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Theoretical assessment of the impact of awareness programs on cholera transmission dynamic

  • Daudel Tchatat , Gabriel Kolaye EMAIL logo , Samuel Bowong and Anatole Temgoua
Published/Copyright: October 13, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 5

Abstract

In this paper, we propose and analyse a mathematical model of the transmission dynamics of cholera incorporating awareness programs to study the impact of socio-media and education on cholera outbreaks. These programs induce behavioural changes in the population, which divide the susceptible class into two subclasses, aware individuals and unaware individuals. We first provide a basic study of the model. We compute the Disease-Free Equilibrium (DFE) and derive the basic reproduction number R 0 0 that determines the extinction and the persistence of the disease. We show that there exists a threshold parameter ξ such that when R 0 0 ≤ ξ < 1 , the DFE is globally asymptotically stable, but when ξ ≤ R 0 0 < 1 , the model exhibits the phenomenon of backward bifurcation on a feasible region. The model exhibits one endemic equilibrium locally stable when R 0 0 > 1 and in that condition the DFE is unstable. Various cases for awareness proportions are performed using the critical awareness rate in order to measure the effect of awareness programs on the infected individuals over time. The results we obtained show that the higher implementation of strategies combining awareness programs and therapeutic treatments increase the efficacy of control measures. The numerical simulations of the model are used to illustrate analytical results and give more precision on critical values on the controls actions.

Keywords: awareness programs; behavioural changes; cholera; disease-free equilibrium; stable
Corresponding author: Gabriel Kolaye, Department of Mathematics and Computer Science, Faculty of Science, University of Maroua, P.O. Box 814, Maroua, Cameroon; UMI 209 IRD/UPMC UMMISCO, Bondy, France; Project GRIMCAPE, LIRIMA, University of Yaounde I, Yaounde, Cameroon; and Laboratory of Mathematics, Department of Mathematics and Computer Science, Faculty of Science, Bondy, France; and Project GRIMCAPE, LIRIMA, University of Yaounde I, University of Douala, P.O. Box 24157,Douala, Cameroon, Email:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Proof of Lemma 1

The number of positive roots of (25) determines the number of endemic equilibrium of system (3). Clearly, H 4 is non negative and H 0 is positive or negative depending whether R 0 is less than or greater than unity, respectively. In order to identify the number of endemic equilibrium, we require the Descartes rule of signs on the polynomial function g ( λ * ) = H 4 ( λ * ) 4 + H 3 ( λ * ) 3 + H 2 ( λ * ) 2 + H 1 ( λ * ) + H 0 , which is given by the following Table 4.

Table 4:

Roots signs.

H 4 H 3 H 2 H 1 H 0 R 0 Solutions
+ + + + + R 0 < 1 0
+ + + + − R 0 > 1 1
+ + + − + R 0 < 1 2
+ + + − − R 0 > 1 1
+ + − + + R 0 < 1 2
+ + − + − R 0 > 1 3
+ + − − + R 0 < 1 2
+ + − − − R 0 > 1 1
+ − + + + R 0 < 1 2
+ − + + − R 0 > 1 3
+ − + − + R 0 < 1 4
+ − + − − R 0 > 1 3
+ − − + + R 0 < 1 2
+ − − + − R 0 > 1 3
+ − − − + R 0 < 1 2
+ − − − − R 0 > 1 1

Appendix B: Proof of Theorem 4

In this Appendix, we present the proof of Theorem 4 on the local stability of the endemic equilibrium point of system (3) when R 0 > 1 . To do so, the following simplification and change of variables are made first to all. Let x 1 = S u, x 2 = S a, x 3 = I, x 4 = R, x 5 = B and x 6 = M. Further, by using the vector notation x = (x 1, x 2, x 3, x 4, x 5, x 6), the model (3) can be re-written in the form x ̇ = f ( x ) with f = (f 1, f 2, f 3, f 4, f 5, f 6) as follows:

(29) x 1 ̇ = Λ 1 + q α x 4 + δ x 2 − ( ρ x 6 + μ + λ ) x 1 , x 2 ̇ = ρ x 6 x 1 + ( 1 − q ) α x 4 − ( μ + δ + ε λ ) x 2 , x 3 ̇ = λ ( x 1 + ε x 2 ) − ( μ + γ + d ) x 3 , x 4 ̇ = γ x 3 − ( α + μ ) x 4 , x 5 ̇ = b x 5 1 − x 5 T + θ x 3 − μ B x 5 , x 6 ̇ = Λ 2 + η x 3 − ν x 6 .

where

(30) λ = β h x 3 + β e x 5 x 5 + K .

System (29) has a DFE, which is given by Q 0 = S u 0 , S a 0 , 0,0,0 , M 0 where S u 0 = Λ 1 ( μ + δ ) μ ( μ + δ + ρ M 0 ) , S a 0 = Λ 1 ρ M 0 μ ( μ + δ + ρ M 0 ) and M 0 = Λ 2 ν . The Jacobian of system (29) at the DFE Q 0 is

J ( Q 0 ) = − ( μ + ρ M 0 ) δ β h S u 0 α q β e S u 0 K ρ S u 0 ρ M 0 − ( μ + δ ) β h ε S a 0 α ( 1 − q ) β e ε S a 0 K ρ S u 0 0 0 β h S u 0 + ε S a 0 0 β e S u 0 + ε S a 0 K 0 0 0 γ − ( μ + α ) 0 0 0 0 θ 0 b − μ B 0 0 0 η 0 0 − ν .

The basic reproduction number of the transformed (linearised) model system (29) is the same as that of the original model given by Eq. (9). Therefore, choosing β h as a bifurcation parameter, solving for β h from R 0 = 1 , one obtains

(31) β h * = μ + γ + d S u 0 + ε S a 0 − β e θ K ( μ B − b )

It follows that the Jacobian J(Q 0) of system (29) at the DFE Q 0, with β h = β h * , denoted by J β h * has a simple zero eigenvalue (with all other eigenvalues having negative real parts). Hence, the Centre Manifold theory [44] can be used to analyse the dynamics of system (29). In particular, the theorem in Castillo and Song [45], reproduced below for convenience, will be used to show that when R 0 > 1 , there exists an endemic equilibrium of system (29), which is locally asymptotically stable for R 0 near 1 under certain conditions.

Theorem 5

(Castillo-Chavez and Song [45]). Consider the following general system of ordinary differential équations with a parameter Φ:

(32) d z d t = f ( x , Φ )   f : R n × R → R   and   f ∈ C 2 ( R n × R )

where 0 is an equilibrium point of the system (that is, f(0, Φ) ≡ 0 for all Φ) and assume

  1. A = D z f ( 0,0 ) = ∂ f i ∂ z j ( 0,0 ) is the linearisation matrix of system (32) around the equilibrium 0 with Φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts;

  2. Matrix A has a right eigenvector u and a left eigenvector v (each corresponding to the zero eigenvalue). Let f k be the kth component of f and

    a * = ∑ k , i , j = 1 n v k u i u j ∂ 2 f k ∂ x i ∂ x j ( 0,0 )    and    b * = ∑ k , i = 1 n v k u i ∂ 2 f k ∂ x i ∂ Φ ( 0,0 ) ,

    then, the local dynamics of the system around the equilibrium point 0 is totally determined by the signs of a and b.

  1. a* > 0, b* > 0. When Φ < 0 with |Φ| ≪ 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < Φ ≪ 1, 0 is unstable and there exists a negative, locally asymptotically stable equilibrium;

  2. a* < 0, b* < 0. When Φ < 0 with |Φ| ≪ 1, 0 is unstable; when 0 < Φ ≪ 1, 0 is locally asymptotically stable equilibrium, and there exists a positive unstable equilibrium;

  3. a* > 0, b* < 0. When Φ < 0 with |Φ| ≪ 1, 0 is unstable and there exists a locally asymptotically stable negative equilibrium; when 0 < Φ ≪ 1, 0 is stable, and a positive unstable equilibrium appears;

  4. a* < 0, b* > 0. When Φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.

In order to apply the above theorem, the following computations are required (it should be noted that we used β h * as the bifurcation parameter, in place of Φ in Theorem 5).

Eigenvectors of J β h * : For the case when R 0 = 1 , it can be shown that the Jacobian of system (29) has a right eigenvector (corresponding to the zero eigenvalue), given by U = ( u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ) T , where

(33) u 1 = ( μ + α ) ( μ + δ ) S u 0 + ε δ S a 0 [ β e θ ν + β h ν K ( μ B − b ) ] + α γ ν K ( μ B − b ) ( q μ + δ ) μ ν K ( μ + α ) ( μ B − b ) ( μ + δ + ρ M 0 ) u 3 + K ρ ( μ B − b ) ( μ + α ) ( μ + 2 δ ) S u 0 μ ν K ( μ + α ) ( μ B − b ) ( μ + δ + ρ M 0 ) u 3 , u 2 = ν ( μ + α ) ρ M 0 S u 0 + ε ( μ + ρ M 0 ) S a 0 [ β e θ + β h K ( μ B − b ) ] μ ν K ( μ + α ) ( μ B − b ) ( μ + δ + ρ M 0 ) u 3 + α γ ν K ( μ B − b ) [ μ ( 1 − q ) + ρ M 0 ] + η K ( μ B − b ) ( μ + α ) ( μ + 2 ρ 2 M 0 ) S u 0 μ ν K ( μ + α ) ( μ B − b ) ( μ + δ + ρ M 0 ) u 3 , u 3 = u 3 > 0 , u 4 = γ μ + α u 3 , u 5 = θ μ B − b u 3 , u 6 = η ν u 3 .

In a similar way, the components of the left eigenvectors of J β h * (corresponding to the zero eigenvalue), denoted by V = ( v 1 , v 2 , v 3 , v 4 , v 5 , v ) T , are given by

(34) v 1 = v 2 = v 4 = v 6 = 0 , v 3 = v 3 > 0   and   v 5 = β e S u 0 + ε S a 0 K ( μ B − b ) v 3 .

Computation of b*: For the sign of b*, it can be shown that the associated non-vanishing partial derivatives of f are:

∂ 2 f 3 ∂ x 3 ∂ β h * ( 0,0 ) = x 1 0 + ε x 2 0 = S u 0 + ε S a 0 .

It follows that

(35) b = v 3 S u 0 + ε S a 0 > 0 .

Computation of a: For system (29), the associated non-zero partial derivatives of f (at the DFE Q 0) are given by

∂ 2 f 3 ∂ x 1 ∂ x 3 ( 0,0 ) = β h * , ∂ 2 f 3 ∂ x 1 ∂ x 5 ( 0,0 ) = β e K , ∂ 2 f 3 ∂ x 2 ∂ x 3 ( 0,0 ) = β h * ε , ∂ 2 f 3 ∂ x 2 ∂ x 5 ( 0,0 ) = β e ε K ,

∂ 2 f 3 ∂ x 5 2 ( 0,0 ) = − 2 β e S u 0 + ε S a 0 K 2   and   ∂ 2 f 5 ∂ x 5 2 ( 0,0 ) = − 2 b T .

Therefore, it follows that

(36) a = v 3 ∑ i , j = 1 6 u i u j ∂ 2 f 3 ∂ x i ∂ x j ( 0,0 ) + v 5 ∑ i , j = 1 6 u i u j ∂ 2 f 5 ∂ x i ∂ x j ( 0,0 ) , = u 1 u 2 β h * + u 1 u 5 β e K + u 2 u 3 β e ε + u 2 u 5 β e ε K − 2 u 5 2 S u 0 + ε S a 0 1 K + b T ( μ B − b ) v 3

Therefore, according to the parameters of the model system (3), the value of a can be positive or negative. So, if b > 0, if a > 0, model system (3) undergoes the phenomenon of backward bifurcation (see Theorem 5, item (1)). Also, if a < 0 (by Theorem 5, item (4)), we have established the result about the local stability of the endemic equilibrium of model system (3) for R 0 > 1 but close to 1. This concludes the proof of Theorem 4.

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Received: 2021-08-29
Revised: 2022-09-06
Accepted: 2022-09-18
Published Online: 2022-10-13

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2021-0341
Keywords for this article
awareness programs; behavioural changes; cholera; disease-free equilibrium; stable

Articles in the same Issue

  1. Frontmatter
  2. Original Research Article
  3. Hybrid solitary wave solutions of the Camassa–Holm equation
  4. Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
  5. A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
  6. Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
  7. Numerical modeling of the dam-break flood over natural rivers on movable beds
  8. A class of piecewise fractional functional differential equations with impulsive
  9. Lie symmetry analysis for two-phase flow with mass transfer
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  15. An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
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