Big Bracket for equations

Add \left( and \right., but using equation and aligned:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation}
\left(
\begin{aligned}
\frac{dS_d}{dt}&= A + \lambda_dR_d + \sigma_d(1-\gamma_d)E_d -\beta_dS_dI_d- (m_d+k_d+c_d)S_d \\[0.5ex]
\frac{dE_d}{dt}&= \beta_dS_dI_d -(m_d+\sigma_d+c_d)E_d \\[0.5ex]
\frac{dI_d}{dt}&= \sigma_d\gamma_dE_d -(m_d+\mu_d+c_d)I_d \\[0.5ex]
\frac{dV_d}{dt}&= k_dS_d -(m_d+\lambda_d)V_d \\[0.5ex]
\frac{dS_h}{dt}&= B + \lambda_hR_h + \sigma_h(1-\gamma_h)E_h -\beta_hS_hI_d- m_hS_h \\[0.5ex]
\frac{dE_h}{dt}&= \beta_hS_hI_d -(m_h+\sigma_h+k_h)E_h \\[0.5ex]
\frac{dI_h}{dt}&= \sigma_h\gamma_hE_h -(m_h+\mu_h)I_h \\[0.5ex]
\frac{dV_h}{dt}&= k_hE_h -(m_h+\lambda_h)V_h
\end{aligned}
\right.
\end{equation}

\end{document}

As you see, the equation number is centered without any guesswork.

Use the empheq package, you can meet your expectation:

\documentclass{article}
\usepackage{empheq}
\begin{document}

\begin{empheq}[left=\empheqbiglparen]{align}
\frac{dS_d}{dt}&= A + \lambda_dR_d + \sigma_d(1-\gamma_d)E_d -\beta_dS_dI_d- (m_d+k_d+c_d)S_d \nonumber \\
\frac{dE_d}{dt}&= \beta_dS_dI_d -(m_d+\sigma_d+c_d)E_d \nonumber \\
\frac{dI_d}{dt}&= \sigma_d\gamma_dE_d -(m_d+\mu_d+c_d)I_d \nonumber \\
\frac{dV_d}{dt}&= k_dS_d -(m_d+\lambda_d)V_d \nonumber \\
\frac{dS_h}{dt}&= B + \lambda_hR_h + \sigma_h(1-\gamma_h)E_h -\beta_hS_hI_d- m_hS_h \\
\frac{dE_h}{dt}&= \beta_hS_hI_d -(m_h+\sigma_h+k_h)E_h \nonumber \\
\frac{dI_h}{dt}&= \sigma_h\gamma_hE_h -(m_h+\mu_h)I_h \nonumber \\
\frac{dV_h}{dt}&= k_hE_h -(m_h+\lambda_h)V_h \nonumber
\end{empheq}

\end{document}

Another variant, also based on empheq (which loads mathtools), but with a less tight rows spacing with the spreadlines environment.

B.t.w., don't end your multilined environments with a \\; if the environment is numbered, it adds a number for an empty row, and an unwanted vertical spacing.

\documentclass{article}

\usepackage{empheq}

\begin{document}

\begin{spreadlines}{8pt}
\begin{empheq}[left =\empheqlparen]{equation}
\begin{aligned}
\frac{dS_d}{dt}&= A + \lambda_dR_d + \sigma_d(1-\gamma_d)E_d -\beta_dS_dI_d- (m_d+k_d+c_d)S_d \\
\frac{dE_d}{dt}&= \beta_dS_dI_d -(m_d+\sigma_d+c_d)E_d \\
\frac{dI_d}{dt}&= \sigma_d\gamma_dE_d -(m_d+\mu_d+c_d)I_d \\
\frac{dV_d}{dt}&= k_dS_d -(m_d+\lambda_d)V_d \\
\frac{dS_h}{dt}&= B + \lambda_hR_h + \sigma_h(1-\gamma_h)E_h -\beta_hS_hI_d- m_hS_h \\
\frac{dE_h}{dt}&= \beta_hS_hI_d -(m_h+\sigma_h+k_h)E_h \\
\frac{dI_h}{dt}&= \sigma_h\gamma_hE_h -(m_h+\mu_h)I_h \\
\frac{dV_h}{dt}&= k_hE_h -(m_h+\lambda_h)V_h 
\end{aligned}
\end{empheq}
\end{spreadlines}

\end{document}