Journal of Applied Mathematics

Volume 2017, Article ID 4264737, 6 pages

https://doi.org/10.1155/2017/4264737

## Analysis of Economic Burden of Seasonal Influenza: An Actuarial Based Conceptual Model

Research and Development Center for Mathematical Modeling, Faculty of Science, University of Colombo, 00300 Colombo, Sri Lanka

Correspondence should be addressed to S. S. N. Perera; kl.ca.bmc.shtam@pnss

Received 15 June 2017; Revised 26 August 2017; Accepted 5 September 2017; Published 11 October 2017

Academic Editor: Winston Garira

Copyright © 2017 S. S. N. Perera. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Analysing the economic burden of the seasonal influenza is highly essential due to the large number of outbreaks in recent years. Mathematical and actuarial models can be considered as management tools to understand the dynamical behavior, predict the risk, and compute it. This study is an attempt to develop conceptual model to investigate the economic burden due to seasonal influenza. The compartment* SIS* (susceptible-infected-susceptible) model is used to capture the dynamical behavior of influenza. Considering the current investment and future medical care expenditure as premium payment and benefit (claim), respectively, the insurance and actuarial based conceptual model is proposed to model the present economic burden due to the spread of influenza. Simulation is carried out to demonstrate the variation of the present economic burden with respect to model parameters. The sensitivity of the present economic burden is studied with respect to the risk of disease spread. The basic reproduction is used to identify the risk of disease spread. Impact of the seasonality is studied by introducing the seasonally varying infection rate. The proposed model provides theoretical background to investigate the economic burden of seasonal influenza.

#### 1. Introduction

Infectious diseases such as Dengue, Ebola, SARS, Zika, and various types of influenza are viral infections that are of major public health concern in tropical and subtropical countries [1]. Influenza, generally known as “the flu,” is a sessional infectious disease caused by virus. High fever, runny nose, sore throat, muscle pains, headache, cough, and feeling lethargic can be considered as common symptoms. Seasonal influenza can cause mild to severe illness and even death, particularly in some high-risk individuals such as pregnant women, children, very old people, and people with chronic underlying medical conditions such as cancer, diabetes mellitus, and heart diseases. Three different types of influenza viruses generally affect humans, namely, Type A, Type B, and Type C. In recent years, the influenza A virus subtypes H1N1 and H3N2 have been in circulation. In addition, there are two types of B viruses that are also circulating as seasonal influenza viruses. However, influenza C may be considered as less of a disease burden [2].

Normally, the influenza virus spreads over a short distance through the air from coughs or sneezes. Human mobility is the main responsible factor for the spread over long distance. Annual influenza outbreaks occur around the world and cause significant morbidity and mortality. Since the virus survives in cold environments with low relative humidity, normally the influenza outbreak can be seen in winter season in the Northern and Southern Hemispheres. Generally, in the tropical and subtropical region outbreaks may occur at any time of the year. However, in most of the tropical countries these peaks of infection are seen mainly during the rainy season [3]. In Sri Lanka, for the last few years, influenza epidemic outbreak has been generally observed during April to June and again in November to January.

From a public health and community point of view, seasonal influenza epidemics spread rapidly and are very difficult to control. Seasonal influenza spreads all over the world as an annual outbreak, making about three to five million severe illness cases and about 250,000 to 500,000 death cases [4]. In the United States approximately, annually, from 5% to 20% of the individuals are infected with influenza. Further, the United States health report indicates that about 3,000 to 49,000 influenza-associated deaths occur every year. According to the WHO and health report, in Sri Lanka, for the last few years, it has been generally observed that there is high trend concerning the spread of influenza. This epidemics risk causes enormous economic loss and serious adverse events leading to hospitalization. In USA, annual economic burden due to seasonal influenza is approximately more than 1 billion US $ [3–5]. However, in Sri Lanka the annual economic losses due to this epidemic risk have not yet been addressed.

Due to significant level of outbreaks in and around the world including Sri Lanka during recent years, investigating the economic burden of annual seasonal influenza is motivated. The economic burden can be classified in two ways, namely, individual level which accounts for medical and death expenditures and public level which accounts for the burden due to loss of human days. This study is an attempt to develop the conceptual model to capture economic losses in an individual level due to influenza. However, the proposed tool is only responsible for identifying losses for personal medical care. Classical compartment model is used to describe the spread of disease; model parameters are used to capture the seasonal/climatic impacts. Epidemiological insurance is designed to represent expenditures due to personal medical care and actuarial based computation tool is proposed to compute the economic burden.

This paper is organized in the following way. Classical compartment model, insurance based model, and actuarial tools are presented in Section 2. Mathematical analysis of the model is also presented in Section 2. In Section 3, numerical results and discussion are presented. Finally, conclusion remarks and further steps are highlighted in Section 4.

#### 2. Mathematical Models

In this section, first, we introduce the classical compartment deterministic model to study the dynamics of influenza. Secondly, mathematical and actuarial analysis of the epidemic model is discussed and a basic insurance model is constructed.

##### 2.1. Classical Compartment Model

Since influenza virus is constantly changing, people can get infected multiple times throughout their lives. This concept is mathematically modeled by using classical* SIS* (susceptible (), infected (), susceptible) compartments [6, 7]. Entire population is divided into two compartments, namely, susceptible as previously unexposed to the disease and infected as currently colonized by the virus. Since the objective is to investigate the annual seasonal economic losses due to individual medical care, impacts of the population demography, births, deaths, and migration, are not significant. Hence, omitting such effects, model is developed by considering transition only from compartment to and to . Let and denote the number of susceptible and infected ones at time , respectively. The increment of rate of change of infected cases is proportional to the number of susceptible individuals and the number of individuals previously infected. It also noted that such a rate is also proportional to number of transitions from infected to susceptible cases. Taking , as a potentially infective contact rate, as the probability of infection per contact, as the average disease duration, and as a total population, simple* SIS* model is given by (1a), (1b), and (1c).withTaking population fraction, , , introducing rate of infection, , as product of and and the rate of recovery, , and considering close environment (i.e., ) and dropping , the system (1a), (1b), and (1c) reads as [8, 9]

##### 2.2. Mathematical Analysis of the Model

Equation (2) can be further simplified and one can obtain an analytical solution given by Equation (2) has two equilibrium points, namely, disease-free equilibrium point () and the endemic equilibrium point . One can see, using linear stability analysis, disease-free equilibrium point, , is locally asymptotically stable if and unstable if , whereas the endemic equilibrium point, , is locally asymptotically stable if and unstable if [9–11]. The product of the rate of infection, , and the average duration of the infectious period, , is the expected number of new infections from one infected individual in a fully susceptible population through the entire duration of the infectious period. This parameter is known as a basic reproduction number, , and if , an introduced infectious individual leads to more than one infection so the disease spreads in the population. One can easily see that if , the infection in one individual cannot replace itself so the disease dies out. Figure 1 displays variation of infected population fraction with respect to different rate of infection, .