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Global Stability and Numerical Analysis of a Compartmental Model of the Transmission of the Hepatitis A Virus (HAV): A Case Study in Tunisia

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Abstract

This paper deals with the transmission and spread of the hepatitis A virus in central west Tunisia (the city of Thala). The target of this framework is to determine the global stability of a SEIRD epidemiological model where the infectious compartment is split into symptomatic and asymptomatic compartments. We study the global stability of the equilibrium state using the Lyapunov function which depends on the value of the basic reproduction number \(R_0\). Therefore, we prove that when \(R_0<1\), the disease-free equilibrium (DFE) is globally stable, but when \(R_0>1\), the DFE is unstable, and therefore the the endemic equilibrium is globally stable.

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In our previous study [6], we developed an age-structured epidemiological model to assess the impact of introducing vaccination. The study was conducted in Thala, a predominantly rural area in west-central Tunisia (Kasserine governorate); covering 34,508 inhabitants; according to the 2014 national census. The study population consisted of 1379 individuals, representing 5% of the target population and selected by a cross-sectional household survey conducted between January and June 2014. Permanent residents aged 5 to 75 were screened for hepatitis A virus antibodies. In [6], we estimated adjusted parameters for seroprevalence, incidence, and strength of infection by a linear compartmental SEIR (Susceptible-Exposed-Infectious-Recovered) model structured by age. Then, a vaccine model was constructed to assess the impact on the epidemiology of hepatitis A virus.

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Authors and Affiliations

  1. Laboratory of Networks Intelligent and Nanotechnologies: LaRINA, University of Carthage, Tunis, Tunisia

    Walid Ben Aribi, Bechir Naffeti & Hamadi Ammar

  2. Laboratory of Bio-(Informatic, Mathematics and Statistics), Pasteur Institute of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Walid Ben Aribi, Bechir Naffeti, Slimane Ben Miled & Amira Kebir

  3. Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Kaouther Ayouni

  4. Laboratory of Clinical Virology, Pasteur Institute of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Kaouther Ayouni & Henda Triki

  5. Faculty of Economic Sciences and Management of Nabeul, University of Carthage, Tunis, Tunisia

    Hamadi Ammar

  6. IPEIT, University of Tunis, Tunis, Tunisia

    Amira Kebir

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  1. Walid Ben Aribi
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Contributions

All authors contributed to the conception and design of the study. Formal analysis and investigation, Writing the first and original draft preparation: [Walid BEN ARIBI]; Writing and editing: [Walid BEN ARIBI, Kaouther AYOUNI]. Conceptualization: [Walid BEN ARIBI, Slimane BEN MILED]. Mathematical methodology: [Walid BEN ARIBI, Amira KEBIR]. Numerical analysis: [Bechir NAFFETI]. Critically revised and commented the manuscript: [Amia KEBIR, Hamadi AMMAR, Slimane BEN MILED]. Data curation: [Henda Triki]. Supervision: [Slimane BEN MILED]. All authors have read and approved the final manuscript.

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A Computation of the Basic Reproductive Number \(R_0\)

A Computation of the Basic Reproductive Number \(R_0\)

Using the system (2) we denote by X(t) a small perturbation about the equilibrium point \(P_{0}^{*}\), defined by

$$\begin{aligned} \begin{array}{l} x(t)=S(t)-S^{*} \\ y(t)=E(t)-E^{*} \\ z(t)=A s(t)-As^{*} \\ T(t)=S y(t)-S y^{*} \end{array} \end{aligned}$$
(19)

Then, by linearizing the system (2) around \(P^{*}\) we have:

$$\begin{aligned} X(t)=A X(t)=F(X(t)) \end{aligned}$$
(20)

Where A is the matrix which is given by:

$$\begin{aligned} \left( \begin{array}{cccc} -\left( diag\left( B\left( As^{*}+Sy^{*}\right) \right) +diag\left( e+\mu ^{S}\right) \right) &{} 0 &{} -diag\left( S^{*}\right) B &{} -diag\left( S^{*}\right) B \\ diag\left( B\left( As^{*}+Sy^{*}\right) \right) &{} -diag\left( \delta +\mu ^{E}\right) &{} diag\left( S^{*}\right) B &{} diag\left( S^{*}\right) B \\ 0 &{} (1-\alpha ) diag(\delta ) &{} -diag\left( \sigma +\mu ^{\varLambda 5}\right) &{} 0 \\ 0 &{} diag(\delta ) &{} 0 &{} -diag\left( \sigma +\mu ^{S y}\right) \end{array}\right) \end{aligned}$$

and X(t) is a function of \({\mathbf {R}}_{+}^{4 n},\) where diag(E) is defined by the matrix, where the diagonal is equals to the components of the vector E and \(\forall i \ne j\) is the coefficient equals to 0,  and equivalent to

$$\begin{aligned} {\begin{array}{llll} {\dot{x}}(t)=-\left( diag\left( B\left( A s^{*}+s y^{*}\right) \right) +diag\left( e+\mu ^{s}\right) \right) x(t)-diag\left( s^{*}\right) B z(t)-diag\left( s^{*}\right) B T(t) \\ {\dot{y}}(t)=diag\left( B\left( A s^{*}+s y^{*}\right) \right) x(t)-diag\left( \delta +\mu ^{E}\right) y(t)+diag\left( s^{*}\right) B z(t)+diag\left( s^{*}\right) B T(t) \\ {\dot{z}}(t)=(1-\alpha ) diag(\delta ) y(t)-diag\left( \sigma +\mu ^{A s}\right) z(t) \\ {\dot{T}}(t)=diag(\delta ) y(t)-diag\left( \sigma +\mu ^{s y}\right) T(t) \end{array}} \end{aligned}$$
(21)

Lets \(F_{a}(y, z, T)\) the rate of appearance of new infected in the infectious compartment, \(F_{t}(y, z, T)\) the transfer rate of the individuals in to and out of the infectious compartment of the system (21) defined by Van Den Driessche [17].

$$\begin{aligned} F_{a}(y, z, T)=\left( \begin{array}{c} diag\left( B\left( A s^{*}+s y^{*}\right) \right) x(t)+diag\left( s^{*}\right) B z(t)+diag\left( s^{*}\right) B T(t) \\ 0 \\ 0 \end{array}\right) \end{aligned}$$

And

$$\begin{aligned} F_{t}(y, z, T)=\left( \begin{array}{c} -diag\left( \delta +\mu ^{E}\right) y(t) \\ (1-\alpha ) diag(\delta ) y(t)-diag\left( \sigma +\mu ^{A s}\right) z(t) \\ diag(\delta ) y(t)-diag\left( \sigma +\mu ^{5 y}\right) \tau (t) \end{array}\right) \end{aligned}$$

Then

$$\begin{aligned} F=\left( \begin{array}{ccc} 0 &{} diag\left( s^{*}\right) B &{} diag\left( s^{*}\right) B \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array}\right) \end{aligned}$$

And

$$\begin{aligned} V=\left( \begin{array}{ccc} -diag\left( \delta +\mu ^{E}\right) &{} 0 &{} 0 \\ (1-\alpha ) diag(\delta ) &{} -diag\left( \sigma +\mu ^{A s}\right) &{} 0 \\ diag(\delta ) &{} 0 &{} -diag\left( \sigma +\mu ^{s y}\right) \end{array}\right) \end{aligned}$$

Consequently

$$\begin{aligned} M=-F V^{-1}=\left( \frac{\beta _{k j} S_{k}^{0} \delta \left( (1-\alpha )\left( \sigma _{k}+\mu _{k}^{S y}\right) +\alpha \left( \sigma _{k}+\mu _{k}^{A s}\right) \right) }{\left( \delta +\mu _{k}^{E}\right) \left( \sigma _{k}+\mu _{k}^{A s}\right) \left( \sigma _{k}+\mu _{k}^{S y}\right) }\right) _{1 \le k, j \le n} \end{aligned}$$

Then, \(R_{0}=\rho (M),\) where \(S_{k}^{0}=\frac{\varLambda _{k}}{e_{k}+\mu _{k}^{S}}\)

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Ben Aribi, W., Naffeti, B., Ayouni, K. et al. Global Stability and Numerical Analysis of a Compartmental Model of the Transmission of the Hepatitis A Virus (HAV): A Case Study in Tunisia. Int. J. Appl. Comput. Math 8, 126 (2022). https://doi.org/10.1007/s40819-022-01326-0

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