Mathematical Biosciences 249 (2014) 102–109

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Mathematical Biosciences

journal homepage: www.elsevier .com/locate /mbs

A multi-stage compartmental model for HIV-infected individuals:II – Application to insurance functions and health-care costs

http://dx.doi.org/10.1016/j.mbs.2014.01.0090025-5564/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +1 706 542 3281.E-mail address: lynne@stat.uga.edu (L. Billard).

L. Billard a,⇑, P.W.A. Dayananda b

a Department of Statistics, University of Georgia, Athens, GA, United Statesb Department of Mathematics, University of St. Thomas, St. Paul, MN, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 March 2012Received in revised form 20 November 2013Accepted 31 January 2014Available online 11 February 2014

Keywords:Compartmental modelsDeath from non-HIV/AIDS causesInsurance premiumsLife annuity policiesLife expectanciesHealth-care functions

Stochastic population processes have received a lot of attention over the years. One approach focuses oncompartmental modeling. Billard and Dayananda (2012) [1] developed one such multi-stage model forepidemic processes in which the possibility that individuals can die at any stage from non-disease relatedcauses was also included. This extra feature is of particular interest to the insurance and health-careindustries among others especially when the epidemic is HIV/AIDS. Rather than working with numbersof individuals in each stage, they obtained distributional results dealing with the waiting time any oneindividual spent in each stage given the initial stage. In this work, the impact of the HIV/AIDS epidemicon several functions relevant to these industries (such as adjustments to premiums) is investigated. The-oretical results are derived, followed by a numerical study.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Over the century since Ross [2] first developed his mathemati-cal model for a malaria process, there has been an extensive arrayof models introduced for numerous population processes, somebeing deterministic such as Ross’ model, some being stochasticsuch as Bartlett’s [3] early stochastic models for evolutionary pro-cesses followed by Bailey’s [4] simple stochastic epidemic process.As a general rule (though obviously not true universally), equationsgoverning deterministic models are ’easier’ to solve than the moremathematically intractable equations governing stochastic pro-cesses. This is particularly so for stochastic models with nonlineartransition rates/probabilities. Yet, these nonlinearities are quite en-demic to epidemic/disease processes. Our focus is with stochasticepidemic models; and in particular the impact of the HIV/AIDS epi-demic on insurance and health-care functions.

Typically, many researchers have established the relevant set ofdifferential-difference equations describing a given stochastic pro-cess (be these epidemic, queuing, biological, etc. models), and thenexplored techniques to solve these (largely intractable) equations.In more recent times, other researchers have set up their processesin a compartmental modeling framework. An extensive, but by no

means exhaustive, review of some of the many approachesadopted over the years is in [1].

Billard and Dayananda [1] considered a compartmental modelwith transitions from one compartment (stage) to another and inaddition allowed for death at any stage to occur. While their gen-eral approach is applicable to many population processes, theirwork was motivated by the need to establish guidelines for theinsurance industry in particular (with parallel applications tothe health-care industry, among others) as to the impact of theHIV/AIDS epidemic on insurance functions. Therefore, in thatwork, several probabilities and waiting time distributions forthe compartmental process were derived. In this work, we deriveinsurance functions of interest. For example, insurance companiesare vitally interested in future payouts (when and how much) forpolicies issued to their clients. A major concern relates to the im-pact that the HIV/AIDS epidemic has on these payments. These inturn are related to the premium levels clients are required to payfor continuous t-year life annuity and insurance policies (e.g.) andtheir long term effect (t !1). Clearly, these entities are impactedby whether or not a potential policy-holder is healthy (i.e., isuninfected) or not (i.e., is already infected with HIV) at the timethe policy is issued. Life expectancies and how, if at all, theseare affected by the HIV/AIDS process are other functions of inter-est to insurers.

Accordingly, we give first, in Section 2, a description of the basiccompartmental model, along with a summary of the approach used

L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109 103

in Billard and Dayananda [1] to obtain the necessary distributionsetc. Then, in Section 3, we derive insurance functions, against thebackdrop that policy holders can become infected with HIV andbe diagnosed with AIDS and how this diagnosis impacts the insur-ance entities. Section 4 looks briefly at how the ideas can be ex-tended to health-care functions. The special case of exponentialwaiting times (usually not applicable for HIV/AIDS but can be validfor other diseases) is included, in Section 5. How these functionsare affected by changing values of the model parameters is inves-tigated in Section 6. This numerical study includes consideration ofthe impact of HIV/AIDS treatment to those known to have thedisease.

2. Compartmental model – waiting time approach

In general, a compartmental process can be modeled as a multi-stage process with XjðtÞ individuals in stage j ¼ 0; . . . ; J at timet > 0. In the framework of epidemic processes, stage j ¼ 0 corre-sponds to individuals who are initially susceptibles (i.e., uninfectedby whatever disease is under study). Stages j ¼ 1; . . . ;mþ 1 corre-spond to different levels of infection through which individualssuccessively pass. (Processes which allow a return to an earlierstage are accommodated by interpreting passage to a given stageas the last such visit to that stage.) Ultimately, individuals candie when they move into stage j ¼ mþ 2. In the context of HIV/AIDS, the stage j ¼ 1 is when initially infected with the HIV virus,and j ¼ mþ 1 corresponds to diagnosis with AIDS. Transition rates,and/or probabilities, of moving from stage to stage are functions ofthe number of individuals in each stage; most such functions arenonlinear and so produce differential-difference equations thatare mathematically intractable.

Instead of writing the model in terms of the variables XjðtÞ, Bil-lard and Dayananda [1] focused on waiting time distributions tomove across stages. They also allowed for individuals to die fromnon-disease (such as non-HIV/AIDS) causes at any time; i.e., indi-viduals could move directly to stage j ¼ mþ 2. This feature is miss-ing from most models even though it readily exists in manyrealistic situations (especially in insurance and health-care amongother applications).

To summarize the Billard and Dayananda [1] (B&D) approach,and using their notation in the sense they defined it, let usdefine Vj to be the time an individual stays in stage j until movingout into stage ðjþ 1Þ; j ¼ 0; . . . ;mþ 1. Let us further defineUj; j ¼ 0; . . . ;mþ 1, as the time an individual stays in stage j untildeath, i.e., until he moves directly into stage mþ 2. It is assumedthat Vj and Uj are independent random variables. Suppose at timet an individual is in stage SðtÞ. A key entity (see Sections 3–5) is theprobability qijðtÞ that an individual is in stage j at time t given thathe started in stage, i.e.,

qijðtÞ ¼ PfSðtÞ ¼ jjSð0Þ ¼ ig; i 6 j; i; j ¼ 0; . . . ;mþ 2; t > 0: ð1Þ

From B&D (Theorem 1, Eq. (34)), we have, for i 6 j; i; j ¼0; . . . ;mþ 2,

qijðtÞ ¼ ½PðYij > tjWk > 0; k ¼ i; . . . ; j� 1Þ � PðYi;j�1 > tjWk > 0;

k ¼ i; . . . ; j� 1Þ� � PðWk > 0; k ¼ i; . . . ; j� 1Þ; t > 0; ð2Þ

where

Wj ¼ Uj � Vj; ð3Þ

Yij ¼Xj

k¼i

Hk ð4Þ

with

Hi ¼minðUi;ViÞ: ð5Þ

Here, when an individual survives stage j (without dying) tomove into stage ðjþ 1Þ; Wj > 0; but when an individual died whilein stage j;Wj < 0. The variable Hj is the actual time an individualspends in stage j, by moving to the next stage after time Vj or bydying and moving to stage mþ 2 after time Uj, whichever happensfirst. Thence, the variable Yij in (4) is the total time an individualwho started in stage i spends in stages i; . . . ; j before moving intostage jþ 1.

We need expressions for the probabilitiesPðWk > 0; k ¼ i; . . . ; j� 1Þ and the conditional probabilitiesPðYij > tjWk > 0; k ¼ i; . . . ; j� 1Þ of (2). Given its importance, theseare derived in B&D (Sections 3.2 and 3.4, respectively) explicitly forthe 4-stage model (m ¼ 1Þ. In those derivations, susceptibles weremodeled as becoming infected with HIV at a Poisson rate withmean k; hence the waiting time random variable V0 is exponen-tially distributed with parameter k. The waiting time distributionV1 corresponds to the incubation period between infection withHIV and diagnosis with AIDS; thus a Weibull distribution withparameters (a; b ¼ 2) was used. The waiting time distribution V2

corresponding to the time of diagnosis to death from AIDS was as-sumed to follow an exponential distribution with parameter h. Thewaiting times for death from non-AIDS causes, Uj, were assumed tobe exponentially distributed random variables with parameterslj; j ¼ 0;1;2.

Thence, the survival probabilities qijðtÞ can be obtained as inB&D (Section 3.5). The qijðtÞ are summarized in Appendix A.1. Aswe shall see in Sections 3,4, these waiting time distributions arefundamental entities in the derivation of many insurance andhealth-care functions. While the derivations in Sections 3,4 focuson the four compartmental model (m ¼ 1) with the numericalstudy of Section 6 including parameter values that easily accom-modate the influence and impact of individuals undergoing HIV/AIDS treatment, an alternative model could include an additionalstage (such as a ‘‘treatment’’ stage, so that m ¼ 2). Another alterna-tive model would be to adjust the current (m ¼ 1) model’s distri-butions appropriately so as to capture the effect of treatment.Thus, the model presented in the paper can be modified to accom-modate the effects of drug therapies.

3. Insurance functions

There are numerous insurance functions of interest. Tradition-ally, insurance companies charge premiums, at a given percentagehigher than the normal rates, to individuals or groups that arelikely to be considered as having ‘‘higher risk’’ than normal. Sinceindividuals exposed to HIV/AIDS are perceived to be in higher riskcategories, insurers are particularly interested in those insurancefunctions relating to premium rates, expected payouts and lifeexpectancies. Therefore, we limit our focus to but these illustra-tive few functions. The basic theory behind these entities canbe found in any of a number of actuarial sources (e.g., Bowerset al. [5]).

3.1. Premiums for t-year pure endowment policy

Payouts and hence premium rates are typically expressed interms of $1 units. A $1 (future) payout to an individual who hassurvived t units of time is currently worth $fexpð�dtÞg where d isthe force of interest. This $1 is paid only if the individual has sur-vived, with that person being in stage j; j 6 mþ 1, at time t. If thepresent expected value for an individual currently in stage i is de-noted by �EiðtÞ, then

�EiðtÞ ¼Xmþ1

j¼i

ðpresent value of $1ÞqijðtÞ:

104 L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109

Hence, the net single premium for a t-year pure endowment forsomeone in stage i when the policy is issued is

�EiðtÞ ¼Xmþ1

j¼i

e�dtqijðtÞ: ð6Þ

Therefore, by substituting for the qijðtÞ from (A.1)–(A.9), the netsingle premium for a pure t-year term endowment policy, can beshown to equal, for i ¼ 0;1;2, respectively,

�E0ðtÞ ¼ke�dt

ðkþ l0 � h� l2Þe�ðhþl2Þt � ðhþ l2 � l0Þ

ke�ðkþl0Þt

�þ l1 � h� l2ÞfGðt;l1; kþ l0; kþ l0Þ�� Gðt;l1; hþ l2; hþ l2Þg

�; t > 0; ð7Þ

�E1ðtÞ ¼ e�dt þ e�ðhþl2Þt ; t > 0; ð8Þ�E2ðtÞ ¼ e�ðdþhþl2Þt; t > 0; ð9Þ

where

Gðx; a; b; cÞ ¼ e�cxþða�bÞ2=4a

�ffiffiffiffiffiffiffiffiffip=a

pU x

ffiffiffiffiffiffi2ap

þ a� bffiffiffiffiffiffi2ap

� ��U

a� bffiffiffiffiffiffi2ap

� �� ð10Þ

with UðuÞ being the usual standard normal distribution function

UðuÞ ¼Z u

�1

1ffiffiffiffiffiffiffi2pp e�

12w2

dw:

3.2. Continuous t-year life annuity policy

A t-year continuous life annuity and insurance policy issued toan individual in stage i has a value of, for t > 0,

�aiðtÞ ¼Xmþ1

j¼i

Z t

0e�duqijðuÞdu: ð11Þ

This can be written as

�aiðtÞ ¼Xmþ1

j¼i

Q �ijðtÞ ð12Þ

where

Q �ijðtÞ ¼Z t

0e�duqijðuÞdu: ð13Þ

Expressions for Q �ijðtÞ can be derived from simple to algebrai-cally tedious but routine derivations by substituting for qijðtÞ givenin Appendix A.1 and by utilizing B&D (Lemma 1 and integralsIk; k ¼ 1; . . . ;4). Therefore, setting q1 ¼ dþ kþ l0 and q2 ¼ dþhþ l2, we can show that

Q �00ðtÞ ¼ ½1� e�q1t �=q1; ð14ÞQ �01ðtÞ ¼ ðk=q1Þ½Gðt; dþ l1;0;0Þ � Gðt;l1; kþ l0;q1Þ�; ð15ÞQ �02ðtÞ ¼ ½k=fq1q2ðq1 � q2Þg�½ðq1 � q2 þ q2e�q1t � q1e�q2t

þ q1ðl1 � h� l2ÞGðt;l1; hþ l2;q2Þ� q2ðl1 � k� l0ÞGðt;l1; kþ l0;q1Þ� ðq1 � q2Þðdþ l1ÞGðt; dþ l1;0;0Þ�; ð16Þ

Q �11ðtÞ ¼ Gðt; dþ l1;0;0Þ; ð17ÞQ �12ðtÞ ¼ ð1=q2Þ½1� e�q2t þ ðl1 � h� l2ÞGðt;l1; hþ l2;q2Þ

� ðdþ l1ÞGðt; dþ l1; 0;0Þ�; ð18ÞQ �22ðtÞ ¼ 1� ð1=q2Þe�q2t: ð19Þ

Thence, we can show that, for i ¼ 0,

�a0ðtÞ ¼ ½ðdþ kþl0Þðdþ hþl2Þ��1

ðdþ kþ hþl2Þ þðl1 � h�l2Þðkþl0 � h�l2Þ

a0ðtÞ�

; t > 0; ð20Þ

where

a0ðtÞ ¼ ½ðhþ l2 � l0Þðdþ hþ l2Þe�ðdþkþl0Þt � kðdþ k

þ l0Þe�ðdþhþl2Þt �=ðl1 � h� l2Þ þ kðdþ kþ l0ÞGðt;l1; h

þ l2; dþ hþ l2Þ � kðdþ hþ l2ÞGðt;l1; kþ l0; dþ kþ l0Þ� kðkþ l0 � h� l2ÞGðt;l1 þ d;0;0Þ;

and, for i ¼ 1;2,

�a1ðtÞ¼1

ðdþhþl2Þ1�e�ðdþhþl2Þtþðl1�h�l2Þ

� Gðt;l1;hþl2;dþhþl2Þ�Gðt;dþl1;0;0g� �

; t>0; ð21Þ

�a2ðtÞ ¼1

ðdþ hþ l2Þ1� e�ðdþhþl2Þt �

; t > 0: ð22Þ

The net single premium for these continuous t-year insurancepolicies can be obtained from

�AiðtÞ ¼Xmþ1

j¼i

l0jZ t

0e�duqijðuÞdu ¼

Xmþ1

j¼1

l0jQ�ijðtÞ ð23Þ

where l0j is the death rate parameter for an individual in stage j, i.e.,l00 � l0;l01 � l1 and l02 � hþ l2, and where the Q �ijðtÞ are given in(14)–(19).

Hence, the net single premium for a t-year (t > 0Þ insurancepolicy for an uninfected, i.e., in stage i ¼ 0, individual when thepolicy is taken out is, from (23),

�A0ðtÞ ¼ ðhþ l2Þ=q2

��1� e�q2t þ ðl1 � h� l2ÞGðt;l1; hþ l2;q2Þ

þ dðl1 � h� l2Þðhþ l2Þ

Gðt; dþ l1;0;0Þ�; ð24Þ

the premium necessary for an individual who is infected at the timethe policy is issued is, for i ¼ 1,

�A1ðtÞ¼ðhþl2Þðdþhþl2Þ

�1�e�ðdþhþl2Þtþðl1�h�l2Þ

� Gðt;l1;hþl2;dþhþl2Þþd

ðhþl2ÞGðt;dþl1;0;0Þ

� �; ð25Þ

and for stage i ¼ 2,

�A2ðtÞ ¼ ½ðhþ l2=q2�½1� e�q2Þt�: ð26Þ

It can easily be shown that the fundamental relationship

�AiðtÞ ¼ 1� d�aiðtÞ; t > 0; ð27Þ

holds for all i and all time t.Further, the net continuously paid premium per unit time for a

t-year continuous term annuity and insurance policy issued tosomeone in stage i is

�PiðtÞ ¼ �AiðtÞ=�aiðtÞ: ð28Þ

Given these expressions for the values of the policies written,�aiðtÞ, and the total (single) premium values, �AiðtÞ, a straightforwardapplication of (28) allows calculation of the continuous premiumper unit time �PiðtÞ required from an individual who was in stagei when the policy was issued. The details are omitted. Note that,because of (27), these �PiðtÞ can be expressed in terms of �aiðtÞ only.

Insurance companies are vitally interested in the long term; i.e.,interest is centered on the �aiðtÞ; �AiðtÞ and �PiðtÞ, as t !1. Takingthis limit on each of these terms in turn, we can show that, if wewrite �aið1Þ ¼ limt!1�aiðtÞ and likewise for �Aið1Þ and �Pið1Þ, wehave

�a0ð1Þ ¼ ½ðdþ kþ l0Þðdþ hþ l2Þ��1½dþ kþ hþ l2

� kðl1 � h� l2ÞIð0; dþ l1;0Þ�; ð29Þ

L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109 105

�a1ð1Þ ¼ ðdþ hþ l2Þ�1ðl1 � h� l2ÞIð0; dþ l1; 0Þ; ð30Þ

�a2ð1Þ ¼ 1=ðdþ hþ l2Þ; ð31Þ

�A0ð1Þ ¼ ½ðdþ kþ l0Þðdþ hþ l2Þ��1½l0ðdþ hþ l2Þ þ kðhþ l2Þ

þ ðl1 � h� l2ÞIð0; dþ l1;0Þ�; ð32Þ

�A1ð1Þ ¼ ðdþ hþ l2Þ�1½hþ l2 þ dðl1 � h� l2ÞIð0; dþ l1;0Þ�;

ð33Þ

�A2ð1Þ ¼ ðhþ l2Þ=ðdþ hþ l2Þ ð34Þ

where

Iðt; a; bÞ ¼ ea2=4affiffiffiffiffiffiffiffiffip=a

p1�U t

ffiffiffiffiffiffi2ap

þ a� bffiffiffiffiffiffi2ap

� �� : ð35Þ

Details of these derivations are omitted. It can easily be shownthat the fundamental relationship (27) holds for t ¼ 1, for each ivalue.

It then follows that the long-term annual premium rates, �Pið1Þ,satisfy, from (28),

�P0ð1Þ ¼ ½ðkþ l0Þðhþ l2Þ þ dl0 þ dkðl1 � h� l2ÞIð0; dþ l1;0Þ�=½dþ kþ hþ l2 � kðl1 � h� l2ÞIð0; dþ l1;0Þ�; ð36Þ

�P1ð1Þ ¼ ½hþ l2 þ dðl1 � h� l2ÞIð0; dþ l1;0Þ�=½1� ðl1 � h� l2ÞIð0; dþ l1; 0Þ�; ð37Þ

�P2ð1Þ ¼ ðhþ l2Þ: ð38Þ

3.3. Life expectancies

Finally, the life expectancy for someone presently in stage i is

ei ¼Xmþ1

j¼i

l0jZ 1

0tqijðtÞdt: ð39Þ

The life expectancies, given in (39), can be rewritten as

ei ¼Xmþ1

j¼i

l0jQ ij ð40Þ

where

Q ij ¼Z 1

0tqijðtÞdt: ð41Þ

Expressions for these integrals Qij are derived by utilizingextensively the integral I5 of B&D (Appendix A.1). It can thencebe shown that

Q 00 ¼ 1=ðkþ l0Þ2; ð42Þ

Q 01 ¼k

kþ l0

12af1� l1Ið0;l1;0Þg þ

1kþ l0

� �Ið0;l1; 0Þ

� �; ð43Þ

Q 02¼k

ðkþl0Þðhþl2Þkþl0þhþl2

ðkþl0Þðhþl2Þ�l1

2a

�

þ ðkþl0Þðhþl2Þ�l1ðkþl0þhþl2Þðkþl0Þðhþl2Þ

þl21

2a

� Ið0;l1;0Þ

�; ð44Þ

Q 11 ¼1

2a½1� l1Ið0;l1;0Þ�; ð45Þ

Q 12 ¼1

hþ l2

� �1� l1

2aþ l2

1

2a� ðl1 � h� l2Þ

ðhþ l2Þ

� Ið0;l1;0Þ

� �; ð46Þ

Q22 ¼ 1=ðhþ l2Þ2: ð47Þ

Substituting back into (39), we can show that

e0 ¼ 1þ l1

ðhþ l2ÞIð0;l1;0Þ; ð48Þ

e1¼1

ðkþl0Þ2

�l0þkðkþl0þhþl2Þ

� kðhþl2Þ

ðkþl0Þðhþl2Þ�l1ðkþl0þhþl2Þ�

Ið0;l1;0Þ�; ð49Þ

e2 ¼ ðhþ l2Þ�1: ð50Þ

4. Health-care costs

It is clear that some insurance entities are equivalent to varioushealth-care counterparts. Thus, for example, the value of the t-yearcontinuous term life annuity policy, �aiðtÞ, of Eq. (11) could insteadbe the expected cost of health care for someone currently in stage iassuming cost is $1 per unit time in the future.

These costs can be adjusted to allow for stage dependent costs.For example, the health cost for an individual currently in stage i(e.g., i ¼ 0, without HIV infection) who develops AIDS at a futuretime during the period ð0; tÞ could be adjusted to

Hðt; iÞ ¼Z t

0ðc1i þ c2iehiuÞe�duqi;mþ1ðuÞdu ð51Þ

where c1i; c2i and hi are constants. In (51), it is assumed there issome fixed care cost c1i for stage i (e.g., cost of basic medical testsor etc.) along with a variable cost c2iehiu;0 < u < t, related to the ini-tial stage i, over time (e.g., continuing cost of treatment includingmedications, say). The term e�du is used to discount the cost to timet ¼ 0 so that costs evaluated over time are comparable; this com-pares with the future $1 value payout having a current value of$(e�du). Clearly, interest centers on the overall time 0 6 u 6 t and(51) is concerned with the overall cost of going from the presentstage i (at t ¼ 0) to the AIDS stage i ¼ mþ 1 (at t ¼ t). Other formu-lations can be validated; for example, fc1iehiug could be replaced byðc1iuÞ. Or, interest may center on the care costs up to some otherspecific stage j, and not necessarily the j ¼ mþ 1 stage of (51).

We can then show that, for the health-care cost function of (51),

Hðt;0Þ ¼X2

k¼1

ck0kðkþ l0 � h� l2Þ

� 1dk0 þ hþ l2

� �e�ðdk0þkþl0Þt � e�ðdk0þhþl2Þt� �

þ l1 � h� l2

dk0 þ hþ l2

� �Gðt;l1; hþ l2; dk0 þ hþ l2Þ

� l1 � k� l0

dk0 þ kþ l0

� �Gðt;l1; kþ l0; dk0 þ kþ l0Þ

þ ðl1 þ dk0Þðkþ l0 � h� l2Þðdk0 þ kþ l0Þðdk0 þ hþ l2Þ

Gðt;l1 þ dk0;0;0Þ�; ð52Þ

Hðt;1Þ ¼X2

k¼1

ck1

ðdk1 þ hþ l2Þ

� 1� e�ðdk1þhþl2Þt � ðl1 þ dk1ÞGðt;l1 þ dk1;0;0Þ� l1 � h� l2ÞGðt;l1; hþ l2; dk1 þ hþ l2Þ� �

; ð53Þ

Hðt;2Þ ¼X2

k¼1

ck2

ðdk2 þ hþ l2Þ1� e�ðdk2þhþl2Þt�

ð54Þ

106 L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109

where d1i ¼ d and d2i ¼ d� hi, and where Gðt; a; b; cÞ was defined in(10). The details are omitted but the derivations use the integrals ofB&D (Appendix A.1) repeatedly.

Another formulation of total health care costs, for someone instage i at time T, is given by

CðT; iÞ ¼ hið1þ kiÞZ 1

Tðt � TÞe�dðt�TÞqi;iðt � TÞdt

þXmþ1

j¼iþ1

Z 1

T

ddt

qijðt � TÞ�

e�dðt�TÞCðt; jÞdt ð55Þ

for i ¼ 0; . . . ;mþ 1. The first term in (55) represents the cost perunit time, hið1þ kiÞ, of being in stage i at a future time t multipliedby the expected time of tenure in this i stage; as before, there is theneed to adjust a future $1 to the current $(e�du), u > T , so that theevaluated costs are comparable. The second term in (55) representsthe differential cost of care for each future stage ðj ¼ iþ 1; . . . ;

mþ 1Þ. Thus, for example, the health-care costs for an individualwho has just been diagnosed with AIDS is, from (55), CðT;2Þ, foundto be

CðT;2Þ ¼ h2ð1þ k2Þðdþ hþ l2Þ

2 e�ðhþl2ÞT ; T > 0: ð56Þ

For an individual found to be HIV infected at time T, the overallhealth-care cost is

CðT;1Þ ¼ h1ð1þ k1Þ2k

e�l1T�kT2 � edT IkðT; dþ l1;0Þh i

þ h1ð1þ k2Þe�ðhþl2ÞT

ðdþ hþ l2Þðdþ 2hþ 2l2Þ� 1� ðdþ l1 þ hþ l2ÞIð0; dþ l1 þ hþ l2;0Þ �

ð57Þ

where Iðx; a; bÞ is defined in (35) and where Ikðx; a; bÞ is the same asIðx; a; bÞ but with a everywhere replaced by k.

The range of possible functions of interest is large and dependson the cost structure per unit time assumed. In general thesehealth care cost functions, as for the insurance functions, dependon knowledge of the conditional probabilities qijðtÞ defined in (1).

5. Exponential waiting times for Ui;Vi

When all Ui and Vi; i ¼ 1; . . . ;mþ 1, waiting times are exponen-tially distributed, Ui � ExpðliÞ and Vi � ExpðkiÞ, the derivations forthe conditional probabilities qijðtÞ simplify. This exponentialassumption does not hold for the HIV/AIDS process in general,although Bailey [6] in effect has proposed exponential times butwith m > 6 or 7 as an estimated process; likewise, for other epi-demic processes with nonlinear transition rates. There can be how-ever other processes (especially those whose underlying transitionrates are linear, e.g., in queues) for which this exponential assump-tion pertains.

We give without proof the basic result for qijðtÞ. Thus, we canshow that

qijðtÞ ¼ ½Fi;j�1ðtÞ � Fi;jðtÞ�Yj�1

k¼1

kk

kk þ lk; i; j ¼ 0; . . . ;m; ð58Þ

where

FijðtÞ ¼ PðYij 6 tÞ ¼ 1�Xj

k¼1

Yj

r¼ir–k

ak

ar � ak

0@

1Ae�akt;

j ¼ i; iþ 1; . . . ;m;

with ak ¼ lk þ kk, andQi

r¼ir–kð�Þ � 1 when i ¼ j ¼ mþ 1, and where

qmþ1;mþ1ðtÞ ¼ e�amþ1t : ð59Þ

An application of this model to some simple models for insur-ance functions can be found in Dayananda [7]. Derivation of vari-ous insurance functions and health-care costs can proceed alongthe lines outlined in Sections 3,4. Thus, for example, the expectedhealth-care cost for someone currently in stage i can be found from(11). We obtain

�aiðtÞ ¼Xmþ1

s¼i

ls

Ys�1

r¼i

kr

!Xs

k¼i

1� e�ðdþakÞt

akðdþ akÞ

� Ys

r¼ir–k

ðar � akÞ�1: ð60Þ

6. Numerical illustrations

Numerical work can be carried out to study the influence ofmodel parameters on the insurance functions. We focus attentionon two important insurance functions, viz., the value at timet ¼ 0 of a t-year continuous life annuity for an insurance policy is-sued to someone in stage i; �aiðtÞ; i ¼ 0;1;2. We also look at the netsingle premiums for these policies when the policy-holder was in-fected at the time the policy was issued �A1ðtÞ and their long termvalues as t !1, as well as premium increases for uninfected indi-viduals �A0ðtÞ. (Similar comparisons could be made for the otherinsurance functions and for health-care functions.).

Since the many insurance functions developed in Section 3 areexpectations, they depend heavily on the probabilities qijðtÞ;i < j ¼ 0;1;2. Hence, sensitivities to varying parameter values ob-served in these probabilities will also be reflected in the insurancefunctions. Billard and Dayananda [1] investigated the sensitivityover different values of the parameters (k;a;l0;l1;l2; h) on theseqijðtÞ terms; a summary of those findings is in Table 1. Thus, e.g.,all applicable qijðtÞ terms are sensitive to changes in values of a.As in [1], we use values for (k;a;l0;l1;l2; h) to reflect those knownto pertain for these parameters. Thus, death rates for 20-, 30-, 40-,and 50-year-olds are known to produce values for lk; k ¼ 0;1;2,that are on or around l ¼ :001; :0026; :0042; :0057, respectively.Expected incubation times (from infection to diagnosis with AIDS)of about 4, 9.5, 15, and 28 years, translate into the parameter a tak-ing values on or around a ¼ :05; :009; :005; :001, respectively; theexpected time ðh�1Þ from diagnosis to death from AIDS of 2, 10,16, 20 years, equates to the parameter h being on or aroundh ¼ :5; :1; :08; :05, respectively; and k values on or aroundk ¼ :005; :01; :02; :05; :1, correspond to 0.5, 1, 2, 5, and 10 suscepti-bles per 100 being infected with HIV, respectively. We note that animpact of HIV/AIDS drug treatments is that incubation times can beincreased and/or the expected time from diagnosis to death fromAIDS can be increased. Therefore, the numerical study herein in-cludes parameter values that reflect the impact on individualswho are faithful to the treatment regime, viz., a ¼ :001 (which cor-responds to an expected incubation period of 28 years), and h ¼ :05(for an expected time from diagnosis to death from AIDS of20 years). For individuals who do not undertake treatment, theseexpected incubation times are closer to 4 years for young childrenand 9.5 years for adults (i.e., a ¼ :05; :009, respectively), and theexpected time from diagnosis to death is closer to 2 years (i.e.,h ¼ :5). Thus, the range of values investigated run the gamut acrossthe levels of the use of and/or adherence to treatment therapies.

Accordingly, we display in Fig. 1(a) the plots for the continuoustime annuity for a susceptible (i.e., someone who has not been in-fected with the HIV virus at the time the policy was taken out) �a0ðtÞof (20), for changing infection rates k ¼ ð:005; :01; :02; :05; :1Þ ford ¼ :05; h ¼ :08;a ¼ :005;l0 ¼ :0026;l1 ¼ :0042 and l2 ¼ :0057.It is clear from these plots that the different infection rates haveno impact on this annuity (until at least 20–30 years, with onlysmall differences in impact for longer time frames). It is also thecase that this annuity is essentially the same for changing

Table 1Sensitivities of probabilities {qij; i < j ¼ 0;1;2} to parameters values.

q01 q02 q03 q12 q13 q23

k Sensitive Sensitive Sensitive N/A N/A N/Aa Sensitive Sensitive Sensitive Sensitive Sensitive N/Al0 Insensitive Insensitive Insensitive N/A N/A N/Al1 Insensitive Insensitive Insensitive Insensitive Insensitive N/Al2 N/A Insensitive Insensitive Insensitive Insensitive Insensitiveh N/A Sensitive Sensitive Sensitive Sensitive Sensitive

Fig. 1. (a). Annuity rates for uninfectives �a0ðtÞ by time as infection rate k varies. (b). Annuity rates for uninfectives �a0ðtÞ by time as interest rate d varies.

Table 2�a0ð1Þ and �a1ð1Þ for some ðd;aÞ values.

Interest Rate d (a) �a0ð1Þ � a� (b) �a1ð1Þ � a�

.05 .02 .009 .005 .001 .05 .02 .009 .005 .001

.01 (1%) 60.71 61.21 61.86 62.49 65.12 13.71 15.50 17.78 20.01 29.30

.025 (2:5%) 32.47 32.69 32.95 33.19 34.06 11.75 13.16 14.86 16.45 22.15

.05 (5%) 18.18 18.26 18.36 18.44 18.68 9.46 10.44 11.55 12.49 15.28

.075 (7:5%) 12.58 12.62 12.67 12.70 12.79 7.89 8.60 9.35 9.95 11.44

.1 (10%) 9.61 9.63 9.67 9.67 9.71 6.75 7.29 7.81 8.20 9.06

� Mean time infection to diagnosis of 4–28 years.

L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109 107

a; h;l0;l1 and l2 values (plots not shown). This is not surprisingfor the lk; k ¼ 0;1;2, values as Billard and Dayananda [1] has al-ready shown that the functions qijðtÞ (used in the derivation of�aiðtÞ, see (11)) were not sensitive to changes in these death ratesfrom non-HIV/AIDS related causes. Collectively, then, for a giveninterest rate d, the impact on annuities for someone who has notbeen infected at the time of taking out the policy for at least a20–30 year term is negligible.

There is an impact, as would be expected, on annuity values asthe interest rate d varies; see Fig. 1(b) where d ¼ ð:01; :025;:05; :075; :1Þ for a ¼ :005; k ¼ :005; h ¼ :08;l0 ¼ :0026;l1 ¼ :0042and l2 ¼ :0057. This result is also evident when studying the longterm annuity values as t !1. Table 2(a) provides these long termvalues as d;a change over ð:01; :025; :05; :075; :1Þ � ð:05; :02;:009; :005; :001Þ. Thus, as the mean incubation period increases(i.e., a decreases), the long term annuity increases; while as theinterest rate d increases, the long term annuity decreases asexpected. Fig. 2 shows plots for the continuous time annuity,�a1ðtÞ, for someone already infected with HIV initially as h; d anda vary for l1 ¼ :0026 and l2 ¼ :0057. Thus, Fig. 2(a) takesh ¼ ð:5; :1; :08; :05Þ (i.e., reflecting increases in the expected timefrom diagnosis with AIDS to death) for interest rate d ¼ :05 (i.e.,5%) and a ¼ :009 (i.e., incubation period of 9.5 years). We firstobserve there is no significant difference in these annuities for a6–8 year policy (6 6 t� 6 8 approximately). When h ¼ :5 (i.e., theexpected time to death from AIDS diagnosis is 2 years), the presentvalue of �a1ðtÞ is perforce low with a limiting value of �a1ð1Þ ¼ 8:15;

whereas when h ¼ :05 (i.e., the expected time to death from AIDSdiagnosis is 20 years), a higher present value pertains with�a1ð1Þ ¼ 12:98. Plots for d ¼ ð:01; :025; :05; :075; :1Þ for h ¼ :08and a ¼ :009 are shown in Fig. 2(b). There are no real differencesacross d for a 3–4 year policy (t� near 3.5). After that as d increases,then the present values �a1ðtÞ must decrease for any t value. In thiscase, �a1ð1Þ ¼ 11:66 when the interest rate is 5%. Thea ¼ ð:05; :02; :009; :005; :001Þ plots when h ¼ :08 and d ¼ :05 aredisplayed in Fig. 2(c). Again for a 6–8 year policy, there is no signif-icant difference in the annuity �a1ðtÞ for different a values. Sinceannuity values represent the present values of future paymentsof $1, they are expected to increase as the respective h; d;a valuesdecrease; see Figs. 2(a)-(c). All increase with time with the higherannuities occurring for lower a values since the later payouts willtherefore be higher. Thus we have �a1ð1Þ ¼ 9:51 when a ¼ :05 (i.e.,when the mean incubation period is 4 years) compared with�a1ð1Þ ¼ 15:579 when a ¼ :001 (i.e., when the mean incubationperiod is 28 years). Table 2(b) gives these long term results for�a1ð1Þ for several ðd;aÞ values. Table 3(a) displays the correspond-ing results for several ðd; hÞ values.

Since the �aiðtÞ represent the amount needed now for a futurepayout at time t, then we expect that a policy for an individualwho is infection free will need a larger payout than one for some-one who is now infected (and so has a lower life expectancy).Therefore, e.g., when d ¼ :025 and a ¼ :02, we see from Table 2,that �a0ð1Þ ¼ 32:69 > 13:16 ¼ �a1ð1Þ. Plots for �a2ðtÞ as d; h and l2

change are similar to those for �a1ðtÞ in shape and trends but with

Fig. 2. (a). Annuity rates for infectives �a1ðtÞ by time as expected time from diagnosis to death h�1 varies. (b). Annuity rates for infectives �a1ðtÞ by time as interest rate d varies.(c). Annuity rates for infectives �a1ðtÞ by time as expected incubation time (via a) varies.

Table 3�a1ð1Þ and �a2ð1Þfor some ðd; hÞ values.

Interest Rate d (a) �a1ð1Þ � h� (a) �a2ð1Þ � h�

.5 .1 .08 .05 .5 .1 .08 .05

.01 (1%) 5.66 12.00 13.71 18.22 1.94 8.64 10.45 15.22

.025 (2:5%) 5.37 10.55 11.75 14.74 1.88 7.65 9.03 12.39

.05 (5%) 4.94 8.69 9.46 11.15 1.80 6.42 7.37 9.46

.075 (7:5%) 4.56 7.38 7.89 8.94 1.72 5.53 6.22 7.65

.1 (10%) 4.22 6.39 6.75 7.45 1.65 4.86 5.39 6.42

� Mean time to death after diagnosis 2–20 years.

108 L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109

lower annuity values; see, e.g., Fig. 3 which shows the changes ash ¼ ð:05; :08; :1; :5Þ for d ¼ :05 and l2 ¼ :0057. This is also reflectedby a comparison of the annuity values for the long term, �a2ð1Þ, dis-played in Table 3(b) with those for �a1ð1Þ in Table 3(a), forl2 ¼ :0057. For example, when d ¼ :025 and h ¼ :08, we see that�a1ð1Þ ¼ 11:75 > 9:03 ¼ �a2ð1Þ.

Now consider the influence of different parameters on �A1ðtÞ, thesingle premium for an infected individual taking out a $1 policy.

Fig. 3. Annuity rates for diagnosed �a2ðtÞ by time as expected time from diagnosis todeath h�1 varies.

Given the relationship (27), these plots (not shown) behave simi-larly to those for �a1ðtÞ, except for a ‘‘y-axis’’ scale change. The effectof the multiplier d is that there is an immediate impact even atsmall time points (e.g., t� ¼ 1–2 years) rather than the t� ¼ 6–8 years observed for �a1ðtÞ in Fig. 2. Note that the longer a personis expected to live, then the higher is the value of the single pre-mium of the policy. Table 4 provides the limiting values ast !1 for combinations of ðd ¼ ð:01; :025; :05; :075; :1Þ anda ¼ ð:05; :02; :009; :005; :001Þ, which values can be compared withthose in Table 2(b) for �a1ð1Þ.

Finally, in Table 5, we give the percentage increase in single pre-miums for interest rates d ¼ ð:03; :04; :05Þ needed for an individual

Table 4�A1ð1Þ for some ðd;aÞ values.

Interest Rate d a�

.05 .02 .009 .005 .001

.01 (1%) 0.86 0.85 0.82 0.80 0.71

.025 (2:5%) 0.71 0.67 0.63 0.59 0.45

.05 (5%) 0.53 0.48 0.42 0.32 0.24

.075 (7:5%) 0.41 0.35 0.30 0.25 0.14

.1 (10%) 0.33 0.27 0.22 0.18 0.09

� Mean time infection to diagnosis of 4–28 years.

Table 5Increase in (�A0) for some ðd; kÞ values.

Interest Rate d k�

.020 .021 .022 .023 .024 .025

.03 (3%) 13.8 16.9 19.9 22.3 25.6 27.4

.04 (4%) 29.2 33.1 36.8 40.4 43.8 47.2

.05 (5%) 53.4 58.2 62.9 67.5 71.9 76.3

� 2–2.5 per 100 susceptibles become infected.

L. Billard, P.W.A. Dayananda / Mathematical Biosciences 249 (2014) 102–109 109

who is exposed to HIV at age 20, with a ¼ :01 and l0 ¼ :0019, forthe case that k ¼ ð:020; :021; :022; :023; :024; :025Þ. Thus, e.g., whenthe rate of susceptibles becoming infected is 2 per 100 (k ¼ :02)when the interest rate is 3% (d ¼ :03), the increase in the single pre-mium (�A0) is 13.8% over what is expected were there no exposureto HIV (as calculated from the Institute of Actuaries tables [8]).From these comparisons, it is clear that the impact of the presenceof HIV is substantial on premium rates.

7. Conclusion

Insurance companies and healthcare organizations are particu-larly interested in the impact of high risk individuals and groupswhen it comes to determining the premium rates and related func-tions for life policies and healthcare costs. In this paper, we havetaken fundamental probabilities in a compartmental model devel-oped in [1] to derive a number of basic insurance functions, viz.,premium rates, annuities, and life expectancies, for individuals ex-posed to HIV/AIDS. In this work, HIV/AIDS development and inter-est rates have been considered to be a continuous time process.The numerical study on a range of parameter values which coversinfected and/or diagnosed individuals who are fully faithful to theirtreatment regimes to those who do not undertake treatment ther-apies was conducted.

Thus, it is seen that for a given interest rate, there is a negligibleimpact on a person uninfected when taking out a 20–30 year termpolicy. On the other hand, as interest rates increase, the long termannuity decreases. For someone infected when taking out a policy,there is no real effect on annuities for the first 6–8 years; also asinterest rates increase, there is no effect for a 3–4 year policy, butthere is a decrease in annuities for longer term policies. The overallconclusion is that the impact on premium rates (and hence also onother insurance functions) is substantial for individuals infectedwith HIV. Similar conclusions can be made for correspondinglyequivalent health-care functions.

Appendix A

A.1. Probabilities qijðtÞ

The set of probabilities qijðtÞ; i 6 j ¼ 0;1;2;3, derived in [1], aregiven by:

q00ðtÞ ¼ e�ðkþl0Þt; t > 0; ðA:1Þq01ðtÞ ¼ kGðt;l1; kþ l0; kþ l0Þ; t > 0; ðA:2Þ

q02ðtÞ ¼k

ðkþ l0 � h� l2Þe�ðhþl2Þt � e�ðkþl0Þt

� ðl1 � h� l2ÞGðt;l1; hþ l2; hþ l2Þþ l1 � k� l0ÞGðt;l1; kþ l0; kþ l0Þ� �

; t > 0; ðA:3Þ

q03ðtÞ ¼ 1� kðkþ l0 � h� l2Þ

e�ðhþl2Þt � ðhþ l2 � l0Þk

e�ðkþl0Þt�þ ðl1 � h� l2Þ Gðt;l1; kþ l0; kþ l0Þ

�� ðt;l1; hþ l2; hþ l2Þ

�; t > 0; ðA:4Þ

q11ðtÞ ¼ e�l1t�at2; t > 0; ðA:5Þ

q12ðtÞ ¼ e�ðhþl2Þt � e�l1t�at2

� ðl1 � h� l2ÞGðt;l1; hþ l2; hþ l2Þ; t > 0; ðA:6Þq13ðtÞ ¼ 1� e�ðhþl2Þt

þ ðl1 � h� l2ÞGðt;l1; hþ l2; hþ l2Þ; t > 0; ðA:7Þq22ðtÞ ¼ e�ðhþl2Þt; t > 0; ðA:8Þq23ðtÞ ¼ 1� e�ðhþl2Þt ; t > 0: ðA:9Þ

References

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