Some Mathematical Models and Applications Used in Epidemics

2.1. Epidemiological Models: SI and SIS

The mathematical epidemiological model that is probably the most widely used for theorizing about and emulating epidemics is the SIS model [10]. In this model which is implemented by an iterative calculation and is effective for epidemic diseases, the simplest version without demographic factors is defined as follows: It is assumed that in a closed society taken as the system there are no deaths, no births in the population, no migration into or out of the population and infected people are not re-infected after they have recovered. Of course these assumptions are valid for a limited interval of time. In SI model, over the course of the epidemic almost everyone eventually becomes infected. In the SIS model, each individual belongs to either the set of susceptible (S) or the set of infected people (I). Thus, the model can be shown as SIS, that is, a person is either susceptible or infected. In other words, a person does not gain immunity. After being infected, he/she can be susceptible again (i.e. after getting recovered). When a susceptible individual and an infected individual interact (by some manner), the former may be infected at a rate of infection, denoted by a constant 𝛽 which can be positive or negative depending on the values for two successive years, as can be seen in Eq (7). In this way, susceptibility is the same for each individual. Having accepted these rules the following equations are obtained [10]: Here we consider the SI and SIS models where the total population;

is a constant and is assumed not to change with time for our closed system. For the SI model;

In Eq.[2] the prod uct S(t)I(t) represents the possible encounters between the susceptible and infected people. The larger the term S t  and/or I t  , the larger the number of encounters between the susceptible and the infected people and therefore the greater the risk of spreading the disease. By using Eq.[4] and Eq.[5], firstly  is found as follows:

where R represents the recovered people. Since there are two groups, namely S and I, there is no R in the SI model, however, there are three groups, S, I and R in the SIR model. For Eq’s. [26], we accept the number of people leaving the susceptible group is equal to incoming people to the infected group (I). In the meantime infected people can recover and this average recovering 1 ratio is symbolized by 𝛾, while denotes the average infectious period. As time 𝑡 approaches 𝛾

infinity, S and R approach ‘0’ and ‘N’, respectively. For solving the differential Eq.[2] and Eq.[3], the Euler Method is used. A susceptible individual may get in contact with an infected individual with probability 𝛽 within a short time interval ∆t. With high probability 1– 𝛽∆𝑡, the susceptible individual stays in the susceptible state throughout the interval ∆t and this interval ∆t can be identified with a discrete simulation time step.

2.2. Euler Method

Euler equations are given by discretization of the differential Eq’s [2-5] and Eq’s [4-5]; for SI and SIS respectively; For the SI Model; 𝛽

Population in the next step can easily be determined in very short time intervals. To solve these equations, firstly the ratios 𝛽 and 𝛾 (used in SIS model) had to be known. An approximate prediction on infected and healthy people can be made by knowing these 𝛽 and 𝛾 ratios for a fixed number of populations in defined time intervals [10, 11, and 2].

3.1. Cases of Tuberculosis and Predictions in Norway

In this section, tuberculosis is considered as an epidemic disease. Norway was chosen as a developed country. Tuberculosis patients’ estimation in Norway between 2014 and 2016 was given in Table 1. First, using the patients’ population of the previous years, predictions of patients’ population for the following years are obtained. These predictions and exact results are given in Table 1. Although it is seen that tuberculosis case estimation (Estimated Number) for 2013 in SIS model is greater than the real number of cases (Exact Number), it is closer to real number of cases when compared to SI model’s estimation. In other words, exact number lies between the SIS and SI models’ estimations. After the validation and verification we could have predicted the number of cases for 2014, 2015 and 2016, it can be said that estimated numbers are between the predicted values by the two models. Not all models are appropriate for all kinds of diseases. Hence, the SI model is not convenient for tuberculosis, as is seen from Table 1. However, the SIS model is quite appropriate for this disease. Table 1. Number of Cases of Tuberculosis and Predictions in Norway* *Population for Norway was taken as 5 million. Year

3.2. Cases of Malaria and Predictions in Nigeria

90% of all malaria deaths occur in sub-Saharan Africa where Nigeria is a country in that region [12]. Malaria cases were estimated for the year 2013 by the SI and the SIS models with 0.030 % and 0.056 % error respectively. Table 2 shows Malaria cases and predictions for Nigeria for the years 2014-2017. Here, the SI model gave a better estimation according to SIS model for the mentioned disease and estimated number by the SI model is greater than that of the SIS model. It is convenient to use both the SI and the SIS models in the estimation for malaria, because the values found are very near the real data, as is seen from Table 2.

3.3. Cases of HIV/AIDS and Predictions in Ghana

In Table 3, the prediction of the HIV/AIDS patients’ population in Ghana was given by SI and SIS models between 2014 and 2017. Prediction for the cases in 2013 was made with a 0.004 20

% error by the SIS model. Real numbers of cases for 2012 and 2013 were given in Table 3 and it is seen that they are decreasing. Both SI and SIS models are capable of showing this trend. However, SIS model predictions for 2014 and beyond 2014 were smaller than those of the SI model. Predictions by SI and SIS were very close to each other and estimated numbers are also very close to exact numbers. Therefore, these models give us a chance to infer an excellent prediction from the data. For the case of HIV/AIDS both the SI and SIS models yield values near the real data. Table 2. Number of Cases of Malaria and Predictions in Nigeria* *

Table 3. Number of patients in Cases of HIV/AIDS and Predictions in Ghana* *Population for Ghana taken as 26 million. Year

3.4. Cases of CCHF and Predictions in Bulgaria

This model is mainly based on the data obtained 10 to 12 years ago. Here it is emphasized that the estimates found by the models exhibit anomalies when faced with extraordinary situations. The unusual increase in the diseases within the period 2007-2008 cannot be observed in the estimations. In Table 4, CCHF exact cases were given [13]. Awareness of being bitten by infected ticks can lead people to go to medical centres without spreading the disease and contaminate others. In CCHF (Crimean-Congo Hemorrhagic Fever), contamination occurs by way of blood and body fluids. Therefore it is much likely that medical personnel are at high risk compared with ordinary people. Inconsistency of these models’ estimation can be attributed to this fact. Since in this model the estimations are carried out by the data belonging to a very early period, the model cannot be expected to be sensitive for the abrupt increase in the number of cases between 2007-2008 tabulated in Table 4.

Table 4. Number of patients of CCHF Cases and Predictions in Bulgaria* *Population for Bulgaria taken as 7683000. Year

3.5. Cases of Measles and Predictions in Afghanistan

Measles used to account for estimated 30000-35000 deaths each year in Afghanistan [14]. Therefore, to reduce measles-related mortality, the nationwide measles vaccination campaign was conducted throughout 2002 and measles immunization coverage among 1-year-old babies was 75% in 2013 [15]. Table shows the exact number [16] and estimated numbers between 20052010. Here, SIS model gave the exact value. Pretty close values have been obtained by both models. As can be seen from this table, the SI and SIS models are appropriate for the measles. Table 5. Number of patients of Measles Cases and Predictions in Afghanistan* *Population for Afghanistan taken as 31411743. Year