Modelling the Influence of Dynamic Social Processes on COVID-19 Infection Dynamics

Abstract

Human behaviour was tipped as the mainstay in the control of further SARS-CoV-2 (COVID-19) spread, especially after the lifting of restrictions by many countries. Countries in which restrictions were lifted soon after the first wave had subsequent waves of COVID-19 infections. In this study, we develop a deterministic model for COVID-19 that includes dynamic non-pharmaceutical interventions known as social dynamics with the goal of simulating the effects of dynamic social processes. The model steady states are determined and their stabilities analysed. The model has a disease-free equilibrium point that is locally asymptotically stable if ${\mathcal{R}}_0<1.$ The model exhibits a backward bifurcation, implying that reducing the reproduction number below one is not sufficient for the elimination of the disease. To ascertain the range of parameters that affect social dynamics, numerical simulations are conducted. The only wave in South Africa in which interventions were purely based on human behavior was the first wave. The model is thus fitted to COVID-19 data on the first wave in South Africa, and the findings given in this research have implications for the trajectory of the pandemic in the presence of evolving societal processes. The model presented has the potential to impact how social processes can be modelled in other infectious disease models.

1. Introduction

COVID-19 has infected nearly 500 million people, and 6.2 million people have died as a result of the pandemic as of 8 April 2022 [1]. The COVID-19 outbreak is likely to be the largest pandemic of the 21st century, in terms of human lives lost [2]. While vaccines have resulted in some notable disease control, with declines in infected cases globally, there are currently very few specific medical interventions, and these are only in developed countries; see, for instance, [3]. For developing countries such as South Africa, classes of drugs that are mainly used include antiviral agents, inflammation inhibitors, low-molecular-weight heparins, plasma, and hyperimmune immunoglobulins [4]. In the absence of effective infection prevention and control measures, human behaviour that reduces or prevents exposure to the virus is key [5]. The challenges of COVID-19 to the global population are detailed in [6].

It is impossible to undervalue the part that mathematical models played in the dynamics of the COVID-19 infection. In order to control the global COVID-19 epidemic, many public health measures have been guided by mathematical models that, to a considerable degree, were regional or country-specific; see, for instance, [7,8] for Wuhan, China; [9] for Italy; [10,11] for Spain; [12,13] for the United Kingdom; [14] for Scotland; and [15,16] for the United States of America, to mention a few. Several scholars have also looked at the difficulties in modeling the worldwide pandemic; see, for instance, [17,18,19,20,21,22]. One of the biggest problems in modelling infectious diseases is the inclusion of human behaviour in the models. The role of human behaviour in modulating the spread and prevalence of COVID-19 was presented in [23] using the imitation dynamic approach of evolutionary game theory. Human behavior is a major factor in the spread of infectious illnesses, and creating and enhancing control methods requires an understanding of how behavior affects disease transmission [18,24,25]. Mathematical models that involve the determination of optimal control measures were considered in [26,27,28]. Environmental transmission and cost-effectiveness analysis for the epidemic in Ghana was considered in [29].

The interaction of community psycho-social processes and infection dynamics has been linked to the formation of multiple waves of COVID-19 [30,31]. The first waves of COVID-19, often referred to as the “herald waves”, see [32], have been followed by the second and third waves of the viral infection months after the initial episodes. As seen in earlier pandemics such as the influenza pandemics of 1918, 1957, 1968, and 2009, the succeeding waves have been more severe than the initial one [30].

In this paper, we explore the role of two important aspects in the dynamics of the pandemic that have evolved over time, i.e., information and non-pharmaceutical interventions (NPIs). NPIs that have involved aspects such as social distancing, the wearing of masks, and restrictions, generally known as "lockdowns", have been touted as the best measures to reduce the spread of COVID-19 during the start of the epidemic. The possible resurgence of other respiratory endemic infections due to the NPIs has been predicted in [33]. The aim of this paper is to investigate how time-dependent information translates into the growth or decline of NPIs that in turn impact the transmission of COVID-19. Many mathematical models have considered NPIs modelled by constant parameters; see, for instance, [34]. The work presented in this paper follows the research in [35], but unlike in [35] we focus on the long-term dynamics of the infection, the role of information, and its potential impact in influencing NPIs. The results presented in this paper have the potential to influence how dynamic social processes can be included in other epidemic models.

This paper is arranged as follows: In Section 1, the introduction to the paper is presented followed by the model formulation in Section 2. The model analysis is presented in Section 3. Numerical simulations are presented in Section 4 and Section 5 concludes the paper.

2. Mathematical Model

We consider a classical human $SEIR$ model, coupled with behavioural state variables of susceptible individuals $S\left(t\right),$ exposed individuals who are not yet infectious $E\left(t\right)$, infectious individuals who are either detected $I_d\left(t\right)$ or undetected $I_u\left(t\right)$, and recovered individuals who are either recorded $R_d\left(t\right)$ or unrecorded $R_u\left(t\right)$. The total human population is given by

$$N\left(t\right)=S\left(t\right)+E\left(t\right)+I_d\left(t\right)+I_u\left(t\right)+R_d\left(t\right)+R_u\left(t\right).$$

The social processes at any time t are modelled by two state variables, $ϵ\left(t\right)$ and $\rho \left(t\right),$ where $ϵ\left(t\right)$ models the information on the disease, mainly driven by the number of detected cases that are then reported in the media, and $\rho \left(t\right)$ models the NPIs (personal hygiene, travel bans, mass gatherings bans, social distancing, curfews, and lockdowns) that are used to reduce transmission and are driven by the changes in the information on COVID-19. We assume that $0<ϵ\left(t\right),\rho \left(t\right)\le 1.$

The human population increases through births at a rate $\mu N,$ with recruits assumed susceptible and $\mu$ being the birth/death rate. Those that are susceptible are infected through interaction with the infected in the compartments $I_d$ and $I_u$ at a rate $\lambda$ defined by

$$\lambda =\beta (1-\rho \left(t\right))\frac{(I_d\left(t\right)+\eta I_u\left(t\right))}{N\left(t\right)},$$

where $\beta$ is the effective contact rate and $\eta$ is the relative infectivity rate of those in the compartment $I_u$ when compared to those in compartment $I_d.$ A value of $\eta >1$ corresponds to a higher infectivity rate for individuals in the $I_u$ class than for individuals in the $I_d$ class, whereas the case $0<\eta <1$ corresponds to a higher infectivity rate for individuals in the $I_d$ than for individuals in the $I_u$ class. Once infected, susceptible individuals move to the exposed class where they progress to become infectious after 5 to 6 days [36] at a rate $\kappa .$ A proportion q will be undetected infectives, while the remaining proportion becomes detected. The detected/undetected infectives recover at rates ${\sigma }_d/{\sigma }_u,$ and die due to COVID-19 at rates ${\delta }_d/{\delta }_u.$

The information growth rate is driven by the number of detected cases at a rate ${\alpha }_2$, and the information decreases at a rate ${\nu }_2$, with ${ϵ}_0$ being the constant rate of information inflow into the population. The NPIs grow at a rate ${\alpha }_1$ and decrease at a rate ${\nu }_1$, and the constant rate at which NPIs are practiced within the population is modeled by ${\rho }_0.$ The model flow diagram is presented in Figure 1 below.

The model formulation summary above, together with the model flow diagram (Figure 1), leads to the following system of equations:

$$\left\{\begin{array}{cccccc}\dot{S}& =& \mu N-\lambda S-\mu S,\hfill & {\dot{R}}_u& =& {\sigma }_u{I}_u-\mu R_u,\hfill \\ \dot{E}& =& \lambda S-(\mu +\kappa )E,\hfill & {\dot{R}}_d& =& {\sigma }_d{I}_d-\mu R_d,\hfill \\ {\dot{I}}_d& =& (1-q)\kappa E-(\mu +{\sigma }_d+{\delta }_d)I_d,\hfill & \dot{\rho }& =& {\rho }_0+{\alpha }_1ϵ-{\nu }_1\rho ,\hfill \\ {\dot{I}}_u& =& q\kappa E-(\mu +{\sigma }_u+{\delta }_u)I_u,\hfill & \dot{ϵ}& =& {ϵ}_0+{\alpha }_2{I}_d-{\nu }_2ϵ.\hfill \end{array}\right.$$

(1)

Given that the human population and the social process have different dimensions, we re-scale the human population by setting

$$s=\frac{S}{N},e=\frac{E}{N},i_u=\frac{I_u}{N},i_d=\frac{I_d}{N},r_u=\frac{R_u}{N}\mathrm{and}{r}_d=\frac{R_d}{N}$$

and obtain the following system of differential equations:

$$\left.\begin{array}{ccc}\dot{s}& =& \mu -\beta (1-\rho )(i_d+\eta i_u)s-\mu s,\hfill \\ \dot{e}& =& \beta (1-\rho )(i_d+\eta i_u)s-(\mu +\kappa )e,\hfill \\ \dot{i_d}& =& (1-q)\kappa e-(\mu +{\sigma }_d+{\delta }_d)i_d,\hfill \\ \dot{i_u}& =& q\kappa e-(\mu +{\sigma }_u+{\delta }_u)i_u,\hfill \\ {\dot{r}}_d& =& {\sigma }_d{i}_d-\mu r_d,\hfill \\ {\dot{r}}_u& =& {\sigma }_u{i}_u-\mu r_u,\hfill \end{array}\right\}\cdots \mathrm{Human}\mathrm{dynamics}$$

(2)

$$\left.\begin{array}{ccc}\dot{\rho }& =& {\rho }_0+{\alpha }_1ϵ-{\nu }_1\rho ,\hfill \\ \dot{ϵ}& =& {ϵ}_0+{\alpha }_2{i}_d-{\nu }_2ϵ,\hfill \end{array}\right\}\cdots \cdots \cdots \cdots \cdots \cdots \mathrm{Social}\mathrm{dynamics}$$

(3)

with the initial conditions $s\left(0\right)>0$, $e\left(0\right)\ge 0$, $i_d\left(0\right)\ge 0$, $i_u\left(0\right)\ge 0$, $r_d\left(0\right)\ge 0$, $r_u\left(0\right)\ge 0$, $\rho \left(0\right)>0$ and $ϵ\left(0\right)>0.$

3. Model Analysis

3.1. Positivity of the Solutions

Lemma 1.

Suppose that $s\left(0\right)>0,e\left(0\right)\ge 0,i_d\left(0\right)\ge 0,i_u\left(0\right)\ge 0,r_d\left(0\right)\ge 0,r_u\left(0\right)\ge 0,\rho \left(0\right)>0$ and $ϵ\left(0\right)>0$, then the solution

$\left(s\left(t\right),e\left(t\right),i_d\left(t\right),i_u\left(t\right),r_d\left(t\right),r_u\left(t\right),\rho \left(t\right),ϵ\left(t\right)\right)\in {\mathbb{R}}^8$ of system (2) and (3) is positive for all time $t\ge 0$.

Proof. 

Let $\left(s\left(t\right),e\left(t\right),i_d\left(t\right),i_u\left(t\right),r_d\left(t\right),r_u\left(t\right),\rho \left(t\right),ϵ\left(t\right)\right)$ be the solution of system (2) and (3). From the first equation of system (2) and (3), we have that

$$\dot{s}=\mu -\beta (1-\rho )(i_d+\eta i_u)s-\mu s,\mathrm{where}\tilde{\lambda }=\beta (1-\rho )(i_d+\eta i_u).$$

Thus,

$$\dot{s}=\mu -(\tilde{\lambda }+\mu )s\Rightarrow \dot{s}\ge -(\tilde{\lambda }+\mu )s.$$

(4)

Integrating both sides and applying the initial conditions leads to

$${\displaystyle s\left(t\right)\ge s\left(0\right)e^{-\left({\int }_0^t\tilde{\lambda }\left(\tau \right)d\tau +\mu t\right)}}$$

(5)

at any value of $t\ge 0.$ Since $s\left(0\right)\ge 0$ then $s\left(t\right)\ge 0.$

Additionally, from the second equation of system (2) and (3) we have that

$$\dot{e}=\beta (1-\rho )(i_d+\eta i_u)s-(\mu +\kappa )e\Rightarrow \dot{e}\ge -(\mu +\kappa )e.$$

(6)

Integrating both sides and applying the initial conditions gives

$$e\left(t\right)\ge e\left(0\right)e^{-(\mu +\kappa )t}$$

(7)

at any value of $t\ge 0$. Since $e\left(0\right)\ge 0$, then $e\left(t\right)\ge 0$.

Similarly, it can be shown that $i_d\left(t\right)\ge 0$, $i_u\left(t\right)\ge 0$, $r_d\left(t\right)\ge 0$, $r_u\left(t\right)\ge 0$, $\rho \left(t\right)\ge 0$ and $ϵ\left(t\right)\ge 0$. □

3.2. Feasible Region

Consider the biological feasible region given by

$\Omega =\left\{\mathcal{X}\in {\mathbb{R}}_{+}^8:n\le 1,\rho \le \frac{{\rho }_0}{{\nu }_1}+\frac{{\alpha }_1}{{\nu }_1{\nu }_2}\left({ϵ}_0+{\alpha }_2\right),ϵ\le \frac{1}{{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)\right\},$ where $\mathcal{X}=\left(s\left(t\right),e\left(t\right),i_u\left(t\right),i_d\left(t\right),r_u\left(t\right),r_d\left(t\right),\rho \left(t\right),ϵ\left(t\right)\right),n=s+e+i_u+i_d+r_u+r_d.$

Lemma 2.

The solution of the system (2) and (3) with the non-negative initial conditions are bounded for all $t\ge 0$ in the biological feasible region $\Omega .$

Proof. 

We note from system (2) that

$$\frac{dn}{dt}=\mu -\mu n-{\delta }_d{i}_d-{\delta }_u{i}_u\le \mu (1-n)\Rightarrow n\left(t\right)\le 1+(n\left(0\right)-1)e^{-\mu t}$$

(8)

upon solving the differential inequality and applying the initial condition.

Thus, $n\to 1$ as $t\to \infty$.

Additionally, from system (3) we have that

$$\dot{ϵ}={ϵ}_0+{\alpha }_2{i}_d-{\nu }_2ϵ\le {ϵ}_0+{\alpha }_2-{\nu }_2ϵ\mathrm{since}{i}_d\le 1.$$

(9)

Upon solving the differential inequality (9), we obtain

$$ϵ\left(t\right)\le \frac{1}{{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)+\left(ϵ\left(0\right)-\frac{1}{{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)\right)e^{-{\nu }_{2}t}.$$

(10)

Thus, as $t\to \infty$ we have that $ϵ\left(t\right)\le \frac{1}{{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)$.

Lastly, from system (3) we have that

$$\dot{\rho }={\rho }_0+{\alpha }_1ϵ-{\nu }_1\rho \le {\rho }_0+\frac{{\alpha }_1}{{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)-{\nu }_1\rho \mathrm{since}ϵ\le \frac{1}{{\nu }_2}\left({ϵ}_0+{\alpha }_2\right).$$

(11)

Solving the differential inequality (11), we obtain

$$\rho \left(t\right)\le \frac{{\rho }_0}{{\nu }_1}+\frac{{\alpha }_1}{{\nu }_1{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)+\left(\rho \left(0\right)-\frac{{\rho }_0}{{\nu }_1}+\frac{{\alpha }_1}{{\nu }_1{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)\right)e^{-{\nu }_{1}t}.$$

(12)

Thus, as $t\to \infty$ we have that $\rho \left(t\right)\le \frac{{\rho }_0}{{\nu }_1}+\frac{{\alpha }_1}{{\nu }_1{\nu }_2}\left({ϵ}_0+{\alpha }_2\right)$. Hence, $(s,e,i_u,i_d,r_u,r_d,\rho ,ϵ)$ are all bounded in the region $\Omega$. Therefore, $\Omega$ is biologically feasible. □

3.3. Disease-Free Equilibrium and Model Reproduction Number

The model system (2) and (3) has a disease-free equilibrium given by

$${\mathcal{D}}^0=\left(s^0,e^0,i_u^0,i_d^0,r_u^0,r_d^0,{\rho }^0,{ϵ}^0\right)=\left(1,0,0,0,0,0,\Phi ,\frac{{ϵ}_0}{{\nu }_2}\right)\mathrm{where}\Phi ={\displaystyle \frac{{\rho }_0{\nu }_2+{ϵ}_0{\alpha }_1}{{\nu }_1{\nu }_2}}.$$

Following the next-generation matrix approach in van den Driesche and Watmough [37], we have that

$$F=\left(\begin{array}{ccccccc}0& \beta \left(1-\Phi \right)& \beta \eta \left(1-\Phi \right)& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right)$$

and

$$V=\left(\begin{array}{ccccccc}\kappa +\mu & 0& 0& 0& 0& 0& 0\\ -(1-q)\kappa & h_d& 0& 0& 0& 0& 0\\ -q\kappa & 0& h_u& 0& 0& 0& 0\\ 0& -{\sigma }_d& 0& \mu & 0& 0& 0\\ 0& 0& -{\sigma }_u& 0& \mu & 0& 0\\ 0& 0& 0& 0& 0& {\nu }_1& -{\alpha }_1\\ 0& -{\alpha }_2& 0& 0& 0& 0& {\nu }_2\end{array}\right).$$

Thus, the basic reproduction number denoted by ${\mathcal{R}}_0$ and defined as the spectral radius of the matrix $F{V}^{-1}$ is given by

$$\begin{array}{cc}{\mathcal{R}}_0\hfill & ={\mathcal{R}}_d+{\mathcal{R}}_u\mathrm{with}\hfill \\ \\ {\mathcal{R}}_d\hfill & ={\displaystyle \frac{\beta \kappa (1-\Phi )(1-q)}{(\mu +\kappa )h_d}}\mathrm{and}{\mathcal{R}}_u={\displaystyle \frac{\beta \kappa (1-\Phi )q\eta }{(\mu +\kappa )h_u}},\hfill \end{array}$$

(13)

where $h_d=\mu +{\sigma }_d+{\delta }_d$ and $h_u=\mu +{\sigma }_u+{\delta }_u$. We fix $\Phi <1$ in order for ${\mathcal{R}}_0$ to be biologically meaningful. Here, ${\mathcal{R}}_d$ and ${\mathcal{R}}_u$ represent the contribution of individuals in classes $i_d$ and $i_u$ respectively.

The following theorem follows from van den Driesche and Watmough: [37].

Theorem 1.

The disease-free equilibrium ${\mathcal{D}}^0$ is locally asymptotically stable when ${\mathcal{R}}_0<1$ and is unstable when ${\mathcal{R}}_0>1$.

Proof. 

The Jacobian matrix of system (2) and (3) at ${\mathcal{D}}^0$ is given by

$$J\left({\mathcal{D}}^0\right)=\left(\begin{array}{cccccccc}-\mu & 0& -\beta (1-\Phi )& -\beta \eta (1-\Phi )& 0& 0& 0& 0\\ 0& -\kappa -\mu & \beta (1-\Phi )& \beta \eta (1-\Phi )& 0& 0& 0& 0\\ 0& (1-q)\kappa & -h_d& 0& 0& 0& 0& 0\\ 0& q\kappa & 0& -h_u& 0& 0& 0& 0\\ 0& 0& {\sigma }_d& 0& -\mu & 0& 0& 0\\ 0& 0& 0& {\sigma }_u& 0& -\mu & 0& 0\\ 0& 0& 0& 0& 0& 0& -{\nu }_1& {\alpha }_1\\ 0& 0& {\alpha }_2& 0& 0& 0& 0& -{\nu }_2\end{array}\right)$$

where $h_d$, $h_u$, and $\Phi$ are defined as before. The local stability of the disease-free equilibrium is determined by the following submatrix of $J\left({\mathcal{D}}^0\right)$,

$$\overline{J}\left({\mathcal{D}}^0\right)=\left(\begin{array}{ccccccc}-\kappa -\mu & \beta (1-\Phi )& \beta \eta (1-\Phi )& 0& 0& 0& 0\\ (1-q)\kappa & -h_d& 0& 0& 0& 0& 0\\ q\kappa & 0& -h_u& 0& 0& 0& 0\\ 0& {\sigma }_d& 0& -\mu & 0& 0& 0\\ 0& 0& {\sigma }_u& 0& -\mu & 0& 0\\ 0& 0& 0& 0& 0& -{\nu }_1& {\alpha }_1\\ 0& {\alpha }_2& 0& 0& 0& 0& -{\nu }_2\end{array}\right).$$

Note that all the off-diagonal elements of $\overline{J}\left({\mathcal{D}}^0\right)$ are positive. We claim that $-\overline{J}\left({\mathcal{D}}^0\right)$ is an $M-$matrix. Define two positive $7\times 1$ matrices Y and Z, respectively, as follows:

$$Y={\left(y_1,y_2,y_3,y_4,y_5,y_6,y_7\right)}^{T}andZ={\left(z,0,0,0,0,0,0\right)}^T,$$

where

$$\left\{\begin{array}{cc}{y}_1\hfill & ={\displaystyle \frac{h_d{h}_u}{\kappa }},y_2=(1-q)h_u,y_3=q{h}_d,y_4={\displaystyle \frac{(1-q){\sigma }_d{h}_u}{\mu }},y_5={\displaystyle \frac{q{h}_d{\sigma }_u}{\mu }},\hfill \\ \\ y_6\hfill & ={\displaystyle \frac{{\alpha }_1{\alpha }_2(1-q)h_u}{{\nu }_1{\nu }_2}},y_7={\displaystyle \frac{{\alpha }_2(1-q)h_u}{{\nu }_2}}andz={\displaystyle \frac{h_u{h}_d(\mu +\kappa )\left(1-{\mathcal{R}}_0\right)}{\kappa }}.\hfill \end{array}\right.$$

(14)

Multiplying matrix $-\overline{J}\left({\mathcal{D}}_0\right)$ by Y gives

$$-\overline{J}\left({\mathcal{D}}_0\right)·Y=(1-{\mathcal{R}}_0)·Z.$$

Thus, employing the theory for $M-$matrices, it follows that all the eigenvalues of $\overline{J}\left({\mathcal{D}}^0\right)$ have negative real parts. Therefore, the disease-free equilibrium ${\mathcal{D}}^0$ is locally asymptotically stable for ${\mathcal{R}}_0<1$. On the other hand, we compute the determinant of $\overline{J}\left({\mathcal{D}}^0\right)$ to obtain

$$det\overline{J}\left({\mathcal{D}}^0\right)=z({\mathcal{R}}_0-1).$$

Thus, if ${\mathcal{R}}_0<1$, then matrix $\overline{J}\left({\mathcal{D}}^0\right)$ has eigenvalues with negative real parts, which implies the stability of the disease-free equilibrium ${\mathcal{D}}^0$. This completes the proof. □

3.4. Endemic Equilibrium

The long-term dynamics of the model can be determined by considering the steady states of the model. Solving the fourth equation of system (2) for $i_u^{*}$, we obtain

$$i_u^{*}={\displaystyle \frac{e^{*}\kappa q}{h_u}}.$$

(15)

Solving the third equation of system (2) for $i_d^{*}$, we obtain

$$i_d^{*}={\displaystyle \frac{e^{*}\kappa (1-q)}{h_d}}.$$

(16)

Likewise, solving the first, fifth, and sixth equations of system (2), respectively, for $s^{*}$, $r_u^{*}$ and $r_d^{*}$ together with system (3), respectively, for ${ϵ}^{*}$ and ${\rho }^{*}$ in terms of $e^{*}$ gives

$$\left\{\begin{array}{cc}{s}^{*}\hfill & =1-{\displaystyle \frac{e^{*}(\mu +\kappa )}{\mu }},r_u^{*}={\displaystyle \frac{e^{*}\kappa q{\sigma }_u}{\mu h_u}},\hfill \\ \\ r_d^{*}\hfill & ={\displaystyle \frac{e^{*}\kappa (1-q){\sigma }_d}{\mu h_d}},{ϵ}^{*}={\displaystyle \frac{{ϵ}_0{h}_d+{\alpha }_2{e}^{*}\kappa (1-q)}{{\nu }_2{h}_d}},\hfill \\ \\ {\rho }^{*}\hfill & ={\displaystyle \frac{{\alpha }_1\left({ϵ}_0{h}_d+{\alpha }_2{e}^{*}\kappa (1-q)\right)+{\nu }_2{\rho }_0{h}_d}{{\nu }_1{\nu }_2{h}_d}}.\hfill \end{array}\right.$$

(17)

Substituting the expressions (15)–(17) into the second equation of system (2) leads to the following third order polynomial equation in terms of $e^{*}$:

$$e^{*}\left(a{e}^{{*}^2}+b{e}^{*}+c\right)=0$$

(18)

where

$$\left\{\begin{array}{cc}a\hfill & ={\alpha }_1{\alpha }_2\beta {\kappa }^2(1-q)(\kappa +\mu )\left(\eta q{h}_d+(1-q)h_u\right),\hfill \\ \\ b\hfill & =-\beta \kappa \left(\eta q{h}_d+(1-q)h_u\right)\left(h_d{\nu }_1{\nu }_2(\mu +\kappa )\left(1-\Phi \right)+\mu \kappa {\alpha }_1{\alpha }_2(1-q)\right),\hfill \\ \\ c\hfill & =\mu h_d^2{h}_u{\nu }_1{\nu }_2(\mu +\kappa )\left[1-{\mathcal{R}}_0\right].\hfill \end{array}\right.$$

(19)

Thus, we summarise the results on the existence of the endemic equilibrium of systems (2) and (3) in Theorem 2.

Theorem 2.

$\left({\mathcal{H}}_1\right)$.

Systems (2) and (3) have a unique endemic equilibrium ${\mathcal{D}}^{*}=\left(s^{*},e^{*},i_u^{*},i_d^{*},r_u^{*},r_d^{*},{\rho }^{*},{ϵ}^{*}\right)$ if and only if $a>0$, $b<0$ and ${\mathcal{R}}_0<1$.

$\left({\mathcal{H}}_2\right)$.

Systems (2) and (3) have two endemic equilibria ${\mathcal{D}}_1^{*}=\left(s_1^{*},e_1^{*},i_{u1}^{*},i_{d1}^{*},r_{u1}^{*},r_{d1}^{*},{\rho }_1^{*},{ϵ}_1^{*}\right)$ and ${\mathcal{D}}_2^{*}=\left(s_2^{*},e_2^{*},i_{u2}^{*},i_{d2}^{*},r_{u2}^{*},r_{d2}^{*},{\rho }_2^{*},{ϵ}_2^{*}\right)$ when $a>0$, $b<0$ and ${\mathcal{R}}_0>1$ where $e_1^{*}=(-b+\sqrt{b^2-4ac})/2a$ and $e_2^{*}=(-b-\sqrt{b^2-4ac})/2a$.

Proof. 

Solving (18) gives $e^{*}=0$, which corresponds to the disease-free equilibrium or

$$a{e}^{{*}^2}+b{e}^{*}+c=0.$$

(20)

We solve the quadratic Equation (20). Note that

$$a>0,b<0,c>0\left({\mathcal{R}}_0<1\right)\mathrm{and}c<0\left({\mathcal{R}}_0>1\right).$$

(21)

Various possibilities for the existence of the endemic equilibrium are shown in

Table 1. Shows the number of possible positive solutions for $e^{*}$.

	$a>0$
	$b<0$
	$c>0$	$c<0$
	$(R_0<1)$	$(R_0>1)$
$e^{*}$	2	1

below. Here, $i^{*}$ denotes the number of roots of (20).

□

3.5. Local Stability of the Endemic Equilibrium Point

In this section, we study the local stability of the endemic equilibrium of system (2) and (3). We employ Theorem $4.1$ proven in the work by Castillo–Chavez and Song [38]. We focus on the application of the theorem and refer readers to [38] for more details on the theorem.

Theorem 3.

Define

$$\begin{array}{ccc}\hfill \Delta & =& \frac{{\xi }_2}{{\xi }_3},\hfill \\ \hfill {\xi }_2& =& h_d^2{\nu }_1{\nu }_2(\kappa +\mu )\left(\Phi -1\right)\left(h_u+\eta \kappa q\right),\hfill \\ \hfill {\xi }_3& =& {\alpha }_1{\alpha }_2\eta {\kappa }^2\mu (1-q)q{ϵ}_0{h}_d+{\alpha }_1{\alpha }_2{\kappa }^2\mu {(1-q)}^2{h}_u.\hfill \end{array}$$

If $\Delta <1$, then the endemic equilibrium of system (2) and (3) is locally asymptotically stable for ${\mathcal{R}}_0>1$ but close to one. Otherwise, if $\Delta >1$, then model system (2) and (3) has a backward bifurcation at ${\mathcal{R}}_0=1$.

Proof. 

We introduce the following change of variables for the easy application of Theorem $4.1$ in [38]:

$s=x_1,e=x_2,i_d=x_3,i_u=x_4,r_d=x_5,r_u=x_6,\rho =x_7,ϵ=x_8$, so that ${\displaystyle N=\sum _{n=1}^8{x}_n}$. We now use the vector notation $X={\left(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\right)}^T.$ Then, the model system (2) and (3) can be written in the form

$${\displaystyle \frac{dX}{dt}}=F(t,x\left(t\right))={\left(f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8\right)}^T,$$

where

$$\left.\begin{array}{ccc}{\displaystyle \frac{d{x}_1}{dt}}\hfill & =\hfill & \mu -\beta (1-\rho )(x_3+\eta x_4)x_1-\mu x_1=f_1,\hfill \\ \\ {\displaystyle \frac{d{x}_2}{dt}}\hfill & =\hfill & \beta (1-\rho )(x_3+\eta x_4)x_1-(\mu +\kappa )x_2=f_2,\hfill \\ \\ {\displaystyle \frac{d{x}_3}{dt}}\hfill & =\hfill & (1-q)\kappa x_2-h_d{x}_3=f_3,\hfill \\ \\ {\displaystyle \frac{d{x}_4}{dt}}\hfill & =\hfill & q\kappa x_2-h_u{x}_4=f_4,\hfill \\ \\ {\displaystyle \frac{d{x}_5}{dt}}\hfill & =\hfill & {\sigma }_d{x}_3-\mu x_5=f_5,\hfill \\ \\ {\displaystyle \frac{d{x}_6}{dt}}\hfill & =\hfill & {\sigma }_u{x}_4-\mu x_6=f_6,\hfill \\ \\ {\displaystyle \frac{d{x}_7}{dt}}\hfill & =\hfill & {\rho }_0+{\alpha }_1{x}_8-{\nu }_1{x}_7=f_7,\hfill \\ \\ {\displaystyle \frac{d{x}_8}{dt}}\hfill & =\hfill & {ϵ}_0+{\alpha }_2{x}_3-{\nu }_2{x}_8=f_8.\hfill \end{array}\right\}$$

(22)

Let $\beta$ be the bifurcation parameter, and ${\mathcal{R}}_0=1$ corresponds to

$$\begin{array}{c}\hfill \beta ={\beta }^{*}={\displaystyle \frac{{\nu }_1{\nu }_2{h}_d{h}_u(\kappa +\mu )}{\kappa \left({\nu }_2\left({\nu }_1-{\rho }_0\right)-{\alpha }_1{ϵ}_0\right)\left(\eta q{h}_d+(1-q)h_u\right)}}.\end{array}$$

(23)

The Jacobian matrix of system (2) and (3) at ${\mathcal{D}}^0$ when $\beta ={\beta }^{*}$ is given by

$$J^{*}\left({\mathcal{D}}^0\right)=\left(\begin{array}{cccccccc}-\mu & 0& -{\beta }^{*}\left(1-\Phi \right)& -{\beta }^{*}\eta \left(1-\Phi \right)& 0& 0& 0& 0\\ 0& -\kappa -\mu & {\beta }^{*}\left(1-\Phi \right)& {\beta }^{*}\eta \left(1-\Phi \right)& 0& 0& 0& 0\\ 0& (1-q)\kappa & -h_d& 0& 0& 0& 0& 0\\ 0& q\kappa & 0& -h_u& 0& 0& 0& 0\\ 0& 0& {\sigma }_d& 0& -\mu & 0& 0& 0\\ 0& 0& 0& {\sigma }_u& 0& -\mu & 0& 0\\ 0& 0& 0& 0& 0& 0& -{\nu }_1& {\alpha }_1\\ 0& 0& {\alpha }_2& 0& 0& 0& 0& -{\nu }_2\end{array}\right)$$

where $h_d$ and $h_u$ are defined as before.

It can be noted that for $\beta ={\beta }^{*}$, systems (2) and (3) have a simple eigenvalue. Thus, the center manifold theory can be applied to analyse the dynamics of system (2) and (3) near $\beta ={\beta }^{*}$. The right eigenvector for $J^{*}\left({\mathcal{D}}^0\right)$ is given by $w={(w_1,w_2,w_3,w_4,w_5,w_6,w_7,w_8)}^T$, where

$$\begin{array}{ccc}\hfill w_1& =& {\nu }_1{\nu }_2{h}_d{h}_u(\kappa +\mu ),w_2=-\mu {\nu }_1{\nu }_2{h}_d{h}_u,w_3=-\kappa \mu {\nu }_1{\nu }_2(1-q)h_u,\hfill \\ \hfill w_4& =& -\kappa \mu {\nu }_1{\nu }_{2}q{h}_d,w_5=-\kappa {\nu }_1{\nu }_2(1-q){\sigma }_d{h}_u,w_6=-\kappa {\nu }_1{\nu }_{2}q{h}_d{\sigma }_u,\hfill \\ \hfill w_7& =& -{\alpha }_1{\alpha }_2\kappa \mu (1-q)h_u,w_8=-{\alpha }_2\kappa \mu {\nu }_1(1-q)h_u.\hfill \end{array}$$

The left eigenvector of $J^{*}\left({\mathcal{D}}^0\right)$, associated with the zero eigenvalue at $\beta ={\beta }^{*}$, is given by $v={(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8)}^T$, where

$$\begin{array}{ccc}\hfill v_1& =& v_5=v_6=v_7=v_8=0,v_2=\kappa \left(\eta q{h}_d+(1-q)h_u\right),\hfill \\ \hfill v_3& =& h_u(\kappa +\mu ),v_4=\eta h_d(\kappa +\mu ).\hfill \end{array}$$

Now, we compute a and b as defined in Castillo–Chavez and Song [38] and apply Theorem $4.1$ in Castillo–Chavez and Song [38]. For system (22), the associated non-zero partial derivatives of F at ${\mathcal{D}}^0$ are given below in (24).

$$\left\{\begin{array}{ccc}{\displaystyle \frac{{\partial }^2{f}_1}{\partial x_1\partial x_3}}\hfill & =\hfill & {\displaystyle \frac{{\partial }^2{f}_1}{\partial x_3\partial x_1}}=-{\beta }^{*}\left(1-\Phi \right),{\displaystyle \frac{{\partial }^2{f}_1}{\partial x_1\partial x_4}}={\displaystyle \frac{{\partial }^2{f}_1}{\partial x_4\partial x_1}}=-{\beta }^{*}\eta \left(1-\Phi \right),\hfill \\ \\ {\displaystyle \frac{{\partial }^2{f}_1}{\partial x_3\partial x_7}}\hfill & =\hfill & {\displaystyle \frac{{\partial }^2{f}_1}{\partial x_7\partial x_3}}={\beta }^{*},{\displaystyle \frac{{\partial }^2{f}_1}{\partial x_4\partial x_7}}={\displaystyle \frac{{\partial }^2{f}_1}{\partial x_7\partial x_4}}=\eta {\beta }^{*},\hfill \\ \\ {\displaystyle \frac{{\partial }^2{f}_2}{\partial x_1\partial x_3}}\hfill & =\hfill & {\displaystyle \frac{{\partial }^2{f}_2}{\partial x_3\partial x_1}}={\beta }^{*}\left(1-\Phi \right),{\displaystyle \frac{{\partial }^2{f}_2}{\partial x_1\partial x_4}}={\displaystyle \frac{{\partial }^2{f}_2}{\partial x_4\partial x_1}}={\beta }^{*}\eta \left(1-\Phi \right),\hfill \\ \\ {\displaystyle \frac{{\partial }^2{f}_2}{\partial x_3\partial x_7}}\hfill & =\hfill & {\displaystyle \frac{{\partial }^2{f}_2}{\partial x_7\partial x_3}}=-{\beta }^{*},{\displaystyle \frac{{\partial }^2{f}_2}{\partial x_4\partial x_7}}={\displaystyle \frac{{\partial }^2{f}_2}{\partial x_7\partial x_4}}=-\eta {\beta }^{*},\hfill \\ \\ {\displaystyle \frac{{\partial }^2{f}_1}{\partial x_3\partial {\beta }^{*}}}\hfill & =\hfill & \Phi -1,{\displaystyle \frac{{\partial }^2{f}_1}{\partial x_4\partial {\beta }^{*}}}=-\eta \left(1-\Phi \right),\hfill \\ \\ {\displaystyle \frac{{\partial }^2{f}_2}{\partial x_3\partial {\beta }^{*}}}\hfill & =\hfill & 1-\Phi ,{\displaystyle \frac{{\partial }^2{f}_2}{\partial x_4\partial {\beta }^{*}}}=\eta \left(1-\Phi \right).\hfill \end{array}\right.$$

(24)

It thus follows that

$$\begin{array}{ccc}\hfill \mathbf{a}& =& {\displaystyle \sum _{i=3}^{4}2v_2{w}_1{w}_i\frac{{\partial }^2{f}_2}{\partial x_1\partial x_i}+\sum _{i=3}^{4}2v_2{w}_7{w}_i\frac{{\partial }^2{f}_2}{\partial x_i\partial x_7},}\hfill \\ \hfill & =& {\xi }_1\left({\xi }_2-{\xi }_3\right),\hfill \\ \hfill & =& {\xi }_1{\xi }_3\left(\Delta -1\right)\left(\mathrm{with}\frac{{\xi }_2}{{\xi }_3}=\Delta \right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\xi }_1& =& 2\beta \kappa \mu {\nu }_1{\nu }_2{h}_u\left(\eta q{h}_d+(1-q)h_u\right),\hfill \\ \hfill {\xi }_2& =& h_d^2{\nu }_1{\nu }_2(\kappa +\mu )\left(\Phi -1\right)\left(h_u+\eta \kappa q\right),\hfill \\ \hfill {\xi }_3& =& {\alpha }_1{\alpha }_2\eta {\kappa }^2\mu (1-q)q{ϵ}_0{h}_d+{\alpha }_1{\alpha }_2{\kappa }^2\mu {(1-q)}^2{h}_u.\hfill \end{array}$$

Now, we fix ${\rho }_0>{\nu }_1$. Note that if $\Delta >1$, then $\mathbf{a}>0$ and $\mathbf{a}<0$ if $\Delta <1$. Lastly,

$$\begin{array}{c}\hfill \mathbf{b}={\kappa }^2\mu \left({\alpha }_1+{\nu }_2\left({\rho }_0-{\nu }_1\right)\right)\left(\eta q{h}_d+(1-q)h_u\right){}^2>0.\end{array}$$

This completes the proof. □

4. Numerical Simulations and Model Validation

The population of South Africa in 2020 was estimated to be 59.54 million [39]. We take the initial time of the epidemic to be the time when the first lockdown measures took effect, i.e., on the 27 March 2020. The choice of the initial time is driven by the fact that there were no non-pharmaceutical interventions before then. Additionally, the waves following the first wave had additional interventions that included vaccination, which was not captured in our model. It is thus apparent that we use data from the first wave. We chose the end time of the first wave to be 15 October 2020. It is important that there was no clear time set for the end of the first wave and the start of the second wave. We hypothetically chose a value at the mid-point of the trough between the first and second waves. The initial conditions are given by

$$\begin{array}{cc}\hfill s0=& \frac{59539894}{59540000},e_0=\frac{86}{59540000},I_{d0}=\frac{20}{59540000},I_{u0}=0,\hfill \\ \hfill r_{d0}=& 0,r_{u0}=0,{\rho }_0=0,{ϵ}_0=0.\hfill \end{array}$$

While some of the parameters such as the birth/death rates can be obtained from [39], the majority of the parameters are generated by the curve fitting algorithm, the fminsearch, which uses the Nelder–Mead simplex algorithm described in detail in [40]. The birth/death rate in this case is assumed to be related to the life expectancy in the year 2020. The life expectancy in South Africa was 65.5 years in the year 2020, translating to $\mu =0.015.$ The number of undetected cases is estimated to be seven times higher than the reported figures [41]. So, q is chosen in such a way that the proportion of undetected cases is seven times that of the detected. We thus have $q=0.875.$ Our fitting suggests a value three times the one reported in [41] for South Africa. The remaining parameters are determined from the fitting that is depicted in Figure 2.

4.1. Impact of Varying NPIs

NPIs were impacted greatly due changes in the lockdown levels as the COVID-19 pandemic progressed. Relaxations and the tightening of NPIs have been the norm in many countries depending on the number of cases and changes in hospitalisation levels over time. We consider the relaxation of interventions that are impacted by fear. As the numbers of cases decrease, the fear of COVID-19 decreases, consequently leading to a relaxation of the NPIs. The impact of NPIs is modelled by the parameter ${\nu }_1.$ We note here that while the changes seem to be minimal, at a scale of ${10}^3$ this is significant, and an increase in the variations of the parameters being varied can increase the observations.

Figure 3 and Figure 4 show that the number of cases increases with any increase in the reduction in NPIs. It is thus important to note that such results are key in the quantification of the effects of NPIs. The results presented are thus for illustrative purposes as the changes in parameters such as ${\nu }_1$ are difficult to observe in real life.

4.2. Impact of Varying Fear Levels

The structure of our model is such that fear resulting from increased numbers of detected cases can give rise to increased NPIs. The removal in the levels of fear is modeled by the parameter ${\nu }_2.$ We vary ${\nu }_2$ from 0.1 to 0.7, and the fraction of undetected infectives follows the dynamics in Figure 5. Clearly, the direct impact of fear does not translate into significant changes in the fraction of infectives. We had to zoom in to determine the influence of varying ${\nu }_2.$ The insert on Figure 5 shows that increasing the levels of fear results in an increased number of infected cases.

4.3. Impact of Varying Parameters ${\nu }_1$ and $\kappa$ on $R_0$

The reduction in NPIs, modelled by ${\nu }_1$, and the progression to infectiousness, modelled by $\kappa ,$ is of interest in this paper. We determine the effects of both parameters by a contour plot $R_0$ and the two parameters. Figure 6 shows that an increase in the two parameters leads to an increase in the value of $R_0.$ We add a surface of $R_0=1,$ to establish the parameter values of ${\nu }_1$ and $\kappa$ for which the value of $R_0=1.$ This is key in determining the persistence of the infection when all other parameters are assumed to be constant.

4.4. Impact Varying Parameters ${\alpha }_1$ and $\beta$ on $R_0$

Also of interest are the parameters ${\alpha }_1$; the growth rate of NPIs; and $\beta ,$ the effective contact rate. We investigate the effects of the two parameters on a contour plot $R_0$ and the two parameters. Figure 7 shows that an increase in $\beta$ has a more significant effect on the values of $R_0$ than ${\alpha }_1.$ We also added a surface of $R_0=1,$ to establish the parameter values of ${\alpha }_1$ and $\beta$ for which the value of $R_0=1.$

5. Discussion and Conclusions

The work considered in this study captures the role of human behaviour on the spread of COVID-19 in South Africa influenced by the spread of information. An eight-state system of ODEs split into two sub-systems capturing human dynamics and social dynamics is formulated. As can be observed in the social dynamics sub-system, information on COVID-19 is considered to depend on data for COVID-19 detected cases that in turn influence changes in human behavior. The baseline values for information availability and human behaviour are included to add realism to the system of equations. It is important to note that, due to differences in scales between the two sub-systems, it was important to non-dimensionalise the system of equations. The non-dimensionalised system was then analysed. The model steady states were determined and their stabilities analysed. The model analysis of the endemic equilibrium shows the existence of multiple steady states. The model exhibits a backward bifurcation. The backward bifurcation is established using the center manifold theory.

Given that at the start of the pandemic, the only control measures involved NPIs, and the first wave, in South Africa and in many other countries, passed without the use of vaccines and therapeutics, we consider data on the first wave of COVID-19 in South Africa to calibrate our model. The fminsearch algorithm is used to find the parameters that best fit the model to data.

Mathematical models that include human behaviour are key to the management of diseases. At the beginning of the pandemic, COVID-19 treatment remained part of a bigger global research drive to find therapeutics for the disease. However, the management and control of the disease relied on human behaviour influenced by government policies. The policies also evolved as more information on the pandemic emerged. In South Africa, restrictions to human movement, personal hygiene, the cleaning of surfaces, sanitising, and social distancing were key to minimising transmission before the advent of the vaccines on the market. In particular, regulations on restrictions varied as lockdown levels changed depending on the number of cases and hospitalisation rates.

The work presented in this article can certainly be improved to capture a holistic view of social processes. It is important to note that the modelling of social processes is complex, and the biggest complexity arises from the choice of mathematical functions used to model the processes. In this paper, the growth and decline of NPIs are modelled through mathematical functions. While the functions presented in this paper were chosen for their simplicity, other functions for $f\left(ϵ\right)$ and $g\left(\frac{I_d}{N}\right)$ can be used to model the growth of NPIs and fear. The choice of the functions has the propensity to influence the quality of the model outputs. One of the weaknesses of this article is the non-inclusion of the hospitalised compartment, which can influence the information that impacts the dynamics of the NPIs. The paper can also be extended to incorporate vaccination as many countries to date have considered this as the main preventive form of control for COVID-19. These aspects and some other observed characteristics that influence the trends of the spread of COVID-19 can be considered as possible extensions to this work.

Despite the shortcomings given above, the modelling framework presented here has the potential to influence the modelling of other infections that are significantly influenced by human behaviour. The advent of COVID-19 shows that new emerging infections are a reality, and models such as the one presented in this paper are useful in managing new infections as human behaviour is essential in influencing the course of an emerging infection.