# Mathematical Modelling of the Spatial Epidemiology of COVID-19 with Different Diffusion Coefficients

Authors: Benedict Barnes; Ishmael Takyi; Bright Emmanuel Owusu; Francis Ohene Boateng; Augustine Saahene; Emmanuel Saarah Baidoo; Jennifer Aduko Adombire
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

This paper addresses the discrepancy between model findings and field data obtained and how it is minimized using the binning smoothing techniques: means, medians, and boundaries. Employing both the quantitative and the qualitative methods to examine the complex pattern involved in COVID-19 transmission dynamics reveals model variation and provides a boundary signature for the potential of the disease’s future spread across the country. To better understand the main underlying factor responsible for the epidemiology of COVID-19 infection in Ghana, the continuous inflow of foreigners, both with and without the disease, was incorporated into the classical Susceptible-Exposed-Quarantined-RecoveredSEIQR model, which revealed the spread of the COVID-19 by these foreigners. Also, the diffusion model provided therein gives a threshold condition for the spatial spread of the COVID-19 infection in Ghana. Following the introduction of a new method for the construction of the Lyapunov function for global stability of the nonlinear system of ODEs was observed, overcoming the problem of guessing for the Lyapunov function.


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## Body

## 1. Introduction

In Wuhan, China, in December 2019, a new type of coronavirus which is a member of severe acute respiratory syndrome (SARS) has been identified. The pandemic disease has since spread to other countries, including Ghana. Realizing the COVID-19 pandemic is not only necessary but also imperative given the increasing death toll, labour force decrease, productivity decrease, declining trend in the economy of the country, etc. There have been 6,418,381 deaths and 581,305,772 reported cases of the disease since it originally spread throughout the world. In order of size, the USA reported 93,054,184 cases and 1,055,020 deaths, India reported 44,000,138 cases and 526,312 deaths, and Brazil reported 33,795,192 cases and 678,375 deaths. Ghana, which is rated 123rd in the statistics table, is hardly an exception with 168,007 cases, 1457 deaths [1].It is impossible to look down upon the current state of the economies of the world and those under the productivity of their labour forces as many nations, particularly those in sub-Saharan Africa that are developing, look for long-term solutions to the spread of the COVID-19 pandemic. There are continuous interventions and control efforts everywhere in the world, including the provision of the Johnson and Johnson vaccine and the AstraZeneca vaccine to population subjects. In Ghana, the government has implemented a number of measures to prevent the spread of the COVID-19 infection from one person to another, including lockdown declarations, school closures, wearing of face mask, social withdrawal, limiting attendance at social gatherings, frequent hand washing, and the use of hand sanitizers. With all these measures put in place by the government, people are not adhering to the COVID-19 protocols as they still go by with their normal activities without a sense of fear of the pandemic of COVID-19. People typically attend weddings with their close friends and family in jest, hoping to catch a glimpse of the celebration. Funerals are events that people simply cannot do away with. It is customary for family members and friends to say their final farewell to the cherished (deceased) individual without considering their proximity to one another or, in the worst-case scenario, without wearing a face mask [2].Mathematical models are used to predict the extent of COVID-19 infection outbreaks, the rate at which the disease spreads globally, and the effectiveness of control measures in limiting the spatial spread of the disease. It cannot be overemphasized how important it is to employ mathematical models to help government officials and policymakers make informed decisions about how changes in the geographical mixing of individuals have affected the COVID-19 transmission dynamics in the country. A mathematical model provides a framework for analyzing causes, connections and concepts. A mathematical model is used to evaluate quantitative assertions. As a result, the effectiveness of the measures for preventing the spatial spread of COVID-19 in the country depends heavily on how well the model parameters capture the many characteristics of the disease. Even the trends of COVID-19 cases could be appreciated with a minimal error by the use of reliable or sufficient parameters in the model. Researchers from all across the world have investigated and evaluated the outcomes of various formulations and assumptions. The results of examining various COVID-19 infection formulations offer policymakers insights and are undoubtedly helpful in selecting models for the epidemiology of COVID-19 infection. Without taking into account the major spreaders and the influx of foreigners into the country, the authors in [3–7] modeled the epidemiology of COVID-19 in their respective countries. The COVID-19 pandemic in a country other than China, which involves the important factor responsible for the spatial spread of disease, requires a robust use of mathematics in order to explain and predict the major underlying mechanism and forecast trends. Since they are unable to identify the cause of the epidemiology and calculate the number of people who acquired the COVID-19 infections during the disease outbreak, their conclusions cast dust in the eyes of the general public.The key to any mathematical model is to formulate it as simple as possible but being adequate for the subject under consideration. Identifying the primary underlying factor, the influx of immigrants into the country, the suitable collection of data, and a mathematical model whose analyses result in the solution constitute the most challenging issues in modeling the epidemiology of COVID-19 infections. The spatial spread of COVID-19 has attracted interest in this period as it involves the main underlying factor and threshold condition for the onset of the disease. The authors of [8] have used the reaction-diffusion equation to describe the COVID-19 epidemiology in Spain and Greece. They included diffusion terms for each class in the susceptible-exposed-asymptomatic-infected-hospitalization-recovered model to account for the discrepancy between predictions and observed data. It is interesting to note that the diffusion model for the spatial spread of COVID-19 developed in [9] is very similar to the one developed in [8], with the exception that diffusion terms were added to the susceptible, asymptomatic, infected, and recovered humans instead of the six subgroups of the population size. Similar to [8, 9], the study in [10] applied the susceptible-exposed-infected-recovered-deceased model to describe the spatial spread of COVID-19 in Italy. The authors in [11] considered the diffused susceptible, asymptomatic, and symptomatic infected people. Unfortunately, their model did not take into account the isolated COVID-19 infected individuals or the patients who had received treatment or had recovered from their illnesses. All of these research works have applied the diffusion concept to a group of people from various countries who have recovered from the disease. An individual who has recovered from COVID-19 is frequently not driven to travel to another country because he or she is afraid of contracting the illness again due to the control measures in place there. In terms of the spatial spread of COVID-19, this assumption is not acceptable. Nevertheless, their diffusion models did not account for the quarantined class, a crucial subgroup of the population size for describing the pandemic of COVID-19 infections. This is a major shortcoming that needs immediate attention from the scientific community and has to be addressed immediately.Nevertheless, research works from many settings have developed the Lyapunov function, which is used to assess the overall stability of the system of ODEs. A significant flaw in the global stability of nonlinear systems of ODEs, such as the Lyapunov method, is that there is no systematic method for determining the Lyapunov function to aid in establishing the global stability of the nonlinear system of ODEs. As a result, the Lyapunov method for determining the global stability of a nonlinear system of ODEs is purely. A significant flaw in the global stability of nonlinear systems of ODEs, such as the Lyapunov method, is that there is no systematic method for determining the Lyapunov function to aid in establishing the global stability of the nonlinear system of ODEs. As a result, the Lyapunov method for determining the global stability of a nonlinear system of ODEs is purely a guess of function properties that meet the requirements. For examples, the study [12] observed the following:(1)Lu,v=au−u∗−u∗lnuu∗+v−v∗−v∗lnvv∗,where a∈ℝ+ is a parameter and u,v=u∗,v∗ is a fixed point of the system of ordinary differential equations (ODEs), as a Lyapunov function for the temporal-spatial variations in the interaction of the predator-prey model, and the study in [13] observed(2)Ls,e,i=s−s∗−s∗lnss∗+e−e∗−e∗lnee∗+i−i∗−i∗lnii∗,where s,e,i=s∗,e∗,i∗ is a fixed point of the system of ODEs, as the Lyapunov function for determining the global stability for the susceptible-exposed-infected model describing the epidemiology of COVID-19 infection.According to the theory that the continuous inflow of immigrants into the country constitutes the spatial effects, the diffusion terms are only incorporated in the susceptible, exposed, and infected subgroups of the susceptible-exposed-infected-quarantined-recoveredSEIQR model in this paper. Thus, the major underlying cause of the spatial spread of the COVID-19 infection is the continuous inflow of foreigners into the country. The reliability and robustness of this mathematical model are established through field data assessments of its trend. The main factors that would support robustness when the discrepancy between the model values and the data get reduced over time are the model parameters that have been evaluated. In fact, the validity of the mathematical model was assessed using monthly data on COVID-19 infections in Ghana. The behavior of the immediate nonstationary points around the fixed point of the nonlinear system of ODEs is given by the local stability of the fixed point, but it does not provide complete information on all nonstationary points in the domain of the nonlinear system of ODEs. For establishing the stability of a nonlinear system of ODEs, the Lyapunov criterion provides information on all nonstationary points in the nonlinear system. This method of determining global stability requires the construction of the Lyapunov function. A systematic method for constructing the Lyapunov function for determining the global stability of a nonlinear system of ODEs is presented in this paper.Section1 of this paper comprises an introduction, Section 2 contains analysis and results, and Section 3 contains the key conclusions from the analyses.


## 2. Main Results

In this section, we provide the mathematical model for describing the epidemiology of COVID-19 in Ghana.


### 2.1. Preliminary Results

Theorem 1 (Lyapunov stability).
LetX∗ be an equilibrium point for X′=FX. Let L:O⟶ℝ be a differentiable function defined on an open set O containing X∗. Suppose further that(i)
LX∗=0 and LX>0 if X≠X∗(ii)
L˙≤0 in O−X∗Then,X∗ is stable. Furthermore, if L also satisfies(iii)
L˙<0 in O−X∗, then X∗ is asymptotically stable.A functionL satisfying a and b is called a Lyapunov function for X∗. If c also holds, we call L a strict Lyapunov function [14].


### 2.2. Development of the Mathematical Model and Its Analyses

The population of Ghana,N, is split into five distinct classes based on their epidemiological status: the susceptible class St), the exposed class (Et), the infected class It, the quarantined class (Qt), and the recovered class (Rt). Due to the continuous inflow of immigrants into the country, including those with and without COVID-19 disease, the susceptible, exposed, and infected classes have changed over time and space. The classifications of quarantined and recovered, however, change over time since they include people who have either been isolated or who have been treated for their sickness after contracting the COVID-19 infection by the Ghana Health Service GHS. Partial differential equations PDEs are made up of the susceptible, exposed, and infected classes, whereas ordinary differential equations ODEs are made up of the quarantined and recovered classes.Assumptions of the mathematical model are as follows:(1)
Newborn babies are recruited into the susceptible class.(2)
New COVID-19 infection arises as the susceptible comes in contact with the COVID-19 patient (infective or quarantined).(3)
Renewal of susceptible people through recovery from transient immunity is not taken into consideration.(4)
The foreigner (invader) is categorized as either susceptible, exposed, or infective. Only these groups of people have different diffusion coefficients in one dimension because the COVID-19 infection was brought into the country through their immigration.Based on Figure1, the following equations are obtained to describe the epidemiology of COVID-19 infection in the country:(3)∂S∂t=αN−μS−βSI+QN+D1∂2S∂X2,∂E∂t=βSI+QN−μ+κ+θ1+δ1E+D2∂2E∂X2,∂I∂t=κE−μ+γ1+θ2+δ2I+D3∂2I∂X2,dQdt=θ1E+θ2I−δ3+μ+γ2Q,dRdt=δ1E+δ2I+δ3Q−μR.Figure 1
Compartmental model for COVID-19 transmission dynamics.Thus, the total population size of Ghana is represented by following equation:(4)Nt=St+Et+It+Qt+Rt.Together withSX,0=S0, EX,0=E0, IX,0=I0, Q0=Q0, and R0=0, where α is the birth rate. That is, the rate at which newborn babies are recruited into the susceptible class. The transmission rate, β, is the rate at which a susceptible person contracts the COVID-19 virus, κ is the rate at which exposed people become infectious while the average incubation period is 1/κ. The disease-induced death rate, γ1, is the rate at which an infected person dies from the COVID-19 infection, while the disease-induced death rate, γ2, is the rate at which a quarantined person dies from the disease. θ1 is the rate at which exposed people are quarantined, whereas θ2 is the rate at which an infectious person is quarantined. δ1 is the rate at which exposed people recover from the virus, δ2 is the rate at which infectious people recover from the disease, δ3 is the rate at which quarantined people recover from the disease, and μ is the natural death rate. D1 represents the constant diffusion coefficient of susceptible travelers into the country, D2 is the constant diffusion coefficient of exposed travelers into the country, and D3 is the constant diffusion coefficient of infectious travelers into the country.Settingm1=μ+κ+θ1+δ1, m2=μ+γ1+θ2+δ2 and m3=δ3+μ+γ2, the system of (3) becomes(5)∂S∂t=αN+D1∂2S∂X2−μS−βSI+QN,∂E∂t=βSI+QN+D2∂2E∂X2−m1E,∂I∂t=κE+D3∂2I∂X2−m2I,dQdt=θ1E+θ2I−m3Q,dRdt=δ1E+δ2I+δ3Q−μR,together with SX,0=S0, EX,0=E0, IX,0=I0, Q0=Q0, and R0=0.Nondimensionalizing of (5) is done by using the following system of equations:(6)s=SN,e=EN,i=IN,q=QN,r=RN,τ=αt,x=XαD11/2,a=μα,b=βα,d1=D2D1,n1=m1α,k=κα,d2=D3D1,n2=m2α,p1=θ1α,p2=θ2α,f1=δ1α,f2=δ2α,f3=δ3α,n3=m3α.Using the system of (6), we obtain the following system of equations:(7)∂s∂τ=1+∂2s∂x2−as−bsi+q,∂e∂τ=bsi+q+d1∂2e∂x2−n1e,∂i∂τ=κe+d2∂2i∂x2−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar,together with sx,0=s0, ex,0=e0, ix,0=i0, q0=q0, and r0=0.


#### 2.2.1. Boundedness of the Solution of the System of ODEs

The boundedness of the system of (7) is obtained by setting the diffusion coefficients of the susceptible, exposed, and infective subgroups to zero which yields(8)dsdτ=1−as−bsi+q,dedτ=bsi+q−n1e,didτ=κe−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar.Together withsx,0=s0, ex,0=e0, ix,0=i0, q0=q0, and r0=0.Summing the ordinary derivatives on the left-hand side of the system of equations yields(9)⇒dNdt=αN−μS+E++I+Q+R−γ1I−γ2QdNdt=α−μN−γ1I−γ2QdNdt≤α−μNNt≤Aeα−μt.SettingN0=N0,⇒Nt≤N0eα−μt.The asymptotic behavior of the scaled population size is(10)limt⟶∞Nt=limt⟶∞Noeα−μtlimt⟶∞Nt=No,∀μ>α,limt⟶∞Nt=No,∀μ>α.The result in inequality (7) implies that the solution set is invariant. Thus, the population size of Ghana approaches a fixed finite number.


#### 2.2.2. Existence and Uniqueness of the Solution of the System of Nondimensionalized PDEs

In this section, we show that the system of (7) has a unique solution in a Banach space. Consider the first three equations of the system of (7), which are given by(11)∂s∂τ=1+∂2s∂x2−as−bsi+q,∂e∂τ=bsi+q+d1∂2e∂x2−n1e,∂i∂τ=κe+d2∂2i∂x2−n2i.The existence and uniqueness of solutions of the system of nondimensionalized PDEs (11) with homogeneous Neumann boundary conditions(12)∂s∂τ=∂e∂τ=∂i∂τ=0,on∂Ω×0,+∞and initial conditions(13)sx,0=ϕ1x≥0,ex,0=ϕ2x≥0,ix,0=ϕ3x≥0,∀x∈Ω¯,where Ω is a bounded domain in ℝ3 with a smooth boundary ∂Ω.∂s/∂τ, ∂e/∂τ, and ∂i/∂τ, respectively, are the normal derivatives of sx,τ,ex,τ, and ix,τ on the boundary of the domain ∂Ω. Moreover, 0<s∗<1, 0<e∗<a/n1, and 0<i∗<b.We setLM=uo=ϕ1x,ϕ2x,ϕ3x∈L+:0≤uo≤M,∀x∈Ω¯ with M=1,a/n1,b.Setting a mapF=F1,F2,F3:LM⟶L, the system of (11) is written as(14)F1u0x=1−aϕ1x−bϕ1xϕ3x+qF2u0x=bϕ1xϕ3x+q−n1ϕ2xF3u0x=kϕ2x−n2ϕ3x,where q∈ℝ+ is a parameter. Thus, F satisfies the Lipschitz condition in the subspace of LM, and the system of (11) can be written in the Banach space L=CΩ¯, where Ω¯⊂ℝ3.(15)u′τ=Auτ+Fuτ,τ>0,u0=u0∈LM,where u.,τ=cols.,τ,e.,τ,i.,τ, Au:col△sx,.,d1△ex,.,d2△ix,., u0=colϕ1,ϕ2,ϕ3, and △ is the Laplace-type operator in one dimension. This result implies that the system of (11) has a solution in a Banach space.Theorem 2.
Setting the initial datauo=ϕ1x,ϕ2x,ϕ3x∈LM, there exists a unique nonnegative solution ux,τ of the system of equations (8) defined on 0,∞×Ω¯ with u0,.,ϕ=ϕ. Moreover, uτ,sx,τ,ex,τ,ix,τ∈LM for τ≥0 and usx,τ,ex,τ,ix,τ, ∀τ,s,e,i∈0,∞×Ω¯, is a classical solution of the system of equations (8).Proof 1.
In this proof, the Euclidean norm is used to achieve the results. Setting0≤h<min1/a+b,1/n1,1/n2 and for any sufficiently small h≥0, we obtain(16)uox+hFuox=1−ah−bhϕ3x+qϕ1x+h1−hn1ϕ2x+bhϕ1xϕ3x+q1−hn2ϕ3x+hkϕ2x⇒uox+hFuox≥1−ah−bhϕ1x1−hn1ϕ2x1−hn2ϕ3x≥000.
Also, we can see that(17)u0x+hFu0x≤1−ah−bhϕ1x+h1−hn1ϕ2x+bhϕ1x1−hn2ϕ3x+hkϕ2x,u0x+hFu0x≤1−a+bh+han1−a−bhb1−n2h+akhn1,limh⟶0uox+hFuox=1an1b,⇒uox+hFuox≤1an1b.
This implies thatuo is bounded above by M. System (8) admits a unique nonnegative solution uτ,x on 0,∞×Ω¯ with u0,.,ϕ=ϕ and uτ,.,ϕ∈LM for τ≥0.
We have(18)∂s∂τ−Δs=1−as,∂s∂ν=0,sx,0=ϕ1x≤ϕ1∞=maxx∈Ω¯ϕ1x.
By solvingds1/dτ=1−as1 and s10=ϕ1∞, using the method of integrating factors yields(19)s1τ=1a+ce−aτ.
Atsx,0=ϕ1x, it implies(20)c=ϕ1x−1a,⇒s1τ=ϕ1xeaτ+1a1−e−aτ.
sx,τ≤s1x,τ≤max1/a,ϕ1∞, for τ∈0,∞.(21)⇒sx,τ≤max1a,ϕ1∞,∀x,τ∈Ω¯×0,∞.
To establish theL∞ uniformly boundedness of ex,τ and ix,τ, we have(22)∂s∂ν=∂e∂ν=∂i∂ν=0.
Then,(23)⇒∂∂τs+e−Δs+d1e=1−as−bsi+q+bsi+q−ne,⇒ddτs+e≤1−as+e,s1τ+e1τ=ϕ1xe−aτ+ϕ2xe−aτ+1a1−e−aτ⇒∫Ωs+e+idx≤mesΩmax1a,ϕ1+ϕ2∞⇒supτ≥0∫Ωex,τdx≤k1≔mesΩmax1a,ϕ1+ϕ2∞,where k1 is a finite positive constant. Also,(24)∂∂τs+e−Δs+d2e=1−as+i−bsi+q−ν1+p2+f2i+ke,⇒ddτs+i≤1−as+i.Solving the equation d/dτs1+i1≤1−as1+i1 yields(25)s1τ+i1τ=ϕ1xe−aτ+ϕ3xe−aτ+1a1−e−aτ⇒∫Ωs+idx≤mesΩmax1a,ϕ1+ϕ3∞⇒supτ≥0∫Ωix,τdx≤k2≔=mesΩmax1a,ϕ1+ϕ3∞,where k2 is a finite positive constant.
We can be deduced that there exist positive constantsk1∗ and k2∗ that depend, respectively, on k1 and k2 and on ϕ1+ϕ2 and ϕ1+ϕ3 such that(26)supτ≥0e.τ∞≤k1∗,supτ≥0i.τ∞≤k2∗.
From the above, it has been proved thatsx,τ, ex,τ, and ix,τ are L∞ uniform bounded on Ω¯×0,τmax. Therefore, it follows from the standard theory of semilinear parabolic systems that τmax=+∞ is the maximal existence time for the solution of the system of equations (11). This completes g of the theorem.


#### 2.2.3. Analysis of the Diffusion-Driven Instability

In this subsection, we investigate the epidemiology of the COVID-19 infection globally. The COVID-19 epidemiology in Ghana is presented herein. As a result, there is a lot of rumor about the COVID-19 disease spreading in Ghana, which is reportedly attributed to the increase in international immigration. There is no scientific evidence that support the claim that the inflow of foreigners caused the COVID-19 outbreak. We take into account the two scenarios in order to support this assertion about the emergence of the COVID-19 infection in the country. The qualitative method of the system of equations (7) without the inflow of foreigners is the first situation that is taken into account in this research. In addition, the analysis of (7) is in light of the continuous inflow of foreign immigrants. Setting D1=0,D2=0, and D3=0 leads in the qualitative solution of the system of (7) without the influx of foreigners. As a result, equation system (4) is as follows:(27)dsdτ=1−as−bsi+q,dedτ=bsi+q−n1e,didτ=κe−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar.The eigenvalues of the linearized system of (27) are obtained as(28)J1a,0,0,0,0−λI=−a−λ0−ba−ba00−n1−λbaba00k−n2−λ000p1p2−n3−λ00f1f2f3−a−λ=0,λ1=−a,λ2=−a,λ3+n1+n2+n3λ2+n1n2+n2n3+n1n3−bka−p1baλ+n1n2n3−n3bka−p2k−n2p1=0,⇒λ3+C1λ2+C2λ+C3=0,where, C1=n1+n2+n3, C2=n1n2+n2n3+n1n3−bk/a−p1b/a, and C3=n1n2n3−n3bk/a−p2k−n2p1.We observe thatC1>0, C3>0, and C1C2−C3>0, so all the eigenvalues of characteristic polynomial (28) have negative real parts. All the eigenvalues of the system of linearized ODEs in (27) without the diffusion term are all negative for all positive parameter values, this system of (27) is a linearly asymptotically stable spatial-homogeneous solution at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0. This implies that there would not have been COVID-19 in Ghana without an influx of foreigners in the country.Again, a situation where there is an influx of foreigners in the country is considered as follows. The system of equation (7) can be written as(29)∂u∂τ=Ju+D△u.We also haveux,t=sx,t,ex,t,ix,t,qt,rt.Here,J is a Jacobian matrix of the linear terms, △ is the Laplacian operator in one dimension, and D is a matrix that contains the coefficients of diffusion, that is,(30)D=100000d100000d2000000000000,together with sx,0=so,ex,0=eo,ix,0=io,q0=0,r0=0,n^⋅∇s⋅,t=0,n^⋅∇e⋅,t=0, and n^⋅∇i⋅,t=0.The solution of the system of (29) is of the form(31)ux,t=∑kckukreλt.For the time-independent component,ukr satisfies the following equation with the same boundary conditions as the system of (29):(32)△u+k2u=0withu⋅∇u=0on∂ℝorℝ¯.⇒△+k2u=0,⇒u≠0or△=−k2.Substituting (31) and (32) into (29) yields(33)Js,e,i,q,r=−a−bi−bq−k20−bs−bs0bi+q−n1−d1k2bsbs00k−n2−d2k2000p1p2−n300f1f2f3−a.By the Hartman–Grobman theorem, the system of (7) at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0 is unstable. At a−1,e∗,i∗,q∗,r∗=a−1,0,0,0,0,(34)Js∗,e∗,i∗,q∗,r∗−λI=−a−k2−λ0−ba−ba00−n1−d1k2−λbaba00k−n2−d2k2−λ000p1p2−n3−λ00f1f2f3−a−λ=0⇒λ1=−a+k2λ2=−a,λ3+C1λ2+C2λ+C3=0,where,(35)C1=n1+n2+n3+d1+d2k2C2=n3n1+n2+n3d1+d2k2+n1+d1k2n2+d2k2+bk−11a,C3=n3n1+d1k2n2+d2k2−bkn3+p2+bp1n2+d2k21a.Using the Routh–Hurwitz criterion for determining the sign of the roots of the characteristic polynomial in (35), the system of (7) at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0 is unstable if at least one of the following conditions is not met: C1>0, C3>0, and C1C2−C3>0. Since all the parameters are positive real numbers if bkn3+p2+bp1n2+d2k2⋅1/a≥n3n1+d1k2n2+d2k2, then the characteristic polynomial in (34) has at least one positive root. This implies that the system of (7) is unstable at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0. We obsvered that C3≥0 is not met which implies that there is at least one of the eigenvalues has a positive real part.The stability of the system of (7) revealed that without arrival of foreigners with the COVID-19 virus (SARS virus), there had not been the epidemiology of COVID-19 in the country. On the other hand, the arrival of foreigners, for example, COVID-19 patients from Norway and Turkey, into the country brought about the epidemiology of COVID-19 infection in Ghana. This finding is confirmed by the instability of the system of (7) at the endemic equilibrium point. The instability of the system of (7) is attributed to the assertion that the pandemic COVID-19 infection in the country is brought by the influx of foreigners especially two people: one from Norway and the other from Turkey.


### 2.3. Global Stability of the System of ODEs

In this subsection, we construct a Lyapunov function to assess the overall stability of the system of ODEs. A comprehensive review of related studies indicates that neither a systematic method nor a special function is utilized to assess the overall stability of system of ODEs in the Lyapunov sense. Due to this issue, one must make an informed guess for a function whose properties satisfy the requirements in the Lyapunov criterion for global stability of system of ODEs. In most cases, the guessing is not feasible since there are no guidelines in arriving at this function. In order to overcome this difficulty, we introduce a new function that can be used in determining the global stability of the system of ODEs in the Lyapunov sense. This function serves as a Lyapunov function for any system of ODEs and the result is given in Theorem3.Theorem 3 (Lyapunov function for the system of ODEs).
Suppose a system of ordinary differential equations,(36)dxidt=Fixi,∀xi∈Rn,Fxi>0,Fxi=F1xi,…,Fnxi,is an increasing function on Rn. Then,(37)Vxit=∑i=1n1xitxit−xi∗+xi∗lnxi∗xit,where x∗=x1∗,…,xn∗ is the fixed point of Fxi, is the Lyapunov function for constructing the stability of the systemof ODEs (19).Proof 2.
We can see thatVxit is a monotone increasing function on Rn for xit≠xi∗. At xi=x∗, we observe that(38)Vx∗=1x1tx1∗−x1∗+x1∗lnx1∗x1∗+⋯+1xntxn∗−xn∗+xn∗lnxn∗xn∗,Vx∗=1x1∗×0+⋯+1xn∗×0,Vx∗=∑i=0n1xi∗×0,Vx∗=0.
This implies thatVxi has a minimum value at xi=x∗. Therefore, x∗ is a fixed point of Vxit.
DifferentiatingVxit with respect to t, we have(39)dVdt=∂V∂x1.dx1dt+…+∂V∂xn,dVdt=−1x12x1−x1∗+x∗lnx∗x1+1x11−x∗x1×F1xi+…+−1xnxn−xn∗+xn∗lnxn∗xn+1xn1−xn∗xn×Fnxi,dVdt=−x1∗x1tlnx1∗x1tF1xi+…+−xn∗xntlnxn∗xntFnxi,dVdt=−∑i=1n−1xi∗xitlnxi∗xitFixidVdt<0.
Since the gradient ofVxit is decreasing on [0, ∞], it implies that Vxit=∑i=1n1/xitxit−xi∗+xi∗lnxi∗/xit is a Lyapunov function for the system of (3). This completes the proof.
This implies thatVxi has a minimum value at xi=x∗.


#### 2.3.1. Application of Theorem2 in Determining the Global Stability of the System of ODEs

In this subsection, the global stability of the system of equation is obtained by setting∂s/∂x=0, ∂e/∂x=0, and ∂i/∂x=0. By Theorem 2, the Lyapunov function for the system of equation (8) is given by(40)Vs,e,i,q,r=1ss−s∗+s∗lns∗s+1ee−e∗+e∗lne∗e+1ii−i∗+i∗lni∗i+1qq−q∗+q∗lnq∗q+1rr−r∗+r∗lnr∗r.We can see that(41)Vs,e,i,q,r>0,∀s,e,i,q,r>0.Again,(42)Vs,e,i,q,r=0,∀s=s∗,e=e∗,i=i∗,q=q∗,r=r∗.Lastly, we can see that(43)dVdτ=∂V∂s⋅∂s∂τ+…+∂V∂r⋅∂r∂τ,dVdτ=−1s2s−s∗+s∗lns∗s+1s1−s∗s×1−as−bsi+q+−1e2e−e∗+e∗lne∗e+1e1−e∗e×bsi+q−n1e+…,dVdτ=−s∗slns∗s1−as−bsi+q−e∗elne∗ebsi+q−n1e−i∗ilni∗ike−n2i−q∗qlnq∗qp1e+p2i−n3q−r∗rlnr∗rf1e+f2i+f3q−ar.Since1−as−bsi+q>0, bsi+q−n1e>0, ke−n2i>0, p1e+p2i−n3q>0, and f1e+f2i+f3q−ar>0,(44)⇒dVdτ<0.Since all conditions of the Lyapunov function are met, it implies that the system of (8) is globally asymptotically stable.


### 2.4. Effects of the COVID-19 Infection in the Country

In this subsection, the travelling wave solution of the influx of foreigners into the country is analysed. Thus, the analytic method via Homotopy Perturbation Method (HPM) is used to obtain the solutions for the proportions of susceptible, exposed, and infective people.Settingsx,τ=sz, ex,τ=ez, and ix,τ=iz where z=x−cτ, c is the constant wave speed of the foreigners into the country; the first equation in the system of equation (4) becomes(45)d2sdz2+cdsdz−as+1−bsi+q=0.qt=0, since the movement of a quarantined person is restricted so he or she cannot travel to another country.(46)d2sdz2+cdsdz−as+1−bsi=0.Similarly, we have(47)d2edz2+cd1dedz+bsi+qd1−n1d1e=0.Settingqt=0, we have(48)d2edz2+cd1dedz−n1d1e+bd1si=0,(49)andd2idz2+cd2didz+kd2e−n2d2i=0.For the travelling wave solution, that is, the influx of foreigners into the country, we consider the proportion of the susceptible, exposed, and infective people.Using the HPM, the series approximation solutions of the system of equations (46), (48), and (49) are of the following form:(50)1−plinear terms of the system of equations+plinear+nonlinear terms of the system of equations=0,⇒1−pd2sdz2+cdsdz−as+1+pd2sdz2+cdsdz−as+1−bsi=0,1−pd2edz2+cd1dedz−n1d1e+pd2edz2+cd1dedz−n1d1e+bd1si=0,1−pd2idz2+cd2didz−n2d2i+pd2idz2+cd2didz+kd2e−n2d2i=0.We use the HPM to find the solution of the linearized system of (7) as follows.Equatingp0=0, the following equation is obtained from equation (50).(51)d2sdz2+cdsdz−as=1,where the homogeneous part of (51) is given by(52)d2sdz2+cdsdz−as=0.Settingsz=eλz into (52) yields(53)⇒λ2+cλ−a=0,∀eλz≠0λ1,2=−c±c2+4a2,∀c2+4a>0.The complementary function of (51) is(54)sz=D1e1/2−c+c2+4az+D2e−1/2c+c2+4az,with s0=so and s1=s1.Setting the particular solutionspz=Az2+Bz+C, finding the first and second derivative and substituting the results into (51) yield the particular solution spz=−1/a. Thus, the general solution of the proportion of susceptible is given by(55)sz=D1e1/2−c+c2+4az+D2e−1/2c+c2+4az−1a.Similarly, the proportion for exposed people and infected people is given by(56)ez=D3e1/2−c/d1+c2−4n1d1/d12z+D4e−1/2c/d1+c2−4n1d1/d12z,iz=D5e1/2−c/d2+c2−4n2d2/d22z+D4e−1/2c/d1+c2−4n2d2/d22z,respectively.


### 2.5. Conversion of a System of Mixture of PDEs and ODEs into a System of ODEs

Now, all the PDEs are expressed in their respective ODEs in order to have solutions which are of the same state.Setting(57)u1z=sz,u2z=dsdz,u3z=ez,u4z=dedz,u5z=iz,u6z=didz,u7z=qz,u8z=rz,yields the following system of ODEs:(58)du1dz=u2,du2dz=−cu2+au1+bu1u5+bu1u7−1,du3dz=u3,du4dz=−cd1u4+n1d1u3−bd1u1u5−bd1u1u7,du5dz=u6,du6dz=−cd2u6−kd2u3+nd2u5,du7dz=p1u3+p2u5−n3u7,du8dz=f1u3+f2u5+f3u7−au8.The equilibrium point of the system of ODEs in (58) is obtained by setting the ordinary derivatives on the right hand side of (58) to zero and solving the resulting equations simultaneously which yields the disease-free equilibrium point as(59)u1∗,u2∗,u3∗,u4∗,u5∗,u6∗,u7∗,u8∗=1a,0,0,0,0,0,0,0.Linearizing the system of ODEs (30) yields the following results:(60)Ju1,u2,u3,u4,u5,u6,u7,u8=01000000a+bu5+u7−c00bu10bu1000100000−bu5+u7d10n1d1−cd1−bu1d10−bu1d100000010000−kd20n2d2−cd20000p10p20−n3000f10f20f3−a.Atu1∗,u2∗,u3∗,u4∗,u5∗,u6∗,u7∗,u8∗=1/a,0,0,0,0,0,0,0, the eigenvalues of the Jacobian matrix in (60) are obtained as(61)J1a,0,0,0,0,0,0,0−λI=−λ1000000a−c−λ00ba0ba0001−λ0000000n1d1−cd1−λ−bad10−bad100000−λ10000−kd20n2d2−cd2−λ0000p10p20−n3−λ000f10f20f3−a−λ=0λ1=−a<0,λ2=−c+c2+4a2>0,λ3=−c−c2+4a2<0,λ4=1>0,λ5=−cd1<0,λ6=−n3<0,λ7=−c/d2+c/d22+4n2/d22>0,λ8=−c/d2−c/d22+4n2/d22<0.Sinceλ2, λ4, and λ7 are all greater than zero, it implies that the COVID-19 infection at the disease-free equilibrium point, u1∗,u2∗,u3∗,u4∗,u5∗,u6∗,u7∗,u8∗=1/a,0,0,0,0,0,0,0, is unstable. Hence, a sufficiently small influx of foreigners into the country will result in the spread of COVID-19 in the country.


#### 2.5.1. Analysis of the Equilibria of the System of ODEs

In the system of (8), the basic reproductive number, Ro, is obtained from the spectral radius of FV−1. Thus,(62)Ro=ρFV−1,where(63)F=000000−bad1000000000,V−1=10000−d1c0000cn2−d2n200−10.Ro is obtained as(64)Ro=bcad1n2⇒Ro=αβD1cμμ+γ1+θ2+δ2D2.The basic reproductive number,R1, for the system of ODEs without diffusion is given by(65)R1=ρFV−1,where(66)F=0baba000000,V=n100−kn20−p1−p2n3,R1=αβθ1γ1+δ2+θ2+μ+κγ2+δ3+μ+θ2μγ2+δ3+μγ1+δ2+θ2+μδ1+θ1+κ+μ.We observe that the basic reproductive number,Ro, in the environment of inflow of foreigners into the country is more realistic than the basic reproductive number in diffusive-free environment since Ro involves the coefficient of diffusion of the susceptible, as well as the speed of the influx of the travellers into the country. The coefficient of diffusion of the susceptible and the wave speed is in direct relationship with Ro which are inflows for the infective so as to result in transmission of the COVID-19 infection.


### 2.6. Numerical Results

The data on the COVID-19 patients in Ghana were obtained from [18] and then segregated into exposed people and infected people by summing the counts of COVID-19 cases of the first fourteen days (incubation period) in the month as the number of exposed people in that month and the remaining counts of COVID-19 patients as the number of infective in the month. The results on these exposed and infective people are summarized in Tables 1 and 2. The normalization of the data points was done using(67)X∗=X−minXmaxX−minXnew maxX−new minX+new minX.Table 1
Exposed and infected people from March 2020 to December 2020.
MonthExposed peopleInfected peopleMarch1556April1438475May23353661June57773894July93768384August17667031September9711357October882547November15442068December133851719Table 2
Exposed and infected people from January 2021 to January 2022.
MonthExposed peopleInfected peopleJanuary100292210February81878826March23554205April1017962May655681June1215801July49052200August82048213September36084438October8191776November210633December11075991January (2022)37869743Table3 shows the estimates for the parameter in the model. Initial COVID-19 records in Ghana are shown in Table 4.Table 3
List of parameters for the model.
ParameterValuesα0.2000000000000β0.8000000000000κ0.0714285714300γ10.7500000000000γ20.0008889880194θ10.0140014195900θ20.7800000000000δ10.2937464950000δ20.0195830996700δ30.5874929900000μ0.0000195371580Table 4
Initial conditions for the five subgroups of the population size of Ghana.
CompartmentValueProportionSusceptible304177560.9999966467Exposed1003.287542469×10−6Infectious26.575084939×10−8Quarantined00Recovered00


#### 2.6.1. Numerical Results of the Nondimensionalized System of Mixture of PDEs and ODEs

This section discusses the effects of the movement of proportions of susceptible, exposed, and infectious individuals at a given time. Four cases are examined. Firstly, the same diffusion coefficient of the proportion of the exposed and the infective people are considered, i.e.,d1≠d2.The diffusion coefficientsd1 and d2 are provided in Table 5 and the initial conditions for the first three compartments of system (4) are as follows:(68)sx,0=1−0.6x2,0≤x<0.8,0.6160.8−xx,0.8≤x≤1.Table 5
Varied coefficients of diffusion for the proportion of susceptible, exposed, and infective.
CaseParameterValuekm2/hrd10.01d20.001Figure2 is plotted using the initial condition from (68). We can see that the proportion of the susceptible curve decreases as the scaled time and scaled distance start to rise from zero at the beginning. The proportion of the susceptibility curve decreases quickly as more susceptible are infected with the SARS virus (COVID-19 disease) since there are more susceptible people who are not adhering to the COVID-19 protocols by both public and private organizations in Ghana. The sharp rises in Figures 3 and 4 illustrate how the proportions of exposed and infected individuals rise when more susceptible individuals become infected with the SARS virus. The susceptible proportion continues to monotonically decrease until stabilizing at a specific point (adequately proportion of susceptible not able to contract SARS virus). The behavior of the susceptible curve shown in Figure 2 is concurrently supported by the decreasing patterns after the peaks in Figures 3 and 4.Figure 2
A 3-dimensional plot for the proportion of susceptible in space and time.Figure 3
A 3-dimensional plot for the proportion of exposed in space and time.Figure 4
A 3-dimensional plot for the proportion of infective in space and time.


#### 2.6.2. Numerical Results on the Trends of the COVID-19 Patients

The quantitative solution of the proportion of the infective subgroup system of ODEs in (40) as depicted in Figure 5. Figure 5 reveals a sinusoidal wave of the COVID-19 infection in the country with an amplitude of approximately 0.92.Figure 5
Proportion of the infective curve.The time in Figure6 begins at zero, which corresponds to March 2020, with no data on January and February 2020, since COVID-19 erupted in Ghana on March 14,2020. After the onset of COVID-19 pandemic in the country, the proportion of infective fluctuates roughly every eight months. As a result, the proportion infective curve increases until its peak and then decreases as the month progresses. Even the peak values are inconsistent, with July 2020 having a peak value of roughly 0.9058, February 2021 recording a peak value of roughly 0.8604, and January 2022 recording a peak value of almost 1.000. However, the proportion of the infective by the model (theoretical results) in Figure 5 also fluctuates at regular interval of almost every 8 months, but the peaked values were recorded on August 2020, April 2021, and October 2021, respectively. There are variations of the peak values of the proportions of the infective on the data as in Figure 6 and the peak values of the proportion of infective from the model (see Figure 5). These variations are attributed to the noise in the data, such as outliers (peak values), incomplete data on the COVID-19 patients on the grounds that not every person who contracted COVID-19 reported the issue at the nearby medical facility, or the health professionals were unable to contact trace all the COVID-19 patients. One of the major deterrents for suspected COVID-19 patients not reporting their illness to the nearest medical facility is the fear of panic brought on by the SARS virus (COVID-19) spreading to the susceptible population. However, errors in instrumentation devices used for testing COVID-19 cases across the nation as well as human error in data entry caused by the healthcare personnel cannot be ignored when it comes to the data on COVID-19 patients. Before it can be identified whether someone has COVID-19 (caused by the SARS virus), they frequently need to undergo a number of tests. In Figure 7, the quantitative result (curve) from the model is placed on the observed data curve for comparison. It was revealed that the variations of the model curve and observed data curve decrease as time passes and then fluctuate for the remaining of the period. This disparity is attributed to the unavoidable inaccuracies in the data. This necessitates data cleansing via bin smoothing techniques.Figure 6
Proportion of the infective from March 2020 to January 2022.Figure 7
Proportion of infective curves from normalized data and the model.Due to all the issues observed about data on COVID-19 patients in the country, inevitably the data cleaning was carried out to remove noise from the data. In Table6, the relative errors between the model values and data values are summarized. The relative errors are not different from the observations in Figures 8–10.Table 6
The relative error between the normalized data and the model values.
Time (month)Normalized data valueModel data valueRelative error00.00000.1524910.04810.09410.956340956320.37530.13280.646149746930.39930.33580.159028299540.86040.74970.128661087950.72140.91530.268782922160.13870.63293.563085797070.05550.26833.834234234080.21170.077640.633254605690.17590.14190.1932916430100.22630.32180.4220061865110.90580.50080.4471185696120.43120.67830.5730519481130.09810.59945.1100917430140.06930.39164.6507936510150.08160.23951.9350490200160.22530.22350.0079834753170.84280.32810.6107024205180.45510.45350.0035157108190.18170.54111.9779856910200.06430.51887.0684292380210.10110.41313.0860534120221.0000.32380.6762000000Figure 8
Proportion of the exposed people from March 2020 to January 2022.Figure 9
Proportion of the exposed curve from the model.Figure 10
Proportion of exposed curves from normalized data and the model.


#### 2.6.3. Smoothing of the Normalized Data

In smoothing the normalized data from the Ghana Health Service (GHS), discrepancies in the data were detected, such as human error in the data entry of the number of infectives, because daily cases of COVID-19 patients and accumulated monthly cases of COVID-19 patients were not segregated into the number of infective and the number of exposed people. The two subgroups, the number of infective and the number of exposed people, of the population size of Ghana cannot be lumped together as they play different roles in the epidemiology of COVID-19 pandemic in the country.To remove noise from the data obtained from GHS, the binning smoothing methods, smoothing by bin means, smoothing by bin medians, and smoothing by bin boundaries, were employed. The proportion of the infective curve superimposed on each normalized smoothed set of data is depicted in Figures 12, 14, and 16 These graphs show that the normalized smooth data by bin boundaries are consistent and that its values are robust against noise, especially when there are outliers. As a result, in Figure11, the variation between the infectious curve (model) and the smoothed data is smaller as the time (month) goes up to 4, then it starts to go up to 0.2687829221 (see relative error in Table 7). It is imperative to note that this peculiar trend could be caused by the unsorting of the normalized data:(1)
Comparison of the smoothened normalized data by bin boundaries, bin means, and bin medians and the normalized model for infective(2)
Comparison of the smoothened normalized data by bin boundaries, bin means, and bin medians and the normalized model for exposedFigure 11
The proportion of infective curves from smoothing by bin boundaries data and the model.Table 7
The relative error between the bin boundaries data and the model values.
Time (month)Bin boundaries data valueModel data valueRelative error00.00000.1510.00000.094120.37530.13280.646149746930.39930.33580.159028299540.72140.74970.039229276450.72140.91530.268782922160.13870.63293.563085797070.13870.26830.934390771480.21170.077640.633254605690.17590.14190.1932916430100.17590.32180.8294485503110.90580.50080.4471185696120.43120.67830.5730519481130.06930.59947.6493506490140.06930.39164.6507936510150.08160.23951.9350490200160.08160.22351.1738970588170.84280.32810.6107024205180.45510.45350.0035157108190.06430.54117.4152410580200.06430.51887.0684292380210.10110.41313.0860534120221.0000.32380.6762000000We can see from Figure12 that the smoothing by bin boundaries maintains most of the features of the original normalized data (see Figure 8). Thus, there are peaks in Figure 12 as in Figure 8, whereas in Figure 13 and Figure 14, they were flattened at the top. The curve of the model superimposed on the smoothened (normalized) data by the bin boundaries, bin means, and bin medians in Figures 12, 15, and 16, respectively, exhibited variations between the two curves in each figure. Comparatively, in Figure 11, the variation between the curve and smoothened (normalized) data by the bin boundaries was minimal and more consistent as time (month) increases than the smoothened by bin means and bin medians in Figures 15 and 16. The smoothing normalized data by the bin means using the frequency of three data points assumes the mean value for all the three data points like smoothing by the bin medians. Similar comparison study of the normalized equations (model) and the smoothened data, on the proportion of exposed people, by bin boundaries, bin means, and bin medians are observed in Figures 17–22. Undoubtedly, the variations between the smoothened data by bin boundaries and the model were least as compared with the variations between the smoothened data by the bin means and the model and the smoothened (normalized) data by bin medians and the model. The relative errors for the three smoothing techniques are summarized in Tables 7–9 for easy comparison.Figure 12
Proportion of the infective from March 2020 to January 2022.Figure 13
The proportion of the infective from March 2020 to January 2022.Figure 14
Proportion of the infective from March 2020 to January 2022.Figure 15
The proportion of infective curves from smoothing by bin means data and the model.Figure 16
Proportion of infective curves from smoothing by bin medians data and the model.Figure 17
Proportion of the exposed people from March 2020 to January 2022.Figure 18
Proportion of exposed curves from smoothing by bin boundaries data and the model.Figure 19
Proportion of the exposed people from March 2020 to January 2022.Figure 20
Proportion of exposed curves from smoothing by bin means data and the model.Figure 21
Proportion of the exposed people from March 2020 to January 2022.Figure 22
Proportion of expose curves from smoothing by bin medians data and the model.Table 8
The relative error between the bin means data and the model values.
Time (month)Bin means data valueModel data valueRelative error00.14110.150.0630758327410.14110.09410.3330970943020.14110.13280.0588235294130.66030.33580.4914432834040.66030.74970.1353930032050.66030.91530.3861880963060.13530.63293.6777531410070.13530.26830.9830007391080.13530.077640.4261640798090.43600.14190.67454128440100.43600.32180.26192660500110.43600.50080.14862385320120.19950.67832.40000000000130.19950.59942.00451127819140.19950.39160.96290726817150.38320.23950.23408141962160.38320.22350.41675365344170.38320.32810.14378914405180.23370.45350.94052203679190.23370.54111.31536157460200.23370.51881.21994009413210.55050.41310.24959128065220.55050.32380.41180744777Table 9
The relative error between the bin medians data and the model values.
Time (month)bin medians data valueModel data valueRelative error00.04810.150.2118503118510.04810.09410.9564309563020.04810.13281.7609147609130.86040.33580.6097164109040.86040.74970.1286610870050.86040.91530.0638075313060.05550.632910.403603603670.05550.26833.8342342340080.05550.077640.3989189189090.22630.14190.37295625700100.22630.32180.42202616865110.22630.50081.21429916040120.09810.67835.91437308800130.09810.59945.11009174300140.09810.39162.99184505600150.22530.23950.06302707501160.22530.22350.07983475300170.22530.32810.46528051480180.18170.45351.49587231700190.18170.54111.97798569100200.18170.51881.85360484310210.55050.41310.24959128065220.55050.32380.41180744777


## 2.1. Preliminary Results

Theorem 1 (Lyapunov stability).
LetX∗ be an equilibrium point for X′=FX. Let L:O⟶ℝ be a differentiable function defined on an open set O containing X∗. Suppose further that(i)
LX∗=0 and LX>0 if X≠X∗(ii)
L˙≤0 in O−X∗Then,X∗ is stable. Furthermore, if L also satisfies(iii)
L˙<0 in O−X∗, then X∗ is asymptotically stable.A functionL satisfying a and b is called a Lyapunov function for X∗. If c also holds, we call L a strict Lyapunov function [14].


## 2.2. Development of the Mathematical Model and Its Analyses

The population of Ghana,N, is split into five distinct classes based on their epidemiological status: the susceptible class St), the exposed class (Et), the infected class It, the quarantined class (Qt), and the recovered class (Rt). Due to the continuous inflow of immigrants into the country, including those with and without COVID-19 disease, the susceptible, exposed, and infected classes have changed over time and space. The classifications of quarantined and recovered, however, change over time since they include people who have either been isolated or who have been treated for their sickness after contracting the COVID-19 infection by the Ghana Health Service GHS. Partial differential equations PDEs are made up of the susceptible, exposed, and infected classes, whereas ordinary differential equations ODEs are made up of the quarantined and recovered classes.Assumptions of the mathematical model are as follows:(1)
Newborn babies are recruited into the susceptible class.(2)
New COVID-19 infection arises as the susceptible comes in contact with the COVID-19 patient (infective or quarantined).(3)
Renewal of susceptible people through recovery from transient immunity is not taken into consideration.(4)
The foreigner (invader) is categorized as either susceptible, exposed, or infective. Only these groups of people have different diffusion coefficients in one dimension because the COVID-19 infection was brought into the country through their immigration.Based on Figure1, the following equations are obtained to describe the epidemiology of COVID-19 infection in the country:(3)∂S∂t=αN−μS−βSI+QN+D1∂2S∂X2,∂E∂t=βSI+QN−μ+κ+θ1+δ1E+D2∂2E∂X2,∂I∂t=κE−μ+γ1+θ2+δ2I+D3∂2I∂X2,dQdt=θ1E+θ2I−δ3+μ+γ2Q,dRdt=δ1E+δ2I+δ3Q−μR.Figure 1
Compartmental model for COVID-19 transmission dynamics.Thus, the total population size of Ghana is represented by following equation:(4)Nt=St+Et+It+Qt+Rt.Together withSX,0=S0, EX,0=E0, IX,0=I0, Q0=Q0, and R0=0, where α is the birth rate. That is, the rate at which newborn babies are recruited into the susceptible class. The transmission rate, β, is the rate at which a susceptible person contracts the COVID-19 virus, κ is the rate at which exposed people become infectious while the average incubation period is 1/κ. The disease-induced death rate, γ1, is the rate at which an infected person dies from the COVID-19 infection, while the disease-induced death rate, γ2, is the rate at which a quarantined person dies from the disease. θ1 is the rate at which exposed people are quarantined, whereas θ2 is the rate at which an infectious person is quarantined. δ1 is the rate at which exposed people recover from the virus, δ2 is the rate at which infectious people recover from the disease, δ3 is the rate at which quarantined people recover from the disease, and μ is the natural death rate. D1 represents the constant diffusion coefficient of susceptible travelers into the country, D2 is the constant diffusion coefficient of exposed travelers into the country, and D3 is the constant diffusion coefficient of infectious travelers into the country.Settingm1=μ+κ+θ1+δ1, m2=μ+γ1+θ2+δ2 and m3=δ3+μ+γ2, the system of (3) becomes(5)∂S∂t=αN+D1∂2S∂X2−μS−βSI+QN,∂E∂t=βSI+QN+D2∂2E∂X2−m1E,∂I∂t=κE+D3∂2I∂X2−m2I,dQdt=θ1E+θ2I−m3Q,dRdt=δ1E+δ2I+δ3Q−μR,together with SX,0=S0, EX,0=E0, IX,0=I0, Q0=Q0, and R0=0.Nondimensionalizing of (5) is done by using the following system of equations:(6)s=SN,e=EN,i=IN,q=QN,r=RN,τ=αt,x=XαD11/2,a=μα,b=βα,d1=D2D1,n1=m1α,k=κα,d2=D3D1,n2=m2α,p1=θ1α,p2=θ2α,f1=δ1α,f2=δ2α,f3=δ3α,n3=m3α.Using the system of (6), we obtain the following system of equations:(7)∂s∂τ=1+∂2s∂x2−as−bsi+q,∂e∂τ=bsi+q+d1∂2e∂x2−n1e,∂i∂τ=κe+d2∂2i∂x2−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar,together with sx,0=s0, ex,0=e0, ix,0=i0, q0=q0, and r0=0.


### 2.2.1. Boundedness of the Solution of the System of ODEs

The boundedness of the system of (7) is obtained by setting the diffusion coefficients of the susceptible, exposed, and infective subgroups to zero which yields(8)dsdτ=1−as−bsi+q,dedτ=bsi+q−n1e,didτ=κe−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar.Together withsx,0=s0, ex,0=e0, ix,0=i0, q0=q0, and r0=0.Summing the ordinary derivatives on the left-hand side of the system of equations yields(9)⇒dNdt=αN−μS+E++I+Q+R−γ1I−γ2QdNdt=α−μN−γ1I−γ2QdNdt≤α−μNNt≤Aeα−μt.SettingN0=N0,⇒Nt≤N0eα−μt.The asymptotic behavior of the scaled population size is(10)limt⟶∞Nt=limt⟶∞Noeα−μtlimt⟶∞Nt=No,∀μ>α,limt⟶∞Nt=No,∀μ>α.The result in inequality (7) implies that the solution set is invariant. Thus, the population size of Ghana approaches a fixed finite number.


### 2.2.2. Existence and Uniqueness of the Solution of the System of Nondimensionalized PDEs

In this section, we show that the system of (7) has a unique solution in a Banach space. Consider the first three equations of the system of (7), which are given by(11)∂s∂τ=1+∂2s∂x2−as−bsi+q,∂e∂τ=bsi+q+d1∂2e∂x2−n1e,∂i∂τ=κe+d2∂2i∂x2−n2i.The existence and uniqueness of solutions of the system of nondimensionalized PDEs (11) with homogeneous Neumann boundary conditions(12)∂s∂τ=∂e∂τ=∂i∂τ=0,on∂Ω×0,+∞and initial conditions(13)sx,0=ϕ1x≥0,ex,0=ϕ2x≥0,ix,0=ϕ3x≥0,∀x∈Ω¯,where Ω is a bounded domain in ℝ3 with a smooth boundary ∂Ω.∂s/∂τ, ∂e/∂τ, and ∂i/∂τ, respectively, are the normal derivatives of sx,τ,ex,τ, and ix,τ on the boundary of the domain ∂Ω. Moreover, 0<s∗<1, 0<e∗<a/n1, and 0<i∗<b.We setLM=uo=ϕ1x,ϕ2x,ϕ3x∈L+:0≤uo≤M,∀x∈Ω¯ with M=1,a/n1,b.Setting a mapF=F1,F2,F3:LM⟶L, the system of (11) is written as(14)F1u0x=1−aϕ1x−bϕ1xϕ3x+qF2u0x=bϕ1xϕ3x+q−n1ϕ2xF3u0x=kϕ2x−n2ϕ3x,where q∈ℝ+ is a parameter. Thus, F satisfies the Lipschitz condition in the subspace of LM, and the system of (11) can be written in the Banach space L=CΩ¯, where Ω¯⊂ℝ3.(15)u′τ=Auτ+Fuτ,τ>0,u0=u0∈LM,where u.,τ=cols.,τ,e.,τ,i.,τ, Au:col△sx,.,d1△ex,.,d2△ix,., u0=colϕ1,ϕ2,ϕ3, and △ is the Laplace-type operator in one dimension. This result implies that the system of (11) has a solution in a Banach space.Theorem 2.
Setting the initial datauo=ϕ1x,ϕ2x,ϕ3x∈LM, there exists a unique nonnegative solution ux,τ of the system of equations (8) defined on 0,∞×Ω¯ with u0,.,ϕ=ϕ. Moreover, uτ,sx,τ,ex,τ,ix,τ∈LM for τ≥0 and usx,τ,ex,τ,ix,τ, ∀τ,s,e,i∈0,∞×Ω¯, is a classical solution of the system of equations (8).Proof 1.
In this proof, the Euclidean norm is used to achieve the results. Setting0≤h<min1/a+b,1/n1,1/n2 and for any sufficiently small h≥0, we obtain(16)uox+hFuox=1−ah−bhϕ3x+qϕ1x+h1−hn1ϕ2x+bhϕ1xϕ3x+q1−hn2ϕ3x+hkϕ2x⇒uox+hFuox≥1−ah−bhϕ1x1−hn1ϕ2x1−hn2ϕ3x≥000.
Also, we can see that(17)u0x+hFu0x≤1−ah−bhϕ1x+h1−hn1ϕ2x+bhϕ1x1−hn2ϕ3x+hkϕ2x,u0x+hFu0x≤1−a+bh+han1−a−bhb1−n2h+akhn1,limh⟶0uox+hFuox=1an1b,⇒uox+hFuox≤1an1b.
This implies thatuo is bounded above by M. System (8) admits a unique nonnegative solution uτ,x on 0,∞×Ω¯ with u0,.,ϕ=ϕ and uτ,.,ϕ∈LM for τ≥0.
We have(18)∂s∂τ−Δs=1−as,∂s∂ν=0,sx,0=ϕ1x≤ϕ1∞=maxx∈Ω¯ϕ1x.
By solvingds1/dτ=1−as1 and s10=ϕ1∞, using the method of integrating factors yields(19)s1τ=1a+ce−aτ.
Atsx,0=ϕ1x, it implies(20)c=ϕ1x−1a,⇒s1τ=ϕ1xeaτ+1a1−e−aτ.
sx,τ≤s1x,τ≤max1/a,ϕ1∞, for τ∈0,∞.(21)⇒sx,τ≤max1a,ϕ1∞,∀x,τ∈Ω¯×0,∞.
To establish theL∞ uniformly boundedness of ex,τ and ix,τ, we have(22)∂s∂ν=∂e∂ν=∂i∂ν=0.
Then,(23)⇒∂∂τs+e−Δs+d1e=1−as−bsi+q+bsi+q−ne,⇒ddτs+e≤1−as+e,s1τ+e1τ=ϕ1xe−aτ+ϕ2xe−aτ+1a1−e−aτ⇒∫Ωs+e+idx≤mesΩmax1a,ϕ1+ϕ2∞⇒supτ≥0∫Ωex,τdx≤k1≔mesΩmax1a,ϕ1+ϕ2∞,where k1 is a finite positive constant. Also,(24)∂∂τs+e−Δs+d2e=1−as+i−bsi+q−ν1+p2+f2i+ke,⇒ddτs+i≤1−as+i.Solving the equation d/dτs1+i1≤1−as1+i1 yields(25)s1τ+i1τ=ϕ1xe−aτ+ϕ3xe−aτ+1a1−e−aτ⇒∫Ωs+idx≤mesΩmax1a,ϕ1+ϕ3∞⇒supτ≥0∫Ωix,τdx≤k2≔=mesΩmax1a,ϕ1+ϕ3∞,where k2 is a finite positive constant.
We can be deduced that there exist positive constantsk1∗ and k2∗ that depend, respectively, on k1 and k2 and on ϕ1+ϕ2 and ϕ1+ϕ3 such that(26)supτ≥0e.τ∞≤k1∗,supτ≥0i.τ∞≤k2∗.
From the above, it has been proved thatsx,τ, ex,τ, and ix,τ are L∞ uniform bounded on Ω¯×0,τmax. Therefore, it follows from the standard theory of semilinear parabolic systems that τmax=+∞ is the maximal existence time for the solution of the system of equations (11). This completes g of the theorem.


### 2.2.3. Analysis of the Diffusion-Driven Instability

In this subsection, we investigate the epidemiology of the COVID-19 infection globally. The COVID-19 epidemiology in Ghana is presented herein. As a result, there is a lot of rumor about the COVID-19 disease spreading in Ghana, which is reportedly attributed to the increase in international immigration. There is no scientific evidence that support the claim that the inflow of foreigners caused the COVID-19 outbreak. We take into account the two scenarios in order to support this assertion about the emergence of the COVID-19 infection in the country. The qualitative method of the system of equations (7) without the inflow of foreigners is the first situation that is taken into account in this research. In addition, the analysis of (7) is in light of the continuous inflow of foreign immigrants. Setting D1=0,D2=0, and D3=0 leads in the qualitative solution of the system of (7) without the influx of foreigners. As a result, equation system (4) is as follows:(27)dsdτ=1−as−bsi+q,dedτ=bsi+q−n1e,didτ=κe−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar.The eigenvalues of the linearized system of (27) are obtained as(28)J1a,0,0,0,0−λI=−a−λ0−ba−ba00−n1−λbaba00k−n2−λ000p1p2−n3−λ00f1f2f3−a−λ=0,λ1=−a,λ2=−a,λ3+n1+n2+n3λ2+n1n2+n2n3+n1n3−bka−p1baλ+n1n2n3−n3bka−p2k−n2p1=0,⇒λ3+C1λ2+C2λ+C3=0,where, C1=n1+n2+n3, C2=n1n2+n2n3+n1n3−bk/a−p1b/a, and C3=n1n2n3−n3bk/a−p2k−n2p1.We observe thatC1>0, C3>0, and C1C2−C3>0, so all the eigenvalues of characteristic polynomial (28) have negative real parts. All the eigenvalues of the system of linearized ODEs in (27) without the diffusion term are all negative for all positive parameter values, this system of (27) is a linearly asymptotically stable spatial-homogeneous solution at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0. This implies that there would not have been COVID-19 in Ghana without an influx of foreigners in the country.Again, a situation where there is an influx of foreigners in the country is considered as follows. The system of equation (7) can be written as(29)∂u∂τ=Ju+D△u.We also haveux,t=sx,t,ex,t,ix,t,qt,rt.Here,J is a Jacobian matrix of the linear terms, △ is the Laplacian operator in one dimension, and D is a matrix that contains the coefficients of diffusion, that is,(30)D=100000d100000d2000000000000,together with sx,0=so,ex,0=eo,ix,0=io,q0=0,r0=0,n^⋅∇s⋅,t=0,n^⋅∇e⋅,t=0, and n^⋅∇i⋅,t=0.The solution of the system of (29) is of the form(31)ux,t=∑kckukreλt.For the time-independent component,ukr satisfies the following equation with the same boundary conditions as the system of (29):(32)△u+k2u=0withu⋅∇u=0on∂ℝorℝ¯.⇒△+k2u=0,⇒u≠0or△=−k2.Substituting (31) and (32) into (29) yields(33)Js,e,i,q,r=−a−bi−bq−k20−bs−bs0bi+q−n1−d1k2bsbs00k−n2−d2k2000p1p2−n300f1f2f3−a.By the Hartman–Grobman theorem, the system of (7) at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0 is unstable. At a−1,e∗,i∗,q∗,r∗=a−1,0,0,0,0,(34)Js∗,e∗,i∗,q∗,r∗−λI=−a−k2−λ0−ba−ba00−n1−d1k2−λbaba00k−n2−d2k2−λ000p1p2−n3−λ00f1f2f3−a−λ=0⇒λ1=−a+k2λ2=−a,λ3+C1λ2+C2λ+C3=0,where,(35)C1=n1+n2+n3+d1+d2k2C2=n3n1+n2+n3d1+d2k2+n1+d1k2n2+d2k2+bk−11a,C3=n3n1+d1k2n2+d2k2−bkn3+p2+bp1n2+d2k21a.Using the Routh–Hurwitz criterion for determining the sign of the roots of the characteristic polynomial in (35), the system of (7) at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0 is unstable if at least one of the following conditions is not met: C1>0, C3>0, and C1C2−C3>0. Since all the parameters are positive real numbers if bkn3+p2+bp1n2+d2k2⋅1/a≥n3n1+d1k2n2+d2k2, then the characteristic polynomial in (34) has at least one positive root. This implies that the system of (7) is unstable at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0. We obsvered that C3≥0 is not met which implies that there is at least one of the eigenvalues has a positive real part.The stability of the system of (7) revealed that without arrival of foreigners with the COVID-19 virus (SARS virus), there had not been the epidemiology of COVID-19 in the country. On the other hand, the arrival of foreigners, for example, COVID-19 patients from Norway and Turkey, into the country brought about the epidemiology of COVID-19 infection in Ghana. This finding is confirmed by the instability of the system of (7) at the endemic equilibrium point. The instability of the system of (7) is attributed to the assertion that the pandemic COVID-19 infection in the country is brought by the influx of foreigners especially two people: one from Norway and the other from Turkey.


## 2.2.1. Boundedness of the Solution of the System of ODEs

The boundedness of the system of (7) is obtained by setting the diffusion coefficients of the susceptible, exposed, and infective subgroups to zero which yields(8)dsdτ=1−as−bsi+q,dedτ=bsi+q−n1e,didτ=κe−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar.Together withsx,0=s0, ex,0=e0, ix,0=i0, q0=q0, and r0=0.Summing the ordinary derivatives on the left-hand side of the system of equations yields(9)⇒dNdt=αN−μS+E++I+Q+R−γ1I−γ2QdNdt=α−μN−γ1I−γ2QdNdt≤α−μNNt≤Aeα−μt.SettingN0=N0,⇒Nt≤N0eα−μt.The asymptotic behavior of the scaled population size is(10)limt⟶∞Nt=limt⟶∞Noeα−μtlimt⟶∞Nt=No,∀μ>α,limt⟶∞Nt=No,∀μ>α.The result in inequality (7) implies that the solution set is invariant. Thus, the population size of Ghana approaches a fixed finite number.


## 2.2.2. Existence and Uniqueness of the Solution of the System of Nondimensionalized PDEs

In this section, we show that the system of (7) has a unique solution in a Banach space. Consider the first three equations of the system of (7), which are given by(11)∂s∂τ=1+∂2s∂x2−as−bsi+q,∂e∂τ=bsi+q+d1∂2e∂x2−n1e,∂i∂τ=κe+d2∂2i∂x2−n2i.The existence and uniqueness of solutions of the system of nondimensionalized PDEs (11) with homogeneous Neumann boundary conditions(12)∂s∂τ=∂e∂τ=∂i∂τ=0,on∂Ω×0,+∞and initial conditions(13)sx,0=ϕ1x≥0,ex,0=ϕ2x≥0,ix,0=ϕ3x≥0,∀x∈Ω¯,where Ω is a bounded domain in ℝ3 with a smooth boundary ∂Ω.∂s/∂τ, ∂e/∂τ, and ∂i/∂τ, respectively, are the normal derivatives of sx,τ,ex,τ, and ix,τ on the boundary of the domain ∂Ω. Moreover, 0<s∗<1, 0<e∗<a/n1, and 0<i∗<b.We setLM=uo=ϕ1x,ϕ2x,ϕ3x∈L+:0≤uo≤M,∀x∈Ω¯ with M=1,a/n1,b.Setting a mapF=F1,F2,F3:LM⟶L, the system of (11) is written as(14)F1u0x=1−aϕ1x−bϕ1xϕ3x+qF2u0x=bϕ1xϕ3x+q−n1ϕ2xF3u0x=kϕ2x−n2ϕ3x,where q∈ℝ+ is a parameter. Thus, F satisfies the Lipschitz condition in the subspace of LM, and the system of (11) can be written in the Banach space L=CΩ¯, where Ω¯⊂ℝ3.(15)u′τ=Auτ+Fuτ,τ>0,u0=u0∈LM,where u.,τ=cols.,τ,e.,τ,i.,τ, Au:col△sx,.,d1△ex,.,d2△ix,., u0=colϕ1,ϕ2,ϕ3, and △ is the Laplace-type operator in one dimension. This result implies that the system of (11) has a solution in a Banach space.Theorem 2.
Setting the initial datauo=ϕ1x,ϕ2x,ϕ3x∈LM, there exists a unique nonnegative solution ux,τ of the system of equations (8) defined on 0,∞×Ω¯ with u0,.,ϕ=ϕ. Moreover, uτ,sx,τ,ex,τ,ix,τ∈LM for τ≥0 and usx,τ,ex,τ,ix,τ, ∀τ,s,e,i∈0,∞×Ω¯, is a classical solution of the system of equations (8).Proof 1.
In this proof, the Euclidean norm is used to achieve the results. Setting0≤h<min1/a+b,1/n1,1/n2 and for any sufficiently small h≥0, we obtain(16)uox+hFuox=1−ah−bhϕ3x+qϕ1x+h1−hn1ϕ2x+bhϕ1xϕ3x+q1−hn2ϕ3x+hkϕ2x⇒uox+hFuox≥1−ah−bhϕ1x1−hn1ϕ2x1−hn2ϕ3x≥000.
Also, we can see that(17)u0x+hFu0x≤1−ah−bhϕ1x+h1−hn1ϕ2x+bhϕ1x1−hn2ϕ3x+hkϕ2x,u0x+hFu0x≤1−a+bh+han1−a−bhb1−n2h+akhn1,limh⟶0uox+hFuox=1an1b,⇒uox+hFuox≤1an1b.
This implies thatuo is bounded above by M. System (8) admits a unique nonnegative solution uτ,x on 0,∞×Ω¯ with u0,.,ϕ=ϕ and uτ,.,ϕ∈LM for τ≥0.
We have(18)∂s∂τ−Δs=1−as,∂s∂ν=0,sx,0=ϕ1x≤ϕ1∞=maxx∈Ω¯ϕ1x.
By solvingds1/dτ=1−as1 and s10=ϕ1∞, using the method of integrating factors yields(19)s1τ=1a+ce−aτ.
Atsx,0=ϕ1x, it implies(20)c=ϕ1x−1a,⇒s1τ=ϕ1xeaτ+1a1−e−aτ.
sx,τ≤s1x,τ≤max1/a,ϕ1∞, for τ∈0,∞.(21)⇒sx,τ≤max1a,ϕ1∞,∀x,τ∈Ω¯×0,∞.
To establish theL∞ uniformly boundedness of ex,τ and ix,τ, we have(22)∂s∂ν=∂e∂ν=∂i∂ν=0.
Then,(23)⇒∂∂τs+e−Δs+d1e=1−as−bsi+q+bsi+q−ne,⇒ddτs+e≤1−as+e,s1τ+e1τ=ϕ1xe−aτ+ϕ2xe−aτ+1a1−e−aτ⇒∫Ωs+e+idx≤mesΩmax1a,ϕ1+ϕ2∞⇒supτ≥0∫Ωex,τdx≤k1≔mesΩmax1a,ϕ1+ϕ2∞,where k1 is a finite positive constant. Also,(24)∂∂τs+e−Δs+d2e=1−as+i−bsi+q−ν1+p2+f2i+ke,⇒ddτs+i≤1−as+i.Solving the equation d/dτs1+i1≤1−as1+i1 yields(25)s1τ+i1τ=ϕ1xe−aτ+ϕ3xe−aτ+1a1−e−aτ⇒∫Ωs+idx≤mesΩmax1a,ϕ1+ϕ3∞⇒supτ≥0∫Ωix,τdx≤k2≔=mesΩmax1a,ϕ1+ϕ3∞,where k2 is a finite positive constant.
We can be deduced that there exist positive constantsk1∗ and k2∗ that depend, respectively, on k1 and k2 and on ϕ1+ϕ2 and ϕ1+ϕ3 such that(26)supτ≥0e.τ∞≤k1∗,supτ≥0i.τ∞≤k2∗.
From the above, it has been proved thatsx,τ, ex,τ, and ix,τ are L∞ uniform bounded on Ω¯×0,τmax. Therefore, it follows from the standard theory of semilinear parabolic systems that τmax=+∞ is the maximal existence time for the solution of the system of equations (11). This completes g of the theorem.


## 2.2.3. Analysis of the Diffusion-Driven Instability

In this subsection, we investigate the epidemiology of the COVID-19 infection globally. The COVID-19 epidemiology in Ghana is presented herein. As a result, there is a lot of rumor about the COVID-19 disease spreading in Ghana, which is reportedly attributed to the increase in international immigration. There is no scientific evidence that support the claim that the inflow of foreigners caused the COVID-19 outbreak. We take into account the two scenarios in order to support this assertion about the emergence of the COVID-19 infection in the country. The qualitative method of the system of equations (7) without the inflow of foreigners is the first situation that is taken into account in this research. In addition, the analysis of (7) is in light of the continuous inflow of foreign immigrants. Setting D1=0,D2=0, and D3=0 leads in the qualitative solution of the system of (7) without the influx of foreigners. As a result, equation system (4) is as follows:(27)dsdτ=1−as−bsi+q,dedτ=bsi+q−n1e,didτ=κe−n2i,dqdτ=p1e+p2i−n3q,drdτ=f1e+f2i+f3q−ar.The eigenvalues of the linearized system of (27) are obtained as(28)J1a,0,0,0,0−λI=−a−λ0−ba−ba00−n1−λbaba00k−n2−λ000p1p2−n3−λ00f1f2f3−a−λ=0,λ1=−a,λ2=−a,λ3+n1+n2+n3λ2+n1n2+n2n3+n1n3−bka−p1baλ+n1n2n3−n3bka−p2k−n2p1=0,⇒λ3+C1λ2+C2λ+C3=0,where, C1=n1+n2+n3, C2=n1n2+n2n3+n1n3−bk/a−p1b/a, and C3=n1n2n3−n3bk/a−p2k−n2p1.We observe thatC1>0, C3>0, and C1C2−C3>0, so all the eigenvalues of characteristic polynomial (28) have negative real parts. All the eigenvalues of the system of linearized ODEs in (27) without the diffusion term are all negative for all positive parameter values, this system of (27) is a linearly asymptotically stable spatial-homogeneous solution at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0. This implies that there would not have been COVID-19 in Ghana without an influx of foreigners in the country.Again, a situation where there is an influx of foreigners in the country is considered as follows. The system of equation (7) can be written as(29)∂u∂τ=Ju+D△u.We also haveux,t=sx,t,ex,t,ix,t,qt,rt.Here,J is a Jacobian matrix of the linear terms, △ is the Laplacian operator in one dimension, and D is a matrix that contains the coefficients of diffusion, that is,(30)D=100000d100000d2000000000000,together with sx,0=so,ex,0=eo,ix,0=io,q0=0,r0=0,n^⋅∇s⋅,t=0,n^⋅∇e⋅,t=0, and n^⋅∇i⋅,t=0.The solution of the system of (29) is of the form(31)ux,t=∑kckukreλt.For the time-independent component,ukr satisfies the following equation with the same boundary conditions as the system of (29):(32)△u+k2u=0withu⋅∇u=0on∂ℝorℝ¯.⇒△+k2u=0,⇒u≠0or△=−k2.Substituting (31) and (32) into (29) yields(33)Js,e,i,q,r=−a−bi−bq−k20−bs−bs0bi+q−n1−d1k2bsbs00k−n2−d2k2000p1p2−n300f1f2f3−a.By the Hartman–Grobman theorem, the system of (7) at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0 is unstable. At a−1,e∗,i∗,q∗,r∗=a−1,0,0,0,0,(34)Js∗,e∗,i∗,q∗,r∗−λI=−a−k2−λ0−ba−ba00−n1−d1k2−λbaba00k−n2−d2k2−λ000p1p2−n3−λ00f1f2f3−a−λ=0⇒λ1=−a+k2λ2=−a,λ3+C1λ2+C2λ+C3=0,where,(35)C1=n1+n2+n3+d1+d2k2C2=n3n1+n2+n3d1+d2k2+n1+d1k2n2+d2k2+bk−11a,C3=n3n1+d1k2n2+d2k2−bkn3+p2+bp1n2+d2k21a.Using the Routh–Hurwitz criterion for determining the sign of the roots of the characteristic polynomial in (35), the system of (7) at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0 is unstable if at least one of the following conditions is not met: C1>0, C3>0, and C1C2−C3>0. Since all the parameters are positive real numbers if bkn3+p2+bp1n2+d2k2⋅1/a≥n3n1+d1k2n2+d2k2, then the characteristic polynomial in (34) has at least one positive root. This implies that the system of (7) is unstable at s∗,e∗,i∗,q∗,r∗=1/a,0,0,0,0. We obsvered that C3≥0 is not met which implies that there is at least one of the eigenvalues has a positive real part.The stability of the system of (7) revealed that without arrival of foreigners with the COVID-19 virus (SARS virus), there had not been the epidemiology of COVID-19 in the country. On the other hand, the arrival of foreigners, for example, COVID-19 patients from Norway and Turkey, into the country brought about the epidemiology of COVID-19 infection in Ghana. This finding is confirmed by the instability of the system of (7) at the endemic equilibrium point. The instability of the system of (7) is attributed to the assertion that the pandemic COVID-19 infection in the country is brought by the influx of foreigners especially two people: one from Norway and the other from Turkey.


## 2.3. Global Stability of the System of ODEs

In this subsection, we construct a Lyapunov function to assess the overall stability of the system of ODEs. A comprehensive review of related studies indicates that neither a systematic method nor a special function is utilized to assess the overall stability of system of ODEs in the Lyapunov sense. Due to this issue, one must make an informed guess for a function whose properties satisfy the requirements in the Lyapunov criterion for global stability of system of ODEs. In most cases, the guessing is not feasible since there are no guidelines in arriving at this function. In order to overcome this difficulty, we introduce a new function that can be used in determining the global stability of the system of ODEs in the Lyapunov sense. This function serves as a Lyapunov function for any system of ODEs and the result is given in Theorem3.Theorem 3 (Lyapunov function for the system of ODEs).
Suppose a system of ordinary differential equations,(36)dxidt=Fixi,∀xi∈Rn,Fxi>0,Fxi=F1xi,…,Fnxi,is an increasing function on Rn. Then,(37)Vxit=∑i=1n1xitxit−xi∗+xi∗lnxi∗xit,where x∗=x1∗,…,xn∗ is the fixed point of Fxi, is the Lyapunov function for constructing the stability of the systemof ODEs (19).Proof 2.
We can see thatVxit is a monotone increasing function on Rn for xit≠xi∗. At xi=x∗, we observe that(38)Vx∗=1x1tx1∗−x1∗+x1∗lnx1∗x1∗+⋯+1xntxn∗−xn∗+xn∗lnxn∗xn∗,Vx∗=1x1∗×0+⋯+1xn∗×0,Vx∗=∑i=0n1xi∗×0,Vx∗=0.
This implies thatVxi has a minimum value at xi=x∗. Therefore, x∗ is a fixed point of Vxit.
DifferentiatingVxit with respect to t, we have(39)dVdt=∂V∂x1.dx1dt+…+∂V∂xn,dVdt=−1x12x1−x1∗+x∗lnx∗x1+1x11−x∗x1×F1xi+…+−1xnxn−xn∗+xn∗lnxn∗xn+1xn1−xn∗xn×Fnxi,dVdt=−x1∗x1tlnx1∗x1tF1xi+…+−xn∗xntlnxn∗xntFnxi,dVdt=−∑i=1n−1xi∗xitlnxi∗xitFixidVdt<0.
Since the gradient ofVxit is decreasing on [0, ∞], it implies that Vxit=∑i=1n1/xitxit−xi∗+xi∗lnxi∗/xit is a Lyapunov function for the system of (3). This completes the proof.
This implies thatVxi has a minimum value at xi=x∗.


### 2.3.1. Application of Theorem2 in Determining the Global Stability of the System of ODEs

In this subsection, the global stability of the system of equation is obtained by setting∂s/∂x=0, ∂e/∂x=0, and ∂i/∂x=0. By Theorem 2, the Lyapunov function for the system of equation (8) is given by(40)Vs,e,i,q,r=1ss−s∗+s∗lns∗s+1ee−e∗+e∗lne∗e+1ii−i∗+i∗lni∗i+1qq−q∗+q∗lnq∗q+1rr−r∗+r∗lnr∗r.We can see that(41)Vs,e,i,q,r>0,∀s,e,i,q,r>0.Again,(42)Vs,e,i,q,r=0,∀s=s∗,e=e∗,i=i∗,q=q∗,r=r∗.Lastly, we can see that(43)dVdτ=∂V∂s⋅∂s∂τ+…+∂V∂r⋅∂r∂τ,dVdτ=−1s2s−s∗+s∗lns∗s+1s1−s∗s×1−as−bsi+q+−1e2e−e∗+e∗lne∗e+1e1−e∗e×bsi+q−n1e+…,dVdτ=−s∗slns∗s1−as−bsi+q−e∗elne∗ebsi+q−n1e−i∗ilni∗ike−n2i−q∗qlnq∗qp1e+p2i−n3q−r∗rlnr∗rf1e+f2i+f3q−ar.Since1−as−bsi+q>0, bsi+q−n1e>0, ke−n2i>0, p1e+p2i−n3q>0, and f1e+f2i+f3q−ar>0,(44)⇒dVdτ<0.Since all conditions of the Lyapunov function are met, it implies that the system of (8) is globally asymptotically stable.


## 2.3.1. Application of Theorem2 in Determining the Global Stability of the System of ODEs

In this subsection, the global stability of the system of equation is obtained by setting∂s/∂x=0, ∂e/∂x=0, and ∂i/∂x=0. By Theorem 2, the Lyapunov function for the system of equation (8) is given by(40)Vs,e,i,q,r=1ss−s∗+s∗lns∗s+1ee−e∗+e∗lne∗e+1ii−i∗+i∗lni∗i+1qq−q∗+q∗lnq∗q+1rr−r∗+r∗lnr∗r.We can see that(41)Vs,e,i,q,r>0,∀s,e,i,q,r>0.Again,(42)Vs,e,i,q,r=0,∀s=s∗,e=e∗,i=i∗,q=q∗,r=r∗.Lastly, we can see that(43)dVdτ=∂V∂s⋅∂s∂τ+…+∂V∂r⋅∂r∂τ,dVdτ=−1s2s−s∗+s∗lns∗s+1s1−s∗s×1−as−bsi+q+−1e2e−e∗+e∗lne∗e+1e1−e∗e×bsi+q−n1e+…,dVdτ=−s∗slns∗s1−as−bsi+q−e∗elne∗ebsi+q−n1e−i∗ilni∗ike−n2i−q∗qlnq∗qp1e+p2i−n3q−r∗rlnr∗rf1e+f2i+f3q−ar.Since1−as−bsi+q>0, bsi+q−n1e>0, ke−n2i>0, p1e+p2i−n3q>0, and f1e+f2i+f3q−ar>0,(44)⇒dVdτ<0.Since all conditions of the Lyapunov function are met, it implies that the system of (8) is globally asymptotically stable.


## 2.4. Effects of the COVID-19 Infection in the Country

In this subsection, the travelling wave solution of the influx of foreigners into the country is analysed. Thus, the analytic method via Homotopy Perturbation Method (HPM) is used to obtain the solutions for the proportions of susceptible, exposed, and infective people.Settingsx,τ=sz, ex,τ=ez, and ix,τ=iz where z=x−cτ, c is the constant wave speed of the foreigners into the country; the first equation in the system of equation (4) becomes(45)d2sdz2+cdsdz−as+1−bsi+q=0.qt=0, since the movement of a quarantined person is restricted so he or she cannot travel to another country.(46)d2sdz2+cdsdz−as+1−bsi=0.Similarly, we have(47)d2edz2+cd1dedz+bsi+qd1−n1d1e=0.Settingqt=0, we have(48)d2edz2+cd1dedz−n1d1e+bd1si=0,(49)andd2idz2+cd2didz+kd2e−n2d2i=0.For the travelling wave solution, that is, the influx of foreigners into the country, we consider the proportion of the susceptible, exposed, and infective people.Using the HPM, the series approximation solutions of the system of equations (46), (48), and (49) are of the following form:(50)1−plinear terms of the system of equations+plinear+nonlinear terms of the system of equations=0,⇒1−pd2sdz2+cdsdz−as+1+pd2sdz2+cdsdz−as+1−bsi=0,1−pd2edz2+cd1dedz−n1d1e+pd2edz2+cd1dedz−n1d1e+bd1si=0,1−pd2idz2+cd2didz−n2d2i+pd2idz2+cd2didz+kd2e−n2d2i=0.We use the HPM to find the solution of the linearized system of (7) as follows.Equatingp0=0, the following equation is obtained from equation (50).(51)d2sdz2+cdsdz−as=1,where the homogeneous part of (51) is given by(52)d2sdz2+cdsdz−as=0.Settingsz=eλz into (52) yields(53)⇒λ2+cλ−a=0,∀eλz≠0λ1,2=−c±c2+4a2,∀c2+4a>0.The complementary function of (51) is(54)sz=D1e1/2−c+c2+4az+D2e−1/2c+c2+4az,with s0=so and s1=s1.Setting the particular solutionspz=Az2+Bz+C, finding the first and second derivative and substituting the results into (51) yield the particular solution spz=−1/a. Thus, the general solution of the proportion of susceptible is given by(55)sz=D1e1/2−c+c2+4az+D2e−1/2c+c2+4az−1a.Similarly, the proportion for exposed people and infected people is given by(56)ez=D3e1/2−c/d1+c2−4n1d1/d12z+D4e−1/2c/d1+c2−4n1d1/d12z,iz=D5e1/2−c/d2+c2−4n2d2/d22z+D4e−1/2c/d1+c2−4n2d2/d22z,respectively.


## 2.5. Conversion of a System of Mixture of PDEs and ODEs into a System of ODEs

Now, all the PDEs are expressed in their respective ODEs in order to have solutions which are of the same state.Setting(57)u1z=sz,u2z=dsdz,u3z=ez,u4z=dedz,u5z=iz,u6z=didz,u7z=qz,u8z=rz,yields the following system of ODEs:(58)du1dz=u2,du2dz=−cu2+au1+bu1u5+bu1u7−1,du3dz=u3,du4dz=−cd1u4+n1d1u3−bd1u1u5−bd1u1u7,du5dz=u6,du6dz=−cd2u6−kd2u3+nd2u5,du7dz=p1u3+p2u5−n3u7,du8dz=f1u3+f2u5+f3u7−au8.The equilibrium point of the system of ODEs in (58) is obtained by setting the ordinary derivatives on the right hand side of (58) to zero and solving the resulting equations simultaneously which yields the disease-free equilibrium point as(59)u1∗,u2∗,u3∗,u4∗,u5∗,u6∗,u7∗,u8∗=1a,0,0,0,0,0,0,0.Linearizing the system of ODEs (30) yields the following results:(60)Ju1,u2,u3,u4,u5,u6,u7,u8=01000000a+bu5+u7−c00bu10bu1000100000−bu5+u7d10n1d1−cd1−bu1d10−bu1d100000010000−kd20n2d2−cd20000p10p20−n3000f10f20f3−a.Atu1∗,u2∗,u3∗,u4∗,u5∗,u6∗,u7∗,u8∗=1/a,0,0,0,0,0,0,0, the eigenvalues of the Jacobian matrix in (60) are obtained as(61)J1a,0,0,0,0,0,0,0−λI=−λ1000000a−c−λ00ba0ba0001−λ0000000n1d1−cd1−λ−bad10−bad100000−λ10000−kd20n2d2−cd2−λ0000p10p20−n3−λ000f10f20f3−a−λ=0λ1=−a<0,λ2=−c+c2+4a2>0,λ3=−c−c2+4a2<0,λ4=1>0,λ5=−cd1<0,λ6=−n3<0,λ7=−c/d2+c/d22+4n2/d22>0,λ8=−c/d2−c/d22+4n2/d22<0.Sinceλ2, λ4, and λ7 are all greater than zero, it implies that the COVID-19 infection at the disease-free equilibrium point, u1∗,u2∗,u3∗,u4∗,u5∗,u6∗,u7∗,u8∗=1/a,0,0,0,0,0,0,0, is unstable. Hence, a sufficiently small influx of foreigners into the country will result in the spread of COVID-19 in the country.


### 2.5.1. Analysis of the Equilibria of the System of ODEs

In the system of (8), the basic reproductive number, Ro, is obtained from the spectral radius of FV−1. Thus,(62)Ro=ρFV−1,where(63)F=000000−bad1000000000,V−1=10000−d1c0000cn2−d2n200−10.Ro is obtained as(64)Ro=bcad1n2⇒Ro=αβD1cμμ+γ1+θ2+δ2D2.The basic reproductive number,R1, for the system of ODEs without diffusion is given by(65)R1=ρFV−1,where(66)F=0baba000000,V=n100−kn20−p1−p2n3,R1=αβθ1γ1+δ2+θ2+μ+κγ2+δ3+μ+θ2μγ2+δ3+μγ1+δ2+θ2+μδ1+θ1+κ+μ.We observe that the basic reproductive number,Ro, in the environment of inflow of foreigners into the country is more realistic than the basic reproductive number in diffusive-free environment since Ro involves the coefficient of diffusion of the susceptible, as well as the speed of the influx of the travellers into the country. The coefficient of diffusion of the susceptible and the wave speed is in direct relationship with Ro which are inflows for the infective so as to result in transmission of the COVID-19 infection.


## 2.5.1. Analysis of the Equilibria of the System of ODEs

In the system of (8), the basic reproductive number, Ro, is obtained from the spectral radius of FV−1. Thus,(62)Ro=ρFV−1,where(63)F=000000−bad1000000000,V−1=10000−d1c0000cn2−d2n200−10.Ro is obtained as(64)Ro=bcad1n2⇒Ro=αβD1cμμ+γ1+θ2+δ2D2.The basic reproductive number,R1, for the system of ODEs without diffusion is given by(65)R1=ρFV−1,where(66)F=0baba000000,V=n100−kn20−p1−p2n3,R1=αβθ1γ1+δ2+θ2+μ+κγ2+δ3+μ+θ2μγ2+δ3+μγ1+δ2+θ2+μδ1+θ1+κ+μ.We observe that the basic reproductive number,Ro, in the environment of inflow of foreigners into the country is more realistic than the basic reproductive number in diffusive-free environment since Ro involves the coefficient of diffusion of the susceptible, as well as the speed of the influx of the travellers into the country. The coefficient of diffusion of the susceptible and the wave speed is in direct relationship with Ro which are inflows for the infective so as to result in transmission of the COVID-19 infection.


## 2.6. Numerical Results

The data on the COVID-19 patients in Ghana were obtained from [18] and then segregated into exposed people and infected people by summing the counts of COVID-19 cases of the first fourteen days (incubation period) in the month as the number of exposed people in that month and the remaining counts of COVID-19 patients as the number of infective in the month. The results on these exposed and infective people are summarized in Tables 1 and 2. The normalization of the data points was done using(67)X∗=X−minXmaxX−minXnew maxX−new minX+new minX.Table 1
Exposed and infected people from March 2020 to December 2020.
MonthExposed peopleInfected peopleMarch1556April1438475May23353661June57773894July93768384August17667031September9711357October882547November15442068December133851719Table 2
Exposed and infected people from January 2021 to January 2022.
MonthExposed peopleInfected peopleJanuary100292210February81878826March23554205April1017962May655681June1215801July49052200August82048213September36084438October8191776November210633December11075991January (2022)37869743Table3 shows the estimates for the parameter in the model. Initial COVID-19 records in Ghana are shown in Table 4.Table 3
List of parameters for the model.
ParameterValuesα0.2000000000000β0.8000000000000κ0.0714285714300γ10.7500000000000γ20.0008889880194θ10.0140014195900θ20.7800000000000δ10.2937464950000δ20.0195830996700δ30.5874929900000μ0.0000195371580Table 4
Initial conditions for the five subgroups of the population size of Ghana.
CompartmentValueProportionSusceptible304177560.9999966467Exposed1003.287542469×10−6Infectious26.575084939×10−8Quarantined00Recovered00


### 2.6.1. Numerical Results of the Nondimensionalized System of Mixture of PDEs and ODEs

This section discusses the effects of the movement of proportions of susceptible, exposed, and infectious individuals at a given time. Four cases are examined. Firstly, the same diffusion coefficient of the proportion of the exposed and the infective people are considered, i.e.,d1≠d2.The diffusion coefficientsd1 and d2 are provided in Table 5 and the initial conditions for the first three compartments of system (4) are as follows:(68)sx,0=1−0.6x2,0≤x<0.8,0.6160.8−xx,0.8≤x≤1.Table 5
Varied coefficients of diffusion for the proportion of susceptible, exposed, and infective.
CaseParameterValuekm2/hrd10.01d20.001Figure2 is plotted using the initial condition from (68). We can see that the proportion of the susceptible curve decreases as the scaled time and scaled distance start to rise from zero at the beginning. The proportion of the susceptibility curve decreases quickly as more susceptible are infected with the SARS virus (COVID-19 disease) since there are more susceptible people who are not adhering to the COVID-19 protocols by both public and private organizations in Ghana. The sharp rises in Figures 3 and 4 illustrate how the proportions of exposed and infected individuals rise when more susceptible individuals become infected with the SARS virus. The susceptible proportion continues to monotonically decrease until stabilizing at a specific point (adequately proportion of susceptible not able to contract SARS virus). The behavior of the susceptible curve shown in Figure 2 is concurrently supported by the decreasing patterns after the peaks in Figures 3 and 4.Figure 2
A 3-dimensional plot for the proportion of susceptible in space and time.Figure 3
A 3-dimensional plot for the proportion of exposed in space and time.Figure 4
A 3-dimensional plot for the proportion of infective in space and time.


### 2.6.2. Numerical Results on the Trends of the COVID-19 Patients

The quantitative solution of the proportion of the infective subgroup system of ODEs in (40) as depicted in Figure 5. Figure 5 reveals a sinusoidal wave of the COVID-19 infection in the country with an amplitude of approximately 0.92.Figure 5
Proportion of the infective curve.The time in Figure6 begins at zero, which corresponds to March 2020, with no data on January and February 2020, since COVID-19 erupted in Ghana on March 14,2020. After the onset of COVID-19 pandemic in the country, the proportion of infective fluctuates roughly every eight months. As a result, the proportion infective curve increases until its peak and then decreases as the month progresses. Even the peak values are inconsistent, with July 2020 having a peak value of roughly 0.9058, February 2021 recording a peak value of roughly 0.8604, and January 2022 recording a peak value of almost 1.000. However, the proportion of the infective by the model (theoretical results) in Figure 5 also fluctuates at regular interval of almost every 8 months, but the peaked values were recorded on August 2020, April 2021, and October 2021, respectively. There are variations of the peak values of the proportions of the infective on the data as in Figure 6 and the peak values of the proportion of infective from the model (see Figure 5). These variations are attributed to the noise in the data, such as outliers (peak values), incomplete data on the COVID-19 patients on the grounds that not every person who contracted COVID-19 reported the issue at the nearby medical facility, or the health professionals were unable to contact trace all the COVID-19 patients. One of the major deterrents for suspected COVID-19 patients not reporting their illness to the nearest medical facility is the fear of panic brought on by the SARS virus (COVID-19) spreading to the susceptible population. However, errors in instrumentation devices used for testing COVID-19 cases across the nation as well as human error in data entry caused by the healthcare personnel cannot be ignored when it comes to the data on COVID-19 patients. Before it can be identified whether someone has COVID-19 (caused by the SARS virus), they frequently need to undergo a number of tests. In Figure 7, the quantitative result (curve) from the model is placed on the observed data curve for comparison. It was revealed that the variations of the model curve and observed data curve decrease as time passes and then fluctuate for the remaining of the period. This disparity is attributed to the unavoidable inaccuracies in the data. This necessitates data cleansing via bin smoothing techniques.Figure 6
Proportion of the infective from March 2020 to January 2022.Figure 7
Proportion of infective curves from normalized data and the model.Due to all the issues observed about data on COVID-19 patients in the country, inevitably the data cleaning was carried out to remove noise from the data. In Table6, the relative errors between the model values and data values are summarized. The relative errors are not different from the observations in Figures 8–10.Table 6
The relative error between the normalized data and the model values.
Time (month)Normalized data valueModel data valueRelative error00.00000.1524910.04810.09410.956340956320.37530.13280.646149746930.39930.33580.159028299540.86040.74970.128661087950.72140.91530.268782922160.13870.63293.563085797070.05550.26833.834234234080.21170.077640.633254605690.17590.14190.1932916430100.22630.32180.4220061865110.90580.50080.4471185696120.43120.67830.5730519481130.09810.59945.1100917430140.06930.39164.6507936510150.08160.23951.9350490200160.22530.22350.0079834753170.84280.32810.6107024205180.45510.45350.0035157108190.18170.54111.9779856910200.06430.51887.0684292380210.10110.41313.0860534120221.0000.32380.6762000000Figure 8
Proportion of the exposed people from March 2020 to January 2022.Figure 9
Proportion of the exposed curve from the model.Figure 10
Proportion of exposed curves from normalized data and the model.


### 2.6.3. Smoothing of the Normalized Data

In smoothing the normalized data from the Ghana Health Service (GHS), discrepancies in the data were detected, such as human error in the data entry of the number of infectives, because daily cases of COVID-19 patients and accumulated monthly cases of COVID-19 patients were not segregated into the number of infective and the number of exposed people. The two subgroups, the number of infective and the number of exposed people, of the population size of Ghana cannot be lumped together as they play different roles in the epidemiology of COVID-19 pandemic in the country.To remove noise from the data obtained from GHS, the binning smoothing methods, smoothing by bin means, smoothing by bin medians, and smoothing by bin boundaries, were employed. The proportion of the infective curve superimposed on each normalized smoothed set of data is depicted in Figures 12, 14, and 16 These graphs show that the normalized smooth data by bin boundaries are consistent and that its values are robust against noise, especially when there are outliers. As a result, in Figure11, the variation between the infectious curve (model) and the smoothed data is smaller as the time (month) goes up to 4, then it starts to go up to 0.2687829221 (see relative error in Table 7). It is imperative to note that this peculiar trend could be caused by the unsorting of the normalized data:(1)
Comparison of the smoothened normalized data by bin boundaries, bin means, and bin medians and the normalized model for infective(2)
Comparison of the smoothened normalized data by bin boundaries, bin means, and bin medians and the normalized model for exposedFigure 11
The proportion of infective curves from smoothing by bin boundaries data and the model.Table 7
The relative error between the bin boundaries data and the model values.
Time (month)Bin boundaries data valueModel data valueRelative error00.00000.1510.00000.094120.37530.13280.646149746930.39930.33580.159028299540.72140.74970.039229276450.72140.91530.268782922160.13870.63293.563085797070.13870.26830.934390771480.21170.077640.633254605690.17590.14190.1932916430100.17590.32180.8294485503110.90580.50080.4471185696120.43120.67830.5730519481130.06930.59947.6493506490140.06930.39164.6507936510150.08160.23951.9350490200160.08160.22351.1738970588170.84280.32810.6107024205180.45510.45350.0035157108190.06430.54117.4152410580200.06430.51887.0684292380210.10110.41313.0860534120221.0000.32380.6762000000We can see from Figure12 that the smoothing by bin boundaries maintains most of the features of the original normalized data (see Figure 8). Thus, there are peaks in Figure 12 as in Figure 8, whereas in Figure 13 and Figure 14, they were flattened at the top. The curve of the model superimposed on the smoothened (normalized) data by the bin boundaries, bin means, and bin medians in Figures 12, 15, and 16, respectively, exhibited variations between the two curves in each figure. Comparatively, in Figure 11, the variation between the curve and smoothened (normalized) data by the bin boundaries was minimal and more consistent as time (month) increases than the smoothened by bin means and bin medians in Figures 15 and 16. The smoothing normalized data by the bin means using the frequency of three data points assumes the mean value for all the three data points like smoothing by the bin medians. Similar comparison study of the normalized equations (model) and the smoothened data, on the proportion of exposed people, by bin boundaries, bin means, and bin medians are observed in Figures 17–22. Undoubtedly, the variations between the smoothened data by bin boundaries and the model were least as compared with the variations between the smoothened data by the bin means and the model and the smoothened (normalized) data by bin medians and the model. The relative errors for the three smoothing techniques are summarized in Tables 7–9 for easy comparison.Figure 12
Proportion of the infective from March 2020 to January 2022.Figure 13
The proportion of the infective from March 2020 to January 2022.Figure 14
Proportion of the infective from March 2020 to January 2022.Figure 15
The proportion of infective curves from smoothing by bin means data and the model.Figure 16
Proportion of infective curves from smoothing by bin medians data and the model.Figure 17
Proportion of the exposed people from March 2020 to January 2022.Figure 18
Proportion of exposed curves from smoothing by bin boundaries data and the model.Figure 19
Proportion of the exposed people from March 2020 to January 2022.Figure 20
Proportion of exposed curves from smoothing by bin means data and the model.Figure 21
Proportion of the exposed people from March 2020 to January 2022.Figure 22
Proportion of expose curves from smoothing by bin medians data and the model.Table 8
The relative error between the bin means data and the model values.
Time (month)Bin means data valueModel data valueRelative error00.14110.150.0630758327410.14110.09410.3330970943020.14110.13280.0588235294130.66030.33580.4914432834040.66030.74970.1353930032050.66030.91530.3861880963060.13530.63293.6777531410070.13530.26830.9830007391080.13530.077640.4261640798090.43600.14190.67454128440100.43600.32180.26192660500110.43600.50080.14862385320120.19950.67832.40000000000130.19950.59942.00451127819140.19950.39160.96290726817150.38320.23950.23408141962160.38320.22350.41675365344170.38320.32810.14378914405180.23370.45350.94052203679190.23370.54111.31536157460200.23370.51881.21994009413210.55050.41310.24959128065220.55050.32380.41180744777Table 9
The relative error between the bin medians data and the model values.
Time (month)bin medians data valueModel data valueRelative error00.04810.150.2118503118510.04810.09410.9564309563020.04810.13281.7609147609130.86040.33580.6097164109040.86040.74970.1286610870050.86040.91530.0638075313060.05550.632910.403603603670.05550.26833.8342342340080.05550.077640.3989189189090.22630.14190.37295625700100.22630.32180.42202616865110.22630.50081.21429916040120.09810.67835.91437308800130.09810.59945.11009174300140.09810.39162.99184505600150.22530.23950.06302707501160.22530.22350.07983475300170.22530.32810.46528051480180.18170.45351.49587231700190.18170.54111.97798569100200.18170.51881.85360484310210.55050.41310.24959128065220.55050.32380.41180744777


## 2.6.1. Numerical Results of the Nondimensionalized System of Mixture of PDEs and ODEs

This section discusses the effects of the movement of proportions of susceptible, exposed, and infectious individuals at a given time. Four cases are examined. Firstly, the same diffusion coefficient of the proportion of the exposed and the infective people are considered, i.e.,d1≠d2.The diffusion coefficientsd1 and d2 are provided in Table 5 and the initial conditions for the first three compartments of system (4) are as follows:(68)sx,0=1−0.6x2,0≤x<0.8,0.6160.8−xx,0.8≤x≤1.Table 5
Varied coefficients of diffusion for the proportion of susceptible, exposed, and infective.
CaseParameterValuekm2/hrd10.01d20.001Figure2 is plotted using the initial condition from (68). We can see that the proportion of the susceptible curve decreases as the scaled time and scaled distance start to rise from zero at the beginning. The proportion of the susceptibility curve decreases quickly as more susceptible are infected with the SARS virus (COVID-19 disease) since there are more susceptible people who are not adhering to the COVID-19 protocols by both public and private organizations in Ghana. The sharp rises in Figures 3 and 4 illustrate how the proportions of exposed and infected individuals rise when more susceptible individuals become infected with the SARS virus. The susceptible proportion continues to monotonically decrease until stabilizing at a specific point (adequately proportion of susceptible not able to contract SARS virus). The behavior of the susceptible curve shown in Figure 2 is concurrently supported by the decreasing patterns after the peaks in Figures 3 and 4.Figure 2
A 3-dimensional plot for the proportion of susceptible in space and time.Figure 3
A 3-dimensional plot for the proportion of exposed in space and time.Figure 4
A 3-dimensional plot for the proportion of infective in space and time.


## 2.6.2. Numerical Results on the Trends of the COVID-19 Patients

The quantitative solution of the proportion of the infective subgroup system of ODEs in (40) as depicted in Figure 5. Figure 5 reveals a sinusoidal wave of the COVID-19 infection in the country with an amplitude of approximately 0.92.Figure 5
Proportion of the infective curve.The time in Figure6 begins at zero, which corresponds to March 2020, with no data on January and February 2020, since COVID-19 erupted in Ghana on March 14,2020. After the onset of COVID-19 pandemic in the country, the proportion of infective fluctuates roughly every eight months. As a result, the proportion infective curve increases until its peak and then decreases as the month progresses. Even the peak values are inconsistent, with July 2020 having a peak value of roughly 0.9058, February 2021 recording a peak value of roughly 0.8604, and January 2022 recording a peak value of almost 1.000. However, the proportion of the infective by the model (theoretical results) in Figure 5 also fluctuates at regular interval of almost every 8 months, but the peaked values were recorded on August 2020, April 2021, and October 2021, respectively. There are variations of the peak values of the proportions of the infective on the data as in Figure 6 and the peak values of the proportion of infective from the model (see Figure 5). These variations are attributed to the noise in the data, such as outliers (peak values), incomplete data on the COVID-19 patients on the grounds that not every person who contracted COVID-19 reported the issue at the nearby medical facility, or the health professionals were unable to contact trace all the COVID-19 patients. One of the major deterrents for suspected COVID-19 patients not reporting their illness to the nearest medical facility is the fear of panic brought on by the SARS virus (COVID-19) spreading to the susceptible population. However, errors in instrumentation devices used for testing COVID-19 cases across the nation as well as human error in data entry caused by the healthcare personnel cannot be ignored when it comes to the data on COVID-19 patients. Before it can be identified whether someone has COVID-19 (caused by the SARS virus), they frequently need to undergo a number of tests. In Figure 7, the quantitative result (curve) from the model is placed on the observed data curve for comparison. It was revealed that the variations of the model curve and observed data curve decrease as time passes and then fluctuate for the remaining of the period. This disparity is attributed to the unavoidable inaccuracies in the data. This necessitates data cleansing via bin smoothing techniques.Figure 6
Proportion of the infective from March 2020 to January 2022.Figure 7
Proportion of infective curves from normalized data and the model.Due to all the issues observed about data on COVID-19 patients in the country, inevitably the data cleaning was carried out to remove noise from the data. In Table6, the relative errors between the model values and data values are summarized. The relative errors are not different from the observations in Figures 8–10.Table 6
The relative error between the normalized data and the model values.
Time (month)Normalized data valueModel data valueRelative error00.00000.1524910.04810.09410.956340956320.37530.13280.646149746930.39930.33580.159028299540.86040.74970.128661087950.72140.91530.268782922160.13870.63293.563085797070.05550.26833.834234234080.21170.077640.633254605690.17590.14190.1932916430100.22630.32180.4220061865110.90580.50080.4471185696120.43120.67830.5730519481130.09810.59945.1100917430140.06930.39164.6507936510150.08160.23951.9350490200160.22530.22350.0079834753170.84280.32810.6107024205180.45510.45350.0035157108190.18170.54111.9779856910200.06430.51887.0684292380210.10110.41313.0860534120221.0000.32380.6762000000Figure 8
Proportion of the exposed people from March 2020 to January 2022.Figure 9
Proportion of the exposed curve from the model.Figure 10
Proportion of exposed curves from normalized data and the model.


## 2.6.3. Smoothing of the Normalized Data

In smoothing the normalized data from the Ghana Health Service (GHS), discrepancies in the data were detected, such as human error in the data entry of the number of infectives, because daily cases of COVID-19 patients and accumulated monthly cases of COVID-19 patients were not segregated into the number of infective and the number of exposed people. The two subgroups, the number of infective and the number of exposed people, of the population size of Ghana cannot be lumped together as they play different roles in the epidemiology of COVID-19 pandemic in the country.To remove noise from the data obtained from GHS, the binning smoothing methods, smoothing by bin means, smoothing by bin medians, and smoothing by bin boundaries, were employed. The proportion of the infective curve superimposed on each normalized smoothed set of data is depicted in Figures 12, 14, and 16 These graphs show that the normalized smooth data by bin boundaries are consistent and that its values are robust against noise, especially when there are outliers. As a result, in Figure11, the variation between the infectious curve (model) and the smoothed data is smaller as the time (month) goes up to 4, then it starts to go up to 0.2687829221 (see relative error in Table 7). It is imperative to note that this peculiar trend could be caused by the unsorting of the normalized data:(1)
Comparison of the smoothened normalized data by bin boundaries, bin means, and bin medians and the normalized model for infective(2)
Comparison of the smoothened normalized data by bin boundaries, bin means, and bin medians and the normalized model for exposedFigure 11
The proportion of infective curves from smoothing by bin boundaries data and the model.Table 7
The relative error between the bin boundaries data and the model values.
Time (month)Bin boundaries data valueModel data valueRelative error00.00000.1510.00000.094120.37530.13280.646149746930.39930.33580.159028299540.72140.74970.039229276450.72140.91530.268782922160.13870.63293.563085797070.13870.26830.934390771480.21170.077640.633254605690.17590.14190.1932916430100.17590.32180.8294485503110.90580.50080.4471185696120.43120.67830.5730519481130.06930.59947.6493506490140.06930.39164.6507936510150.08160.23951.9350490200160.08160.22351.1738970588170.84280.32810.6107024205180.45510.45350.0035157108190.06430.54117.4152410580200.06430.51887.0684292380210.10110.41313.0860534120221.0000.32380.6762000000We can see from Figure12 that the smoothing by bin boundaries maintains most of the features of the original normalized data (see Figure 8). Thus, there are peaks in Figure 12 as in Figure 8, whereas in Figure 13 and Figure 14, they were flattened at the top. The curve of the model superimposed on the smoothened (normalized) data by the bin boundaries, bin means, and bin medians in Figures 12, 15, and 16, respectively, exhibited variations between the two curves in each figure. Comparatively, in Figure 11, the variation between the curve and smoothened (normalized) data by the bin boundaries was minimal and more consistent as time (month) increases than the smoothened by bin means and bin medians in Figures 15 and 16. The smoothing normalized data by the bin means using the frequency of three data points assumes the mean value for all the three data points like smoothing by the bin medians. Similar comparison study of the normalized equations (model) and the smoothened data, on the proportion of exposed people, by bin boundaries, bin means, and bin medians are observed in Figures 17–22. Undoubtedly, the variations between the smoothened data by bin boundaries and the model were least as compared with the variations between the smoothened data by the bin means and the model and the smoothened (normalized) data by bin medians and the model. The relative errors for the three smoothing techniques are summarized in Tables 7–9 for easy comparison.Figure 12
Proportion of the infective from March 2020 to January 2022.Figure 13
The proportion of the infective from March 2020 to January 2022.Figure 14
Proportion of the infective from March 2020 to January 2022.Figure 15
The proportion of infective curves from smoothing by bin means data and the model.Figure 16
Proportion of infective curves from smoothing by bin medians data and the model.Figure 17
Proportion of the exposed people from March 2020 to January 2022.Figure 18
Proportion of exposed curves from smoothing by bin boundaries data and the model.Figure 19
Proportion of the exposed people from March 2020 to January 2022.Figure 20
Proportion of exposed curves from smoothing by bin means data and the model.Figure 21
Proportion of the exposed people from March 2020 to January 2022.Figure 22
Proportion of expose curves from smoothing by bin medians data and the model.Table 8
The relative error between the bin means data and the model values.
Time (month)Bin means data valueModel data valueRelative error00.14110.150.0630758327410.14110.09410.3330970943020.14110.13280.0588235294130.66030.33580.4914432834040.66030.74970.1353930032050.66030.91530.3861880963060.13530.63293.6777531410070.13530.26830.9830007391080.13530.077640.4261640798090.43600.14190.67454128440100.43600.32180.26192660500110.43600.50080.14862385320120.19950.67832.40000000000130.19950.59942.00451127819140.19950.39160.96290726817150.38320.23950.23408141962160.38320.22350.41675365344170.38320.32810.14378914405180.23370.45350.94052203679190.23370.54111.31536157460200.23370.51881.21994009413210.55050.41310.24959128065220.55050.32380.41180744777Table 9
The relative error between the bin medians data and the model values.
Time (month)bin medians data valueModel data valueRelative error00.04810.150.2118503118510.04810.09410.9564309563020.04810.13281.7609147609130.86040.33580.6097164109040.86040.74970.1286610870050.86040.91530.0638075313060.05550.632910.403603603670.05550.26833.8342342340080.05550.077640.3989189189090.22630.14190.37295625700100.22630.32180.42202616865110.22630.50081.21429916040120.09810.67835.91437308800130.09810.59945.11009174300140.09810.39162.99184505600150.22530.23950.06302707501160.22530.22350.07983475300170.22530.32810.46528051480180.18170.45351.49587231700190.18170.54111.97798569100200.18170.51881.85360484310210.55050.41310.24959128065220.55050.32380.41180744777


## 3. Conclusion

The epidemiology of COVID-19 has been modelled by taking into account the main geographical factor, such as the continuous inflow of immigrants into the country, which is the primary underlying factor for so-called pandemic disease, as well as disease-related factors, such as the infectious agent, mode of transmission, latent period, infectious period, susceptibility, and resistance to the disease. The validity and robustness of this mathematical model are established through field data assessments of its trend. The model parameters that have been tested are the major aspects that will support robustness when the disparity between the model values and the data decreases over time. In reality, the validity of the mathematical model was examined using monthly data on COVID-19 infections in Ghana. A newly proposed function for constructing the global stability of the system of ODEs in the Lyapunov sense overcomes the disadvantage of guessing the function for calculating the global stability of a nonlinear ODE system. Furthermore, the Lyapunov function provides information on the magnitude of an asymptotically stable fixed point of a nonlinear system of ODEs. Furthermore, the basic reproductive number in the presence of a continuous inflow of foreigners into the country is more realistic and robust to data noise than the basic reproductive number in the absence of a continuous inflow of foreigners into the country.



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*Source: 7563111-2022-10-19.xml*