An Insurance Based Model to Estimate the Direct Cost ...

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The total health care cost of dengue inward case management in the western province hospitals in Sri Lanka during the last 4 years increased significantly [2].
International Journal of Pure and Applied Mathematics Volume 117 No. 14 2017, 183-189 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue

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An Insurance Based Model to Estimate the Direct Cost of General Epidemic Outbreaks S.S.N. Perera Research & Development Centre for Mathematical Modeling, Faculty of Science, University of Colombo, Colombo 03, Sri Lanka [email protected]

Abstract Epidemic outbreaks, which have been regularly seen in many tropical and subtropical countries, have caused a critical burden on public health sector. Mathematical models with modern financial tools provide new ideas to redefine many existing problems in to different dimension. Considering the fractions of classical compartments as probability densities, the SIR (susceptible, infected, recovered) model is converted into a probabilistic model. Defining the expected values of the future benefit and premium payment, the epidemic insurance is defined. Considering the premium as an average value of future direct medical cost, sensitivity of the present financial burden due to direct medical cost is analysed with respect to the risk of disease spread. Further, such sensitivity is compared with respect to the efficiency of the control strategies. AMS Subject Classification: 91G99, 92D25, 92D30 Key Words and Phrases: Epidemic outbreaks, SIR model, Expected value, Equivalence principle

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Introduction

Dengue, Ebola, Zika, SARS, Chikungunya and avian influenza are among the most critical epidemics of infectious diseases in tropical and subtropical regions of the world. Such diseases represent a significant economic and public health burden in many endemic countries especially in the developing region [1]. In recent decades, it was observed a significant increase in the number of recorded infected cases and hence an increase the public health burden. The incidence of dengue has grown significantly around the world in the past three decades. Over 2.5 billion people, around 40% of the world’s population are now at risk from dengue. According to the World Health Organization (WHO) reports, currently 50-100 million dengue

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infections occurs annually in more than 100 countries in tropics and subtropics. In Sri Lanka, during the first eight months of the year 2017, more than 100,000 suspected dengue cases have been reported from all over the island. The reported cases of the other kind of epidemic infectious diseases such as Ebola, Zika, SARS and influenza were also significantly high during the last decade [1]. Quantifying the burden of infectious diseases is critical for policymakers, health authorities and government bodies to set policy priorities and allocate required funds to control the disease. The total health care cost of dengue inward case management in the western province hospitals in Sri Lanka during the last 4 years increased significantly [2]. Like Sri Lanka, many of the other tropical and subtropical countries face same economic burden due to epidemic outbreaks. Burden of the outbreaks can be broken into two classes, namely direct and in-direct cost. For example, the individual based cost can be identified as medical care related expenditures and the public/society based cost can be viewed in term of the expenditure due to disease controlling program. This study is an attempt to develop an insurance model to estimate the direct medical burden in general epidemic outbreaks. First, using SIR epidemic model we propose insurance method to compute future direct medical expenses. Simulations are carried out to compare the burden due to future direct cost with respect to model parameters and the control strategies. 2

Mathematical Model

First, the entire human population (N ) is divided in to three compartments, namely, Susceptible (S) as previously unexposed to the disease, Infected (I) as currently colonized by the virus and Recovered (R) as already cleared from the infection. Ignoring births, deaths and migration, considering transition only from compartment S to I and I to R, defining S(t), I(t) and R(t) as the number of susceptible, infected and recovered at time t, respectively and taking β > 0, as a contact (infection) rate, γ > 0 as recovery rate, the dynamic of disease spread is given by (1)-(3) [3, 4]. β dS(t) = − S(t)I(t) (1) dt N β dI(t) = S(t)I(t) − γI(t) (2) dt N dR(t) = γI(t), (3) dt with initial condition S(0) > 0, I(0) > 0, R(0) = 0 and S(t) + I(t) + R(t) = N , ∀t. Taking S ∗ (t) as S(t)/N , I ∗ (t) as I(t)/N , R∗ (t) as R(t)/N , introducing u(t) (0 ≤ u(t) ≤ 1) as the control strategy at time t = t, imposing the condition, S(t) + I(t) + R(t) = N , ∀t and dropping ∗, the system (1)-(3) can be reduced to dimensionless two-dimension system and it reads as dS(t) = −β(1 − u(t))S(t)I(t) dt dI(t) = β(1 − u(t))S(t)I(t) − γI(t) dt

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(4) (5)

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with the given initial condition and R(t) = 1 − S(t) − I(t), ∀t. The model parameter, β varies with the type of disease, the season, demography and other external factors whereas, the model parameter, γ depends on mainly biological factors. Further, the ratio between, β and γ is defined as the basic reproduction number and is denoted by R0 and it can be considered as the risk of disease spread. It can be easily notified that, if R0 > 1, the disease spreads out and if R0 < 1, it dies out. By considering the fraction functions, S(t) and I(t) in system 4 - 5 can be interpreted as the probability of an individual being susceptible and infected, respectively, at the time t. The control variable, u(t), represents the reduction in the risk of disease spread. If u(t) = 0 reflects without any control and whereas u(t) = 1 reflects 100% efficiency of it. Further, the control depends basic reproduction number, R0 (u), and is defined as ratio between β(1 − u(t)) and γ. 3

Developing An Insurance Based Model

Different identification can be proposed to this classical compartment SIR model from an insurance point of view. From an actuarial perspective, the three compartments play significantly different roles in an insurance model. The members in the susceptible compartment face the risk of being infected form a market that could contribute premiums to an insurance fund, in return for the coverage for direct medical cost incurred if infected. During the outbreak of an epidemic, the infected policyholders would benefit from the claim payments provided by the insurance fund. Since, S(t) and I(t) denote probability density functions of an individual being susceptible and infected respectively, we now propose actuarial based techniques to develop the quantities of interest for an infectious epidemic disease insurance model. In the infectious disease model, policyholders are committed to paying premiums continuously as long as they remain as susceptible and medical expenses are continuously reimbursed for each infected policyholder during the whole period of treatment. Considering the insurance liability side, the total average (expected) present value of t period unit benefit payment, B(t), is given by (7) [5]. Z

E[B(t)] =

t

exp(−δt) Prob.

of an individual being infected at time t dt

(6)

0

Z

E[B(t)] =

t

exp(−δt)I(t)dt.

(7)

0

On the revenue side of the insurance, the total average (expected) value of the present value of t period premium of payments, P (t), is given by (9) [5].

E[P (t)] =

Z

t

exp(−δt) Prob.

of an individual being susceptible at time t

dt

(8)

0

E[P (t)] =

Z

t

exp(−δt)S(t)dt.

(9)

0

In both (7) and (9), δ denotes the force of interest and it captures the uncertainty of future payments due to external factors. Using the concept of equivalence

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0.7 1.4

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Figure 2: Variation of infected populaFigure 1: Variation of the premium tion fraction with respect to different β with respect to basic reproduction num(0.6 < β < 10.66) corresponding 1.1 < ber without control. R0 < 19.5 and γ = 0.55.

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Figure 3: Variation of the premium with Figure 4: Variation of infected popularespect to the efficiency of control, R0 = tion fraction with respect to u, (0 < u < 1), and β = 2.67, γ = 0.55. 18.0. principle, the level premium, π(u), is defined as (10) π(u) = 4

E[B(t)] . E[P (t)]

(10)

Analysis, Results & Discussion

Numerical simulation is carried out using MATLAB ode45 solver. Figure 1 displays the sensitivity of the premium (present burden of the direct medical cost) with respect to the basic reproduction number (risk of the spread). One can see that when R0 exceeds a threshold point, the present burden exceeds the unit. This means that, in order to have an unit benefit claim it is needed to pay more than unit once the R0 exceeds the critical point. Figure 2 provides evidence that the infected fraction (probability) rises when R0 increases.

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1.5

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Figure 5: Top-left: Sensitivity of the premium with respect to R0 and control efficiency, Top-right: Sensitivity of the premium with respect to R0 (u) and control efficiency, Bottom-left: Corresponding R0 variation, Bottom-right: Corresponding R0 (u) variation. Figure 3 displays the sensitivity of the premium (present burden of the direct medical cost) with respect to the efficiency of the control, u. It is observed that the present burden decreases when the control efficiency increases. Further, it is seen that the present burden can be reduced by imposing higher control efficiency. Though the infected rate and recovery rate are constant, the risk of spread changes since the basic reproduction number now depends on the control efficiency u. Figure 4, summarizes the variation of the infected fraction with respect to the control efficiency. Figure 5 (Top left) displays the sensitivity of premium with respect to both the risk of disease, R0 , and the control efficiency. Further, it summarizes (top-right) the sensitivity of premium with respect to the control efficiency and the control depends R0 . Variation of corresponding basic reproduction number and control depends basic reproduction number are displayed in Figure 5 bottom left/right respectively. 5

Conclusion

The finance burden due to epidemic outbreaks is a key concern in tropical and subtropical regions. We propose an insurance model to estimate the present finance risk due to future direct medical cost in general epidemic diseases. Considering the infected and susceptible fractions as the probability densities of an individual being susceptible and infected, defining the expected value of future benefit payments and premium payments, present financial burden is estimated. The variation of the present premium with respect to model parameters and the efficiency of controls are discussed and presented. This study can be extended to develop feasible premium schemes depending on different seasonal conditions by incorporating reported infected data. Further, the model can be extended by including indirect cost such

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as expenditure due to loss of human hours, death and community expenditures. References [1] D.J. Gubler, M. Meltzer, Impact of dengue/dengue hemorrhagic fever on the developing world, In: Maramorosch K, Murphy FA, Shatkin AJ, editors. Advances in Virus Research, 53, San Diego: Academic Press Inc. 35-70 (1999). [2] N. Thalagala, H. Tissera, P. Palihawadana, A. Amarasinghe, A. Ambagahawita, A. W. Smith, D.S. Shepard, Y. Tozan, Costs of DengueControl Activities and Hospitalizations in the Public Health sector during an Epidemic Year in Urban Sri Lanka, PLOS Neglected Tropical Diseases, 10(2), 1-13 (2016). [3] D.J. Dalley and J. Gani, Epidemic Modelling, Cambridge University Press (1999). [4] S.S.N. Perera, Pricing a Epidemiological Diseases Economic Burden, Proceedings, 6th International Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2017), Singapore 5-9, (2017). [5] N.L. Bowers BOWERS, H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt, Actuarial Mathematics, Schaumburg, IL: Society of Actuaries (1997).

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