Abstract
In this paper, a deterministic model is proposed to perform a thorough investigation of the transmission dynamics of Zika fever. Our model, in particular, takes into account the effects of horizontal as well as vertical disease transmission of both humans and vectors. The expression for basic reproductive number \(R_0\) is determined in terms of horizontal and vertical disease transmission rates. An in-depth stability analysis of the model is performed, and it is shown, that model is locally asymptotically stable when \(R_0 < 1\). In this case, there is a possibility of backward bifurcation in the model. With the assumption that total population is constant, we prove that the disease free state is globally asymptotically stable when \(R_0 < 1\). It is also shown that disease strongly uniformly persists when \(R_0> 1\) and there exists an endemic equilibrium which is unique if the total population is constant. The endemic state is locally asymptotically stable when \(R_0> 1\).
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Appendices
Appendix 1: Backward Bifurcation in Model (1)
The model given in the paper has the variables: \(S_h, E_h, I_h, R_h, S_v, E_v, I_v\).
Now we redefine these equations by assigning them the following values:
Let \(S_h = x_1, E_h= x_2, I_h= x_3, R_h= x_4, S_v= x_5, E_v= x_6, I_v= x_7\). Also let \(\hat{f} = [f_1,\ldots ,f_7]\) denote the vector field of the original model in terms of \(x_i's\).
Then we have the following model in terms of \(x_i's \)
Now the disease free equilibrium (DFE) of the system is given by \({\varepsilon _0}\) as follows:
Consider the case when \(R_0=1\). Take \(C_{hv}=C_{hv}^*\) as a bifurcation parameter.
The Jacobian of the matrix at the DFE is given by
where:
The above Jacobian matrix of the linearized system has a simple zero eigenvalue (with all other eigenvalues having negative real part). Hence, the Centre Manifold Theory can be used to analyze the dynamics of the system (1). We will use the theorem given by Castillo-Chavez and Song [5].
The Right Eigenvector
The right eigenvector of the Jacobian matrix correspond to zero eigenvalue at \(C_{hv}^*\) is given by: w \(={w_1,\ldots ,w_7}\), where
The Left Eigenvector
Similarly, the left eigenvector of the Jacobian matrix correspond to zero eigenvalue at \(C_{hv}^*\) is given by: v \(={v_1,\ldots ,v_7}\), where
The Non-zero Derivatives
The value of
So, that
The value of
is given by:
Since b is always positive, backward bifurcation occurs whenever \(a>0\).
Appendix 2: Stability of Endemic Steady State
Proof
The proof of Theorem 8 is based on using a Krasnoselskii sub-linearity trick [25, 26].
Rewrite (14) as:
Linearizing the system (21) around the endemic equilibrium \(\displaystyle N_1=(S_h^{\varnothing }, E_h^{\varnothing }, I_h^{\varnothing }, R_h^{\varnothing }, S_v^{\varnothing }, E_v^{\varnothing }, I_v^{\varnothing }) \), gives:
It follows that the Jacobian of the system evaluated at \(N_1\) is:
where \(\displaystyle j_1=\frac{C I_h^{\varnothing }}{N_h}\), \(\displaystyle j_2=\frac{C I_v^{\varnothing }}{N_h}\), \(\displaystyle j_3=\frac{C S_h^{\varnothing }}{N_h}\) and \(\displaystyle j_4=\frac{C S_v^{\varnothing }}{N_h}.\)
Assume that the model (22) has solution of the form
where \(Z=(Z_1, Z_2, Z_3, Z_4, Z_5)\). Substituting Z into (22) gives:
We rearranged the above system of equations as follows. First we move the negative terms in the last four equations of (23) to the respective left-hand sides. Secondly, the last four equations are then re-written in terms of \(Z_1\). We have
Substituting the last five equations in the first equation of (23) leads to:
where \(\displaystyle F_i(\omega )\) for \(i=1,\ldots ,5\) are positive functions of parameters and
such that the matrix M has non-negative entries. Define \(F(\omega )=min_i|1 + F_i|\). It is easy to verify that the equilibrium point \(N_1\) satisfies, \(N_1=MN_1.\) The notation \((MZ)_i\) denotes the ith coordinate of the vector MZ. If Z is a solution of (25), then it is possible to find a minimal positive real number r such that \(||Z||\le r N_1\) [25, 26]. We want to show that \(Re (\omega ) < 0\). Assume, \(Re (\omega ) \ge 0\), and consider the following two cases.
Case 1 \(\omega = 0.\) In this case (23) is a homogeneous linear system. It is easy to show that the determinant of this system is negative, and it follows that the system (23) has a unique solution, given by \(Z = 0\), which correspond to disease free steady state of the model (14).
Case 2 \(\displaystyle \omega \ne 0.\) By our assumption in this case \(|1+F_i(\omega )|> 1.\) Since r is a minimal positive real number, it follows that
Where \(F(\omega )\) is minimal of \(|1+F_i(\omega )|.\) From the second equation of (25), we have
which contradicts \(||Z||>\frac{r}{F(w)}N_1.\) Hence, \(Re (\omega ) < 0\). Thus, all eigenvalues of the characteristic equation associated with the linearized system (14) will have negative real parts. This implies local asymptotical stability of endemic state. \(\square \)
Appendix 3: Model Parameters
The estimated parameters are presented in Table 3.
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Imran, M., Usman, M., Dur-e-Ahmad, M. et al. Transmission Dynamics of Zika Fever: A SEIR Based Model. Differ Equ Dyn Syst 29, 463–486 (2021). https://doi.org/10.1007/s12591-017-0374-6
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DOI: https://doi.org/10.1007/s12591-017-0374-6