Dynamics Analysis of a Multi-strain Cholera Model with an Imperfect Vaccine

Abstract

A new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population, is designed. The model has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique, and locally-asymptotically stable, endemic equilibrium when the threshold quantity exceeds unity and another condition holds. Numerical simulations of the model show that, with the expected 50 % minimum efficacy of the first vaccine dose, vaccinating 55 % of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of cholera in the community. Such effective control can also be achieved if 50 % of the first vaccine dose recipients take the second dose. It is shown that a control strategy that emphasizes the use of antibiotic treatment is more effective than one that emphasizes the use of basic (non-pharmaceutical) anti-cholera control measures only. Numerical simulations show that, while the universal strategy (involving all three control measures) gives the best outcome in minimizing cholera burden in the community, the combined basic anti-cholera control measures and treatment strategy also has very effective community-wide impact.

Appendix: Proof of Theorem 4

Proof

Let $\mathcal{R}_{\mathrm{vac}}>1$ and b ₁ b ₂>0 (so that the model (1) has a unique EEP, in line with Theorem 3). It is convenient to make the following change of variables:

Furthermore, let X=(x ₁,x ₂,x ₃,x ₄,x ₅,x ₆,x ₇,x ₈,x ₉,x ₁₀,x ₁₁)^T. Thus, the model (1) can be re-written in the form $\frac{dX}{dt} = F(X)$, with F=(f ₁,f ₂,f ₃,f ₄,f ₅,f ₆,f ₇,f ₈,f ₉,f ₁₀,f ₁₁)^T, as follows:

(19)

The Jacobian of the system (19), at the associated DFE ($\mathcal{E}_{0}$), is given by

$$ J(\mathcal{E}_0)=[M_{11\times6} \ U_{11\times5}] $$

where

$$ M=\left ( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -(\psi+\mu) & \omega&0&0&0&0\\ \psi & -(\omega+\sigma+\mu)&\gamma&0&0&0\\ 0 & \sigma & -(\gamma+\mu) & 0&0&0 \\ 0 & 0 & 0 & -k_1&0&0 \\ 0 & 0 & 0 & 0&-k_2&0 \\ 0 & 0 & 0 & \kappa&0&-k_4\\ 0 & 0 & 0 & \sigma_w&0&\alpha \\ 0&0 & 0 & 0&\sigma_r&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&\theta_a\\ 0&0&0&0&0&\phi_a\\ \end{array} \right ), $$

$$ U=\left ( \begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c} 0&0&0&0&{\frac{-\beta S^*}{K_B}}\\[4pt] 0&0&0&0&{\frac{-r_1\beta V_1^*}{K_B}}\\[4pt] 0&0&0&0&{\frac{-r_2\beta V_2^*}{K_B}}\\[4pt] 0&0&0&0&{\frac{\beta(f_1S^*+f_2r_1V_1^*+f_3r_2 V_2^*)}{K_B}}\\[4pt] 0&0&(1-q)\theta_t&0&{\frac{\beta[(1-f_1)S^*+(1-f_2)r_1V_1^*+(1-f_3)r_2 V_2^*]}{K_B}}\\ 0&0&0&0&0\\ -k_5&0&0&0&0\\ \eta&-k_6&0&0&0\\ \tau&0&-k_7&0&0\\ \theta_s&\theta_r&q\theta_t&-\mu&0\\ \phi_s&\phi_r&\phi_t&0&-k_3\\ \end{array} \right ). $$

Consider the case when $\mathcal{R}_{\mathrm{vac}}=1$. Furthermore, suppose that β is chosen as a bifurcation parameter. Solving for β from $\mathcal{R}_{\mathrm{vac}}=1$ gives

with

The transformed system (19), with β=β ^∗, has a hyperbolic equilibrium point (i.e., the linearized system has a simple eigenvalue with zero real part, and all other eigenvalues have negative real part), so that the center manifold theory (Carr 1981; Castillo-Chavez and Song 2004) can be used to analyze the dynamics of (19) near β=β ^∗.

It can be shown that the right eigenvector of $J(\mathcal{E}_{0})|_{\beta=\beta^{*}}$, denoted by ${\bf w}$, is given by w=(w ₁,w ₂,…,w ₁₀,w ₁₁)^T, with

(20)

It should be noted that w ₄,w ₅,…,w ₁₀ are positive since they can easily be written in terms of w ₁₁ (and not reported here.) Similarly, $J(\mathcal{E}_{0})|_{\beta=\beta^{*}}$ has a left eigenvector, ${\bf v}$, given by v=(v ₁,v ₂,…,v ₁₀,v ₁₁), with

Consequently, it follows that the associated bifurcation coefficients, a and b (defined in Theorem 4.1 of Castillo-Chavez and Song 2004), are given, respectively, by

(21)

(22)

By substituting (20) in (21), and simplifying, it can be shown that the bifurcation coefficient, a, is negative. The proof is concluded using Part (iv) of Theorem 4.1 in Castillo-Chavez and Song (2004). □