Assessing the Impact of Public Compliance on the Use of Non-pharmaceutical Intervention with Cost-Effectiveness Analysis on the Transmission Dynamics of COVID-19: Insight from Mathematical Modeling

Abstract

COVID-19, caused by a lately discovered coronavirus in Wuhan, China, is a respiratory ailment. Since the manifestation in 2019, there are already over 56 million cases of COVID-19 worldwide, with over 1.3 million mortality cases in vulnerable countries with inadequate healthcare infrastructures. The interaction between susceptible humans, infected individuals, and the coronavirus shed in the environment are investigated. A proper insight on the interaction between humans and coronavirus population will provide a comprehensive interplay between susceptible and infected humans, and the virus. The main goal of this chapter is to conduct a rigorous mathematical and statistical analysis of compartmentalized COVID-19 model that takes into consideration public compliance to COVID-19 rules and sanitation, and the interaction of human and the virus population in the environment, for the purpose of obtaining the effective reproduction number $R_{0}$, at the coronavirus-free equilibrium, and in the presence of compliance. A quality analysis is performed to determine the stability of the system equilibrium point. To curtail the spread of the virus, a non-pharmaceutical intervention, such as public adherence and sanitation, is adopted, taking into account the cost effectiveness of the intervention. Our results show that COVID-19 is locally and asymptomatically stable if $R_{0}<1$ and globally stable if $R_{0} \le 1$. Further results show that the control involving both adherence and compliance to COVID-19 rules and sanitation prove to be the most cost-effective strategies.

7 Appendix

1.1 7.1 Positivity of Solutions

In this section, we established that all the dynamical variables of the model system (2.1) are non-negative for all time t. Note that the positivity of S(t) and C(t) depends on the positivity of I(t), we shall start by proving the positivity of I(t). Thus, from the second equation of the model system (2.1), we have

$$\begin{aligned} I^{'} = \frac{d I}{d t} = [\beta _{h} S + \beta _{c} C S - (\gamma + \sigma + \mu )] I + \beta _{c} CS \end{aligned}$$

(7.1)

and $\frac{d I}{d t} - f(S) I(t) = \beta _{c} CS$ where $f(S) = \beta _{h} S - (\gamma + \sigma + \mu )$. This can be suppressed as

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d I}{d t} e^{- \int _{0}^{t} f(S) dS} - f(S) e^{- \int _{0}^{t} f(S) dS} I(t) = e^{- \int _{0}^{t} f(S) dS} \beta _{c} C S \\ \frac{d}{dt} (I(t) e^{- \int _{0}^{t} f(S) dS} ) = e^{- \int _{0}^{t} f(S) dS} \beta _{c} C S \\ I(t) e^{- \int _{0}^{t} f(S) dS} = \int _{0}^{t} e^{- \int _{0}^{t} f(S) dS} \beta _{c} C S dS + I_{0}\\ I(t) = [I_{0} + \int _{0}^{t} e^{- \int _{0}^{t} f(S) dS} \beta _{c} C S dS ] e^{- \int _{0}^{t} f(S) dS} \\ I(t) = I_{0} e^{- \int _{0}^{t} f(S) dS} + e^{- \int _{0}^{t} f(S) dS} \int _{0}^{t} e^{- \int _{0}^{t} f(S) dS} \beta _{c} C S dS \ge 0 \end{array}\right. } {\tiny }\end{aligned}$$

(7.2)

for all $t \ge 0$. Other, from the first equation of system (2.1), we have

$$\begin{aligned} \frac{dS}{ dt} = \Lambda + \gamma I - \left[ \beta _{h}(\alpha ) I(t) + \beta _{c} C(t) +\mu \right] S(t) \end{aligned}$$

(7.3)

(34) can be expressed as

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d S}{d t} e^{ \int _{0}^{t} f_{2}(S) dS} + f_{2}(S) e^{ \int _{0}^{t} f_{2}(S) dS} = (\Lambda + \gamma I(t))e^{\int _{0}^{t} f_{2}(S) dS} \\ \frac{d }{d t} ( S(t)e^{ \int _{0}^{t} f_{2}(S) dS}) =(\Lambda + \gamma I(t))e^{\int _{0}^{t} f_{2}(S) dS} \\ \end{array}\right. } {\tiny }\end{aligned}$$

(7.4)

Thus, we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} S(t)e^{ \int _{0}^{t} f_{2}(S) dS}= \int _{0}^{t}(\Lambda + \gamma I(t))e^{\int _{0}^{t} f_{2}(S) dS} + S_{0}\\ S(t)= e^{ -\int _{0}^{t} f_{2}(S) dS} \int _{0}^{t}(\Lambda + \gamma I(t))e^{\int _{0}^{t} f_{2}(S) dS} + S_{0} e^{ -\int _{0}^{t} f_{2}(S) dS}\\ \end{array}\right. } {\tiny }\end{aligned}$$

(7.5)

Hence, $S(t) > 0$ for all $t > 0$, $f_{2} = \beta _{h} I(t) + \beta _{c} C(t) + \mu$. Also, from the third equation of the model system (2.1), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d C}{dt} + (\phi _{3} -\phi _{2} ) C(t) = \phi _{1} I(t)\\ \frac{d}{dt} (C(t) e^{\int _{0}^{t} (\phi _{3} -\phi _{2}) du}) = \phi _{1} I(t) e^{\int _{0}^{t} (\phi _{3} -\phi _{2}) du}\\ C(t) e^{\int _{0}^{t} (\phi _{3} -\phi _{2} du} = \int _{0}^{t} \phi _{1} I(t) e^{\int _{0}^{u} (\phi _{3} -\phi _{2}) dp} du + C_{0} \\ C(t) = e^{\int _{0}^{t} (\phi _{3} -\phi _{2} du }\int _{0}^{t} \phi _{1} I(t) e^{\int _{0}^{u} (\phi _{3} -\phi _{2}) dp} du + C_{0} e^{- \int _{0}^{t} (\phi _{3} -\phi _{2} du }\\ \end{array}\right. } {\tiny }\end{aligned}$$

(7.6)

for all $t>0$. Thus, the solutions S(t), I(t) and C(t) of model system (2.1) with the initial conditions $S(0) = S_{0} > 0$, $I(0) =I_{0} \ge 0$ and $C(0) = C_{0} \ge 0$ are non-negative for all $t>0$. Hence, the proof.

1.2 7.2 Boundedness of Solution

The total human population of the model denoted by N(t) at time t is given as

$$\begin{aligned} N(t) = S(t) + I(t) \end{aligned}$$

(7.7)

so that

$$\begin{aligned} \frac{dN(t)}{dt} = \Lambda - \mu N(t) - \sigma I(t) \end{aligned}$$

(7.8)

since CI(t) is non-negative, then (A2.2) reduces to

$$\begin{aligned} \frac{dN(t)}{dt} \le \Lambda - \mu N(t) \end{aligned}$$

(7.9)

The above equation can be written as

$$\begin{aligned} \frac{d}{dt} \le \Lambda - \mu N(t) \end{aligned}$$

(7.10)

Integrating the above equation, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} N(t) e^{\mu t} \le \int \Lambda e^{\mu t} dt + N_{0} \\ N(t) e^{\mu t} \le \frac{ \Lambda }{\mu } e^{\mu t} e^{ - \mu t} + N_{0} e^{-\mu t} \\ N(t) \le \frac{ \Lambda }{\mu } e + N_{0} e^{-\mu t} \\ \end{array}\right. } \end{aligned}$$

(7.11)

By applying theory of differential inequality (17), we have

$$\begin{aligned} \limsup \limits _{t \rightarrow 0 } N(t) \le \frac{\Lambda }{\mu } \end{aligned}$$

(7.12)

thus $0 \le N(t) \le \frac{\Lambda }{ \mu }$ as $t \rightarrow \infty$. Now, $I(t) = N(t) - S(t) \ge 0$ $\implies 0 \le I(t) \le N(t) \le \frac{\Lambda }{\mu }$ as $t \rightarrow \infty$.

From the third equation of the model system (2.1), and observing the fact that $I(t) \le \frac{\Lambda }{ \mu }$ for $t \rightarrow \infty$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d C(t)}{dt} + (\phi _{3} - \phi _{2}) C(t) \le \frac{\phi _{1} \Lambda }{\mu } \\ \frac{d }{dt}(C(t) e^{(\phi _{3} - \phi _{2}) t}) \le \frac{\phi _{1} \Lambda }{\mu } e^{(\phi _{3} - \phi _{2})t}) \\ C(t) e^{(\phi _{3} - \phi _{2}) t} \le \frac{\phi _{1} \Lambda }{\mu (\phi _{3} - \phi _{2})} e^{(\phi _{3} - \phi _{2})t}) + C_{1} \\ C(t) \le \frac{\phi _{1} \Lambda }{\mu (\phi _{3} - \phi _{2})} + C_{1} e^{-(\phi _{3} - \phi _{2}) t} \end{array}\right. } \end{aligned}$$

(7.13)

using theory of differential inequality, we obtain $\limsup \limits _{t \rightarrow \infty } C(t) \le \frac{\phi _{1} \Lambda }{\mu (\phi _{3} - \phi _{2})}$ and $0 \le C(t) \le \frac{\phi _{1} \Lambda }{ \mu (\phi _{3} - \phi _{2})}$

1.3 7.3 Basic Reproduction Number ($R_{\alpha })$

The next equilibrium matrix method is used to obtain the basic reproduction ($R_{\alpha }$) of model system (2.1). This method is well documented in [2, 26] and has been implemented by several researchers [9, 10]. First, we obtain the transmission matrix of new infections F(x) and the transition matrix V(x) as follows: Consider the model system (2.1) expressed as

$$\begin{aligned} \frac{dx}{dt} = F(x) - V(x) \end{aligned}$$

(7.14)

where $x = (S, I, C)^{T}$

$$\begin{aligned} F(x)= \begin{pmatrix} \begin{array}{cccc} \beta _{h}(\alpha ) I S + \beta _{c}(\alpha ) C S \\ 0\\ \end{array} \end{pmatrix} \end{aligned}$$

(7.15)

and

$$\begin{aligned} V(x)= \begin{pmatrix} \begin{array}{cccc} (\gamma + \sigma + \mu ) I\\ (\phi _{3} - \phi _{2}) C - \phi _{1} I\\ \end{array} \end{pmatrix} \end{aligned}$$

(7.16)

and the Jacobian matrix of F(x) and V(x) calculated at Covid-19 disease free equilibrium $P_{0} = (\frac{\Lambda }{\mu }, 0, 0)$ are

$$\begin{aligned} J(F(x)P_{0}) = \begin{pmatrix} \begin{array}{cccc} \frac{\beta _{h}(\alpha ) \Lambda }{\mu }&{} \frac{\beta _{c}(\alpha ) \Lambda }{\mu }\\ 0&{} 0\\ \end{array} \end{pmatrix} \end{aligned}$$

(7.17)

and

$$\begin{aligned} J(V(x)P_{0}) = \begin{pmatrix} \begin{array}{cccc} (\gamma + \sigma + \mu )&{} 0\\ -\phi _{1}&{} (\phi _{3} - \phi _{2})\\ \end{array} \end{pmatrix} \end{aligned}$$

(7.18)

The next generation matrix $K = J(F(x), P_{0}) (J(V(x)P_{0}) )^{-1}$ is given by

$$\begin{aligned} K = \begin{pmatrix} \begin{array}{cccc} \frac{\beta _{h} (\alpha ) \Lambda }{\mu }&{} \frac{\beta _{c}(\alpha ) \Lambda }{\mu }\\ 0&{} 0\\ \end{array} \end{pmatrix} \begin{pmatrix} \begin{array}{cccc} \frac{1}{(\gamma + \sigma + \mu )}&{} 0\\ \frac{\phi _{1}}{(\gamma + \sigma + \mu ) (\phi _{3} - \phi _{2})}&{} \frac{1}{(\phi _{3} - \phi _{2})}\\ \end{array} \end{pmatrix} \end{aligned}$$

(7.19)

$$\begin{aligned} K = \begin{pmatrix} \begin{array}{cccc} \frac{\beta _{h} (\alpha ) \Lambda }{\mu (\gamma + \sigma + \mu )} + \frac{\beta _{c}(\alpha ) \Lambda }{\mu (\gamma + \sigma + \mu )(\phi _{3} - \phi _{2})}&{} \frac{\beta _{c} (\alpha ) \Lambda }{\mu (\phi _{3} - \phi _{2})}\\ 0&{} 0\\ \end{array} \end{pmatrix} \end{aligned}$$

(7.20)

1.4 7.4 Model Parameters Estimation

The model parameters were estimated from real life data using statistical techniques. Some of the parameters were assumed based on real life situations and data available for the countries selected for consideration, while some others were extracted from demographic data of the countries considered.

All the parameters are positive numbers, that is, $\Omega > 0$, where $\Omega$ is the parameter space. The natural death rate for human, $\mu$ is directly obtained from demographic data, and it is estimated by

$$\begin{aligned} \hat{\mu } = \frac{1}{\mu _{0}} \end{aligned}$$

(7.21)

where $\mu _{0}$ is the average life expectancy obtained from demographic data.

Without loss of generality, we can denote S(t), I(t) and C(t) as $S_{t}$, $I_{t}$ and $C_{t}$ respectively. The recovery rate of infected individuals, $\gamma$ is estimated by

$$\begin{aligned} \hat{\gamma } = \frac{\sum _{t=1}^{n} R_{t}}{\sum _{t=1}^{n} I_{t} } \end{aligned}$$

(7.22)

where t, $t = 1, 2, . . ., n$ is time measured in days and n is the number of days covered. The variables $R_{t}$ and $I_{t}$ are the number of recovered individuals at time t and number of infected individuals at time t respectively.

COVID-19 induced death rate of infected individuals, $\sigma _{I}$ is estimated by

$$\begin{aligned} \hat{\sigma } = \frac{\sum _{t=1}^{n} D_{t}}{\sum _{t=1}^{n} I_{t} } \end{aligned}$$

(7.23)

The variable $D_{t}$ is the number of COVID-19 induced deaths at time t.

The parameter accounting for compliance, $\alpha$ is assumed to lie be between 0 and 1 and it is assume that the more the compliance level, the more the recovery rate and the less the mortality rate. Thus, $\alpha$ is estimated by

$$\begin{aligned} \hat{\alpha } = \frac{\sum _{t=1}^{n} R_{t}-\sum _{t=1}^{n} D_{t}}{\sum _{t=1}^{n} I_{t}} \end{aligned}$$

(7.24)

The growth rate of virus due to increase in infected individuals of COVID-19, $\phi _{1}$ is estimated by

$$\begin{aligned} \hat{\phi }_{1} = \frac{\sum _{t=1}^{n} I_{t} - sum_{t=1}^{n} R_{t}-\sum _{t=1}^{n} D_{t}}{\sum _{t=1}^{n} I_{t}} \end{aligned}$$

(7.25)

This is because the growth rate of virus is assumed to be the additive inverse of compliance level. If the compliance rate of individuals increases, the growth rate of virus reduces and vice versa.

Natural death rate of virus, $\phi _{3}$ is estimated by adding a fraction constant to recovery rate. The natural death rate of virus is partly constant and partly varies directly as recovery rate. Thus, $\phi _{3}$ is estimated by

$$\begin{aligned} \hat{\phi }_{3} = \kappa + \frac{\sum _{t=1}^{n} R_{t}}{\sum _{t=1}^{n} I_{t}} \end{aligned}$$

(7.26)

where $\kappa$ is the fraction constant selected for all countries, so that $\hat{\phi }_{3} > 0$. Self growth rate of the virus, $\phi _{2}$ is estimated by subtracting a fraction of $\phi _{1}$ from $\phi _{3}$ and it is given by

$$\begin{aligned} \hat{\phi }_{2} = \hat{\phi }_{3} - \frac{1}{q}\hat{\phi }_{1}, \hat{\phi }_{3}>\hat{\phi }_{2} \end{aligned}$$

(7.27)

where $q>0$ is a real number, which is constant for all countries considered. It is also chosen so that $\hat{\phi }_{2}>0$.

The density of the virus population at time t is $C_{t}$ and it is estimated by

$$\begin{aligned} \hat{C}_{t} = \frac{\phi _{1}I_{t}}{\hat{\phi }_{3} - \hat{\phi }_{2}}, \hat{\phi }_{3}>\hat{\phi }_{2}. \end{aligned}$$

(7.28)

But it is restricted by an upper bound given by

$$\begin{aligned} \hat{C}_{m} = \frac{\hat{\phi }_{1} \hat{\Lambda }}{\mu (\hat{\phi }_{3} - \hat{\phi }_{2})}, \hat{\phi }_{3}>\hat{\phi }_{2} \end{aligned}$$

(7.29)

where $\hat{\Lambda }$ is the recruitment rate gotten from demographic data of birth of each country considered, and may be restricted below by a lower bound $S_{t}+I_{t}$.

Minimum rate of transmission of the disease from virus to humans ($\eta _\mathrm{{min}}$) is estimated by 1/N where N is the population of each country considered. And it is also assumed to be directly proportional to minimum transmission rate from an infected person to a susceptible human ($\beta _\mathrm{{min}}$). So, $\eta _\mathrm{{min}}$ and $\beta _\mathrm{{min}}$ can be varied but must always be less than $\eta _\mathrm{{max}}$ and $\beta _\mathrm{{max}}$ respectively. Note that $\eta _\mathrm{{max}}$ and $\beta _\mathrm{{max}}$ are fixed as a supremum. This implies that $\eta _\mathrm{{max}}=\eta _\mathrm{{min}}*w$ and $\beta _\mathrm{{max}}=\beta _\mathrm{{min}}*w$, where w is a positive constant and a multiple of 10.