Modelling SARS-CoV-2 epidemics via compartmental and cellular automaton SEIRS model with temporal immunity and vaccination

Abstract

We consider the SEIRS epidemiology model with such features of the COVID-19 outbreak as: abundance of unidentified infected individuals, limited time of immunity and a possibility of vaccination. The control of the pandemic dynamics is possible by restricting the transmission rate, increasing identification and isolation rate of infected individuals, and via vaccination. For the compartmental version of this model, we found stable disease-free and endemic stationary states. The basic reproductive number is analysed with respect to balancing quarantine and vaccination measures. The positions and heights of the first peak of outbreak are obtained numerically and fitted to simple in usage algebraic forms. Lattice-based realization of this model is studied by means of the asynchronous cellular automaton algorithm. This permitted to study the effect of social distancing by varying the neighbourhood size of the model. The attempt is made to match the quarantine and vaccination effects.

Figures (11)

• Figure 1: The SEIRS epidemiology model for the COVID-19 dissemination.
• Figure 2: (Colour online) Frame (a) shows the discriminant $Q$ of the cubic equation (\ref{['lambda_eqs_EN_2']}), frame (b) the real parts of its roots $\lambda_i$ for the no vaccination case, $\omega=0$. Frames (c) and (d) show the same respective characteristics for the vaccination rate of $\omega=0.01$.
• Figure 3: (Colour online) Numeric results for the time evolutions of the unidentified infected, $E(t)$, and isolated infective, $I(t)$, fractions at various model parameters. The no vaccination case, $\omega=0$, is shown. Frames (a) and (b) demonstrate the effect of variation of the initial value $E_0$ at fixed $\beta$ and $\alpha$; (c) and (d) show the effect of the contact rate $\beta$ at fixed $E_0$ and $\alpha$; (e) and (f) demonstrate the same for the identification rate $\alpha$ at fixed $E_0$ and $\beta$. Dashed vertical lines show approximate positions and heights of the first peak of a pandemic, and are the results of approximate analytic expressions, see explanation further below in the text.
• Figure 4: (Colour online) Example for the $t_{\mathrm{max},E}$, $t_{\mathrm{max},I}$ (a), as well as $E_{\mathrm{max}}$ and $I_{\mathrm{max}}$ (b) as the functions of the identification rate $\alpha$ at a fixed transmission rate $\beta=0.3$ and initial condition $E_0=10^{-5}$. Vertical dashed lines provide the position of the critical identification rate $\alpha_c$, given by equation (\ref{['bc_ac_oc']}).
• Figure 5: (Colour online) The results of fitting of $t_{\mathrm{max},E}(\alpha,\beta,E_0)$ (\ref{['tmaxE']}) and $t_{\mathrm{max},I}(\alpha,\beta,E_0)$ (\ref{['tmaxI']}). (a) shows the exponent $v(\beta,E_0)$ as the function of $E_0$ at fixed $\beta$. (b) shows the same for the exponent $w(\beta,E_0)$. (c) shows the amplitudes $A(\beta)$ and $B(\beta)$. (d) shows the exponents $v_0(\beta)$ and $w_0(\beta)$ entering expressions (\ref{['v0']}) and (\ref{['w0']}), respectively.