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In this paper, dysentery diarrhea deterministic compartmental model is proposed. The local and global stability of the disease-free equilibrium is obtained using the stability theory of differential equations. Numerical simulation of the system shows that the backward bifurcation of the endemic equilibrium exists for

Diarrheal disease is a common threat worldwide, particularly in developing countries. It is preventable and treatable. According to [

In most cases, diarrhea disease occurs in three forms and frequently occurs in children of under five years age [

Stochastic and deterministic mathematical models are widely employed by public health practitioners in order to predict and control disease outbreak as well as determining the cost effectiveness of the available strategies [

Epidemiologically speaking, it is important to have adequate knowledge of the physical characteristic of any given ailment or disease in order to provide accurate model that can be utilized to predict and control its outbreak effectively. The availability of relevant data for the disease will also enhance the model parameters estimation and the usefulness of the model with respect to the disease involved. Model parameters from epidemiological models can be easily estimated using methods involving ordinary least squares estimator, maximum likelihood estimator derivative approximation, moments estimator, Markov chain Monte Carlo (MCMC) strategy, derivative-free optimization algorithms, the Levenberg-Marquardt, and Trust-Region-Reflective [

To the best of our knowledge problem involving estimation of parameters in dysentery diarrhea epidemic model with sensitivity analysis of the basic reproduction number has been scarcely investigated. The main objective of this paper is to tackle this problem by employing constrained optimization technique in order to estimate the values of parameters involved in a deterministic compartmental model for dysentery diarrhea epidemic as a constraint. Real data based on the weekly dysentery diarrhea epidemic that occurred in Ethiopia in 2017 were used to estimate parameters which are not accessible from the characteristic of the disease. In addition, we establish the local and global stability of the disease-free equilibrium and perform the bifurcation analysis. Sensitivity and uncertainty of the basic reproduction number of the system are analyzed. The study estimates the reproduction number of Ethiopia using the estimated parameters to predict the disease spread in the community. In the following Section

The SIRSB (Susceptible-Infected-Recovered-Susceptible-Pathogen population) model for dysentery diarrhea proposed by same authors [

The corresponding flow diagram and the description of the parameters for the model are given in Figure

Description of parameters of the model equation (

Parameters | Interpretation | Units |
---|---|---|

| Recruitment rate of susceptible population | Humans |

| Natural death rate of humans | |

| Natural recovery rate of diarrhea | |

| Relapse rate of the recovered humans to the susceptible class | |

| Disease induced death rate of dysentery diarrhea | |

| Concentration of Shigella | cells |

| Effective transmission rate of diarrhea due to human to human interaction | |

| Effective transmission rate of dysentery diarrhea due to environment to human interaction | |

| Pathogen shedding rate | Cells |

| Shigella Pathogen growth rate | |

| Shigella Pathogen death rate | |

| Net death rate of Shigella Pathogen | |

Schematic diagram for the flow of dysentery diarrhea infection.

System equation (

where

System equation (

If

If

If

If

If

This section is devoted to analytic conditions for the stability of the disease-free equilibrium.

The disease-free equilibrium

Local stability is analyzed by the Jacobian matrix of (

In this section, we use the method used in [

(

(

where

The fixed point

To prove this, we write (

If

To investigate the type of bifurcation, let

If

In this manuscript we solve a dynamics parameter estimation problem. It aims to solve for the unknown parameters of a dynamic model supplied as a system of ordinary differential equations. The problem is setup as a standard nonlinear least squares problem; however the nonlinear function to be fitted involves solving ordinary differential equations using a numerical integration scheme. Least squares are a special case of maximum likelihood in which it is assumed that the data are drawn from a normal distribution with mean given by the solution to the ODE. System equation (

Guess initial parameter values

Using MATLAB

Evaluate error using (

Minimize (

Check convergence criteria. If not converged, go to (2).

On convergence, set

In this section, we first present the existence and nonexistence of the endemic equilibria. Secondly, fitting the data of dysentery diarrhea cases in Ethiopia is performed to estimate the parameters. Next, the sensitivity of the

It is proved in Theorem

(a) Forward bifurcation graph for

Forward bifurcation

Multiple equilibria for

Dysentery diarrhea is one of the most dangerous types of diarrhea which is endemic in Ethiopia. It is reported weekly under Integrated Disease Surveillance and Response System (IDSR) in the Ministry of Health. Ethiopian Weekly Epidemiological Bulletin [

Weekly dysentery diarrhea reports in Ethiopia,

Weeks | Infected | Weeks | Infected | Weeks | Infected | Weeks | Infected | Weeks | Infected |
---|---|---|---|---|---|---|---|---|---|

1 | 4542 | 12 | 7000 | 23 | 7375 | 33 | 7042 | 43 | 5583 |

2 | 4750 | 13 | 7042 | 24 | 7167 | 34 | 6917 | 44 | 5458 |

3 | 4792 | 14 | 6458 | 25 | 7292 | 35 | 6208 | 45 | 5458 |

4 | 5417 | 15 | 7083 | 26 | 7000 | 36 | 5958 | 46 | 5458 |

5 | 5250 | 16 | 7625 | 27 | 6583 | 37 | 6500 | 47 | 5250 |

6 | 5125 | 17 | 6875 | 28 | 6750 | 38 | 5958 | 48 | 4750 |

7 | 5833 | 18 | 6375 | 29 | 6833 | 39 | 5583 | 49 | 4750 |

8 | 6000 | 19 | 6625 | 30 | 6792 | 40 | 5667 | 50 | 4750 |

9 | 5542 | 20 | 6875 | 31 | 6708 | 41 | 5667 | 51 | 4583 |

10 | 5917 | 21 | 7000 | 32 | 6833 | 42 | 5542 | 52 | 4500 |

11 | 6583 | 22 | 7042 |

Weekly time series of dysentery diarrhea data in Ethiopia 2017.

In this model we have ten parameter values to be estimated. Among these parameters, the value of

Estimated parameter values.

Parameter | Value | Bounds | Source |
---|---|---|---|

| 462 | | Estimated |

| 0.69864 | | Estimated |

| 0.04465 | | Estimated |

| 200 | [ | |

| 0.000457 | [ | |

| 0.1106 | [ | |

| 0.55602 | | Estimated |

| 0.0211 | | Estimated |

| 0.01 | Assumed | |

| 46.2086 | | Estimated |

| 216253 | | Estimated |

| 3152 | | Estimated |

| 165 | | Estimated |

| 1356 | | Estimated |

The reproduction number based on the Ethiopian data is therefore

The sensitivity analysis reveals how imperative every parameter is to illness transmission. It is regularly used to decide the robustness of model expectations to parameter values since there are errors in data collection and assumed parameter values [

The normalized forward sensitivity index of

For instance,

Given

This follows immediately from Definition

The sensitivity indices of

Estimated parameters.

Parameter | Sensitivity indices of |
---|---|

| |

| |

| |

| |

| |

| |

| |

| |

| |

Once a model is developed, we use real data to validate it. There are many statistical techniques applied for model validation. The two most frequently used methods are graphical and numerical techniques. Graphical methods demonstrate a comprehensive range of complex features of the relationship between the model and the real data. A numerical technique for model validation, on the other hand, tends to be narrowly targeted on a specific side of the connection between the model and the observed data.

The residuals from a fitted model are the differences between the real data and the solution to the model. If the model fits the real data appropriately, the residuals approximate the random errors that make the association with the real data and the solution to the system [

Fitting of the model equation (

Fitting

Residuals

Residuals plots for the model and the real data in Table

The residual analysis is determined by computing autocorrelations for the residuals at varying time Lags. Residual analysis includes two tests: the whiteness test and the independence test. The whiteness test is about autocorrelation function of the residuals for each output. Based on this test, the residual autocorrelation function of a good model is inside the confidence interval of the corresponding estimates, showing that the residuals are uncorrelated. On the other hand, the independence test checks the cross-correlation between the input and the residuals for each input output pair. The independence test makes evident that a good model has the residuals uncorrelated with past inputs. In this case, correlation shows that the model does not describe how part of the output relates to the corresponding input.

The autocorrelation function plot for the residuals shows that most of the residuals are not statistically significant (Figure

Correlation of residuals for the infected individuals

In the process of parameter estimation, it is reasonable to assume that the parameters are dependent. Particularly, the parameters could be thought to vary together or covary. Covariance is the measure of how the parameter

The covariance between transmission coefficient (

Covariance of the parameters.

Covariance | Direction | Covariance | Direction | Covariance | Direction |
---|---|---|---|---|---|

| +ve | | +ve | | -ve |

| +ve | | +ve | | -ve |

| +ve | | +ve | | +ve |

| -ve | | +ve | | +ve |

| +ve | | -ve |

In this paper, we have proposed a deterministic compartmental dysentery diarrhea model. We have proved the global stability of the disease-free equilibrium. It is established in Theorem

The authors [

Therefore,

As an application, we have used our system to simulate the reported dysentery diarrhea cases in Ethiopia in 2017 (Table

The values of parameters describing the system have been estimated by fitting the integrals of the system to the field data on dysentery diarrhea epidemic. Estimation of parameters, in this case, is a challenging task because of missing a large part of the infectious process. One difficulty is that depending on the initially specified parameter values a local minimum may occur and the minimal value of the sum of squared residuals may be different. In such case, the parameter estimation should be repeated with different initial parameter guesses to achieve a better estimate of the global minimum (rather than local) of the sum of squared residuals.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.