote from the carrier. For instance, during the SARS epidemic in Hong Kong, it was observed that health care workers were at higher risk of infection than most other groups in the population.
To distinguish different levels of vulnerability or infectiousness within different social groups, spatial structures are introduced and developed in epidemiological studies. A typical example of a spatial structure applied to the SARS epidemic in Hong Kong is defined by Chowell et al. (2003) in the following ODE system:
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 129
Figure 6
Percentage of Susceptibles s(t) and Percentage of Infected i(t)
130 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
Figure 7
Cash Values of Great Plague Plan
S(t) S (t) I(t) qE(t) lJ(t),
t 0,
t 0,
(6.2) (6.3)
(6.4)
(6.5) (6.6) (6.7)
11
N
S(t) pS (t) I(t) qE(t) lJ(t),
N
22
E(t) (S1(t) pS2(t)) I(t) qE(t) lJ(t) kE(t), N
I(t)kE(t)(1 )I(t), t0, J(t) I(t) (2 )J(t), t 0,
R(t) 1I(t) 2 J(t), t 0.
t 0,
In this model, there are two distinct susceptible compartments with different levels of exposure to SARS, namely, S1 for the most susceptible urban community and S2 for the less susceptible rural population. Initially, S1(0) N and S2(0) (1 )N, where is the proportion of urban individuals in the total population. An average highly susceptible person (in Class S1) makes an average number of risky contacts (i.e., contacts sufficient to transmit infection) with others per unit time. Because of less frequent visits to public areas where viruses concentrate, an average lower susceptible person
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 131
Figure 8
Transfer Diagram of SARS Epidemic Dynamics
Note: Reprint of Figure 1 in Chowell et al. (2003).
(in Class S2) would be exposed to only an average number of p risky contacts with others per unit time.
Because individuals infected with SARS experience incubation periods of 2–7 days before the onset of any visible symptoms, an infectious class is set up for those infected but not yet symptomatic. The parameter q is used to measure the lower level of infectivity during the incubation. With time, the infected individual develops observable symptoms and becomes fully infectious, in Class I with q 1. To distinguish their potential disease transmission to the general public, Class I is separated for those infectious individuals still undiagnosed. Because almost all diagnosed cases are quarantined in hospi- tals, Class J has a lower infectivity level reflected by a reduction factor l.
The population rates transferring from E, I, and J to their chronologically adjacent compartments I, J, and the recovered Class R, are, respectively, k, , and 2. Considering that even before being diag- nosed, SARS patients may either recover naturally at the rate of 1 or die at the force of death , we also have Class D, tallying deaths as a result of SARS, from sources I and J. The patients under medical treatment in Class J suffer death at the rate assumed to be the same as the mortality rate in Class I.
Notice that both E and I are undiagnosed phases: There is literally no statistical data for estimating their parameters. Therefore, another compartment C for reported probable cases is used to trace back, by a time series, the original time of incidence. Figure 8 illustrates the possible transfer vectors among the different compartments.
Leaving aside the detailed parameter inference analysis, we use the parameter estimates in Chowell et al. (2003), summarized in Table 3. The parameters were chosen to fit observable quantities identified
Table 3
Parameter Values Fitted to the SARS Model for Hong Kong
Parameter
Moving from/to
Value
q l p k 1 2
S1, S2/E Reduced infectiousness Reduced infectiousness Reduced susceptibility E/I
I/J
I/R
J/R
I, J/R Reduced contacts
0.75 0.1 0.38 0.1
1/3 1/3 1/8 1/5 0.006
0.4
Note: See Table 1 in Chowell et al. (2003).
132 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
Figure 9
Individual Numbers in Each Compartment: S1(t), S2(t), E(t), I(t), J(t), and R(t)
in clinical studies. A drawback of this method is that it can lead to negative counts in Classes I and J. In such occurrences, these deficiencies need to be addressed in practical applications and standard techniques are available. An overview of the evolution of individual classes for the SARS epidemic can be seen in Figure 9.
From an insurer’s point of view, this model presents interesting business opportunities. On the one hand, individuals in Classes S1 and S2 are potential buyers facing the risk of infection with SARS. On the other hand, there is an evident need for insurance covering vaccination costs in both S and S ,
12
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 133
Table 4
SARS Insurance Premium Rating (per $100,000 Unit of Benefits)
Plan
P.V. Benefits
P.V. Premiums
Level Premium
AH SH
3.0571 108 1.3231 108
1.71604 108 1.71604 108
1.78 0.77
medical examination expenses for probable cases in Class I, hospitalization and quarantine expenses for Class J, and death benefits for Class D.
Several parties have stakes in our health care systems, such as insurance companies, policyholders, government health agencies, and hospitals. Numerous business models could be designed to reduce the overall financial impact to a minimum level. With a simple illustrative example of an infectious disease insurance, we revisit the two following plans:
1. Annuity for Hospitalization (AH) Plan: Every participant in the mutual insurance fund purchases the coverage by means of an annuity. Insureds in rural areas are charged lower premiums proportional to their reduced susceptibility. From the time of policy issue to the end of the epidemic, insureds can claim a medical examination fee of $100,000, once diagnosed with suspicious symptoms, plus hospitalization expenses of $100,000 per day, in the form of a life annuity for the period under medical treatment in hospital. Specified beneficiaries are entitled to a death benefit of $100,000 if the insured’s death is due to the infectious disease. The protection ends at the earliest of the insured’s time of death or the end of the epidemic.
2. Lump Sum for Hospitalization (SH) Plan: This plan provides all of the same benefits as in the previous AH plan, with the life annuity being replaced by a lump-sum payment of $100,000 when the insured is diagnosed positive with the disease. The protection also ends here at the earliest of the policyholder’s death or the end of the epidemic.
The discounted total benefits and premiums in Table 4 are calculated under the assumption that all Hong Kong residents during the pandemic participate in the fund. This yields surprisingly low net level premiums, determined for both plans by the equivalence principle. This reinforces our assertion that a fairly low-cost insurance plan could cushion the high impact such pandemics would have on our health care systems when they occur.
7. FUTURE WORK
Research in this emerging type of insurance is at the infancy stage. More work is needed to generalize models that could fit different aspects and features of other pandemics. There have been numerous and extensive studies in epidemiological stochastic modeling. We envision that some of these stochastic models can be incorporated in a more pragmatic way for actuarial applications.
APPENDIX
A.1 PROOF OF PROPOSITION 3.2
From (3.1) and (3.2), we obtain that
s(t) i(t) i(t), t 0.
Integrating from 0 to a fixed t gives
s(t) i(t) 1 t 0
i(r) dr, t 0.
134 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 1
Multiplying both sides by et and integrating with respect to t, from 0 to , yields
1 t
eti(t) dt et i(r) dr dt, t 0.
0000
Exchanging the order of integrals and using the integration by parts gives
ets(t) dt
t 1 t
Hence (3.9) is obtained upon rearrangement.
A.2 PROOF OF PROPOSITION 3.2
Substituting (3.1) into (3.11), we have that
A0 It follows from (3.9) that
e s(t)dts(0) ̄ii
e s(t)dtsa ̄.
0 exp(t) 0 i(r) dr dt 0 0 i(r) dr d(exp(t)) 1 1i
̄i t t s
0 exp(r)i(r) dr a ̄
with the fact that
lim et t→
t
0 t→
t i(r) dr lim te
0.
0 0
A ()a ̄ 1s0. A.3 PROOF OF PROPOSITION 5.1
Because s(t) and i(t) are all nonnegative, from (3.1) and (3.3) we know that s(t) s(t)i(t) 0, for t 0, and r(t) i(t) 0. Hence s is a monotonically decreasing function and r is monotonically increasing.
If s0 /, then i(t) i(t)[s(t) ] 0, which means that i(t) is monotonically decreasing. If s0 /, because s is monotonically decreasing, then i(t) i(t)[s(t) ] 0, as long as s(t) /. Thus i(t) reaches a local maximum at the point t* where s(t*) /. As s(t) continues to decrease after reaching /, we must have i(t) 0, and hence i(t) is monotonically decreasing thereafter. □
A.4 PROOF OF PROPOSITION 5.2
To check the concavity of V, consider V (t):
V (t) s(t) i(t) s(t)i(t) s(t)i(t) i(t)
It follows that when
i(t)[ ( 1)s(t)].
1, forallt0, s(t)
(A.1)
V (t) 0 and hence V (t) is concave. Because s is monotonically decreasing, thus condition (5.1) is required. Similarly, we derive the conditions (5.2) and (5.4) according to the changes in the sign of
ACTUARIAL APPLICATIONS OF EPIDEMIOLOGICAL MODELS 135
(A.1). Because the point of inflection tf is where V(t) changes from concave to convex, then it is determined by
V (tf) i(tf)[ ( 1)s(tf)] 0, which implies the condition (5.3).
A.5 PROOF OF PROPOSITION 5.3
To ensure that V(t) 0, we need
i(t) , s(t)
for all t, (A.2)
or equivalently,
Let f(t) ln i(t) ln s(t), then
lnlni(t)lns(t), forallt.
f(t) i(t) s(t) [s(t) i(t)] , by (2.1) and (2.2). i(t) s(t)
Because s(t) i(t) 1 r(t) is monotonically decreasing, we see that f(t) changes from positive to negative at time tm, when
s(tm) i(tm) /,
and f(t) reaches its maximum at time tm. Thus, in order for (A.2) to hold must satisfy
(A.3)
(A.4)
Now, because
i(tm ) . s(tm )
i(t) (s(t))i(t)1 s(t) s(t)i(t)
s(t)
,
integrating both sides to find the orbits of the (s, i)-plane, gives i(t) s(t) ln s(t) c,
(A.5) where c is a constant of integration for each specific orbit, say, c i0 s0 / ln(s0). Combining
(A.3) and (A.5), we can solve for s(tm) and i(tm). The solutions are given by s(tm) exp 1 c,
i(tm) exp 1 c.
Substituting (A.6) and (A.7) into (A.4) gives condition (5.5).
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