PARAMETERS’ ESTIMATION, SENSITIVITY ANALYSIS AND MODEL UNCERTAINTY FOR AN INFLUENZA A MATHEMATICAL MODEL: CASE OF MOROCCO

In this paper, we consider a SEIS model type compartmental model to explore the evolution of the Influenza epidemic in Morocco. To consider a more realistic model we consider the seasonality of parameters, by considering time-dependent infection rate, time-dependent recovery rate, and time-dependent intervention rate. The next-generation matrix method is used to compute the threshold of equilibria’s stability R0. Based on real data of Influenza epidemic A infections in Morocco from the month of December 2018 to the month of March 2019 published by the Influenza Laboratory Surveillance Information system, we estimate the model parameters. In order to identify the most influential parameter in the proposed model, we carry out the local sensitivity analysis. We calculate sensitivity indices based on the estimated parameters which identify the most influential parameters. We perform also the uncertainty analysis to determine the relationship between the different parameters of the model.


INTRODUCTION
Influenza is an infectious disease of birds and mammals caused by RNA viruses of the family the influenza viruses. The most common symptoms are chills, fever, runny nose, sore throat, muscle pain, headache (often severe), coughing, weakness, and general discomfort. Although it is often confused with other influenza-like illnesses, especially colds, flu is an even more severe disease. Influenza may cause nausea and vomiting, especially in children [1].
Usually, the flu is transmitted through the air by coughing or sneezing, which creates aerosols that contain the virus. Influenza can also be transmitted through direct contact with bird droppings or nasal secretions, or by contact with contaminated surfaces. Airborne aerosols are believed to cause most infections, although the most important transportation is not entirely clear. Influenza is a seasonal epidemic disease, which appears every year due to a lack of awareness of basic knowledge [1]. For new infectious diseases, setting epidemiological parameters can help in decision-making.

Seasonal influenza (or "flu") is most often caused by type A or B influenza viruses. Influenza
The main parameter of a new infectious disease is the primary reproductive number (R 0 ), which is defined as the average number of secondary cases resulting from one primary state throughout the entire infection period in a fully susceptible population [4].
Epidemiological models are useful for comparing the effects of prevention or control measures.
One of the most important concerns about any infectious disease is its ability to invade the population. Many epidemiological models have a disease-free equilibrium (DFE) where the population remains in the absence of disease. R 0 directly determines the epidemic growth rate and the final number of infected people and is a dominant factor that must be taken into account in optimal policymaking. In fact, most models typically consider R 0 as a threshold parameter, so that if R 0 < 1, DFE is locally asymptotically stable, and disease cannot invade the population, but if R 0 > 1, DFE is unstable and invasion is always possible, Thus, the stability of epidemiological models can be analyzed with the help of R 0 [4]. There are a few methods available to estimate R 0 . For determination of the basic reproductive number, one of the best approaches is the Next Generation Matrix method [4,5,6].
Mathematical models play an important role in the investigation and the control of different phenomena [7,8], such as the propagation of human infectious diseases, which allowing health policy-makers to predict the impact of particular vaccine and treatment programs or to derive more efficient strategies based on different mathematical methods.
In [10], Kermack and McKendrick devised the Susceptible-Infected-Removed (SIR) model which has presented an interesting contribution to the mathematical theory of epidemics. The mathematical SIR model is in the form of three compartments: susceptible, infected, or removed.
Susceptible populations are healthy and do not carry the epidemic but can contract it from infected individuals which carry the infection and can pass it to susceptible hosts, while the removed people are no longer infected and acquire immunity from future contagion.
Susceptible-Infected-Susceptible (SIS) [15] epidemic models have been applied to situations in which it is supposed that an infected population could move immediately to the susceptible compartment after being recovered from an infection due to the lack of immunization. Susceptible-Exposed-Infected-Susceptible (SEIS) epidemic models have been applied to situations in which it is supposed that there is a period of incubation of the infection and after that, an infected individual could move to the susceptible compartment again after being recovered from an infection due to the lack of immunization like the case of the seasonal Influenza. This kind of compartmental model is also useful to model the evolution of many phenomena in different situations, see as examples, subjects treated in [17,18,19,20,21].
The first work that generalizes the compartments-based models which describe and control the evolution of pandemics and epidemics within a geographical area, is done by Zakary et al in [11,12], where the authors developed a new modeling approach to describe the Spatio-temporal spread of diseases and suggested several strategies of control by adopting different generalized models in discrete and continuous-time [13,14,16,22,23,24,9].
People infected with A (H1N1) passes through an incubation period where they are not infectious and do not show any symptoms. The period of incubation for A (H1N1) virus is 1 to 4 days and the infectious period case is defined as 1 day prior to the onset of symptoms to 7 days after onset [25]. Therefore, and based on all these considerations, we consider in this paper a compartments-based mathematical SEIS model to describe the evolution of the Influenza pandemic in Morocco. First, we present the different parameters of the model and the interactions between its states. We introduce the seasonality of this disease by considering time-dependent parameters such as the infection rate, the recovery rate, and the control policy.
Second, we perform the stability analysis of the equilibria based on values of the threshold R 0 which is calculated with the help of the next-generation matrix method. To validate the model we propose here, our study is based on real data of Influenza A viruses in Morocco [28]. Then we estimate the model's parameters by matching the model's outputs with the actual data. Next, we perform a sensitivity analysis to show the most important parameter in changing the basic reproductive number R 0 and to determine the most influential parameter taking into account the seasonality of parameters, so we define seasonal sensitivity indicators to show that the parameters have an impact period which makes the disease more severe in some months. Finally, we implement the model uncertainty to check and verify the performance of the parameters' estimation.

THE MODEL
We formulate the transmission dynamics model for a single outbreak of the Influenza epidemic in a homogeneously mixing population in Morocco. We divide the population into three main epidemiological classes, susceptible (S), Exposed (E), infected (I). These classes are further interact with each other, different susceptibilities, and infectiousness based on the seasonality of the infection and immunity and control. We take into account the differences in infectivity for diseases such as the Influenza, i.e., latent or incubation period (E), and symptomatic or infectious period (I). We apply the model to the case of Morocco, and assume that the course of the outbreak is short compared with the life of an individual, therefore, births, and natural deaths are equal.
In the SEIS model, it is assumed that the population consists of susceptible (S), exposed (E), infected and infectious (I) individuals. A characteristic feature of the seasonal Influenza is that immunity after infection is temporary, so that the infected individuals become susceptible again. The model depends on several parameters: The parameter µ, which denotes the birth rate, assumed equal to the mortality rate; and α, which is the rate of loss of infectiousness, thus, an infected individuals become susceptible again at a rate αI. The parameter γ denotes the average incubation period, thus , exposed individuals become infectious at a rate γE. Since health authorities and institutions usually interfere in such a situation of pandemics and epidemics to save lives, we thus incorporate the control intervention into the model as a parameter u, thus the treated individuals become susceptible at a rate uI.
The disease transmission is modeled using the standard incidence, given by β IS N Then, the SEIS model for The Influenza epidemic is given by the following system of ordinary differential equations: Where S (0) ≥ 0, I (0) ≥ 0 and E (0) ≥ 0, and N = S + E + I. Moreover, the influence of the seasonality on the transmission and control parameters is modeled by the cosine and sine functions and given by Note that b 0 and a 0 are the averages of the transmission parameters, b 1 and a 1 are the amplitudes of the seasonal fluctuation in the transmission parameters β and γ, respectively, while b 2 , b 3 , a 2 and a 3 are angle parameters that will be chosen later in agreement with the real data for I(t).

STABILITY ANALYSIS
In this section, we find the basic reproduction number and stability of the model. We prove that our model is locally and stable for both disease-free-equilibrium and endemic equilibrium points.
To work in term of proportions we take s = S N , e = E N and i = I N , then we have the following reduced system This system has two equilibria, the disease-free equilibrium e 0 = (1, 0, 0) and the endemic equilibrium In epidemiology, the basic reproduction number R 0 is the average number of secondary infectious cases produced by a single infection in total susceptible population. Based on the next-generation matrix method [4] we compute the basic reproductive number R 0 , which is given by Note that also the basic reproductive number R 0 is a function of time, since the parameters β and γ are seasonal parameters.
Proof. By computing the Jacobian matrix associated with the system (4)-(6) at the disease-free equilibrium, we have Proposition 2. The endemic equilibrium e * is locally asymptotically stable if R 0 > 1.
Proof. By computing the Jacobian matrix associated with the system (4)-(6) at the endemic equilibrium, we have Where the characteristic polynomial of J (e * ) is Where Let and One can easily verify that It is clear that W > 0 and by R 0 > 1 we get also V < 0, thus U < 0, and then c 2 > 0. Using the Routh-Hurwitz stability criterion, we conclude that the equilibrium point e * is locally asymptotically stable.

PARAMETERS ESTIMATION
To verify the reality of the model we propose here, we use nonlinear least-squares regression to fit the model to actual observations for the situation of the Influenza A epidemic in Morocco given [28] from 2018-12-10 to 2019-03-24 and presented in Table 1. Therefore, the following process has been followed for parameters estimation: In this section we present the process of the parameters' estimation [29]. Using MATLAB ode45 routine, the system of ordinary differential equations is solved numerically, with initial chosen values for parameters and state variables. Model outcomes are compared with the field data and the Levenberg-Marquardt optimization algorithm determines a new set of parameters

Data
Data values with the model outcomes in a better fit to the field data [30,31]. After new parameters values are determined by this optimizer, the system of ordinary differential equations is solved numerically using these new parameters value and the model outcomes are compared again with the field data. This iteration process between parameter updating and numerical solutions of the system of ordinary differential equations using Runge-Kutta method [32,33] continue till convergence criteria for the parameters are met. In this process of estimating parameters,   Table 2. Fig. 1 depicts the effectiveness of the parameters' estimation, where we can see that the SEIS model we propose here fits correctly the real data. Based on the estimated parameters given the Table 2, we simulate the seasonal basic reproductive number R 0 in the Fig.3. Where we can see that R 0 rise continuously to exceed 240 by about four weeks, and then it starts to decrease towards 0 by the 7th week. It can be seen from that figure that this first wave of R 0 is followed by an increasing of the infections, which can be interpreted by the incubation period of the Influenza epidemic. When R 0 exceeds 1, the infections will increase after a while by approximately 3 weeks. Again we can see in this figure that R 0 starts to increase up to 200 and then decrease. This means that a second wave of infection is possible.
Notwithstanding this rise in the basic reproductive number, it is fortunate that Morocco did not record any second wave.

SENSITIVITY ANALYSIS
Sensitivity analysis is measured using the sensitivity index. Sensitivity indices allow us to measure the relative change in a state variable when a parameter changes while the other parameters are retained fixed at constant values. The normalized forward sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.  Table 2.
A small perturbation δ x to a parameter x and the corresponding change in R 0 as δ R 0 is given by Thus, the normalized sensitivity index φ x is defined as Therefore normalized sensitivity indices for the 8 parameters are obtained as (a) (a) between the coefficients c and the a 3 is positive (26749.69035). The covariance between the coefficients a 1 and the a 2 is positive (605.2780479). The covariance between the coefficients a 2 and the a 3 is positive (166517.8713). This means that these parameters values vary in a positive way. Table 3 shows the summary of covariance of the Influenza epidemic model parameters.

CONCLUSION
In this paper, a SEIS model type compartmental model has been considered to explore the evolution of the Influenza epidemic in Morocco. To consider a more realistic model we consider the seasonality of parameters, by considering time-dependent infection rate, time-dependent recovery rate, and time-dependent intervention rate. The next-generation matrix method is used to compute the threshold of equilibria's stability R 0 . A real data of Influenza epidemic A infections in Morocco from the Influenza Laboratory Surveillance Information system is used to estimate the model parameters. In order to identify the most influential parameter in the proposed model, sensitivity analysis is carried out. We calculated sensitivity indices based on the estimated parameters to identify the most influential parameters. We found that the seasonal parameters a 0 , a 1 , a 2 , and a 3 are most influential in the first four weeks and the last two weeks. While the parameters b 1 , b 2 , and b 3 are the most influential parameters of the eighth and ninth weeks. Finally, the moderate influence of the µ parameter during the first and last weeks. Uncertainty analysis is carried out to determine the relationship between the different parameters of the model.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.