Mathematical Modelling of the Spatial Epidemiology of COVID-19 with Different Diffusion Coefficients
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-Recovered 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.
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 .
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 .
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  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  is very similar to the one developed in , 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  applied the susceptible-exposed-infected-recovered-deceased model to describe the spatial spread of COVID-19 in Italy. The authors in  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  observed the following:where is a parameter and 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  observedwhere 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-recovered 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.
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). Let be an equilibrium point for . Let be a differentiable function defined on an open set containing . Suppose further that(i) and if (ii) in Then, is stable. Furthermore, if also satisfies(iii) in , then is asymptotically stable.
A function satisfying and is called a Lyapunov function for . If also holds, we call a strict Lyapunov function .
2.2. Development of the Mathematical Model and Its Analyses
The population of Ghana, , is split into five distinct classes based on their epidemiological status: the susceptible class ), the exposed class (), the infected class , the quarantined class (), and the recovered class (). 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 . Partial differential equations are made up of the susceptible, exposed, and infected classes, whereas ordinary differential equations 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 Figure 1, the following equations are obtained to describe the epidemiology of COVID-19 infection in the country:
Thus, the total population size of Ghana is represented by following equation:
Together with , , , , and , where is the birth rate. That is, the rate at which newborn babies are recruited into the susceptible class. The transmission rate, , is the rate at which a susceptible person contracts the COVID-19 virus, is the rate at which exposed people become infectious while the average incubation period is . The disease-induced death rate, , is the rate at which an infected person dies from the COVID-19 infection, while the disease-induced death rate, , is the rate at which a quarantined person dies from the disease. is the rate at which exposed people are quarantined, whereas is the rate at which an infectious person is quarantined. is the rate at which exposed people recover from the virus, is the rate at which infectious people recover from the disease, is the rate at which quarantined people recover from the disease, and is the natural death rate. represents the constant diffusion coefficient of susceptible travelers into the country, is the constant diffusion coefficient of exposed travelers into the country, and is the constant diffusion coefficient of infectious travelers into the country.
Setting , and , the system of (3) becomestogether with , , , , and .
Nondimensionalizing of (5) is done by using the following system of equations:
Using the system of (6), we obtain the following system of equations:together with , , , , and .
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
Together with , , , , and .
Summing the ordinary derivatives on the left-hand side of the system of equations yields
The asymptotic behavior of the scaled population size is
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
The existence and uniqueness of solutions of the system of nondimensionalized PDEs (11) with homogeneous Neumann boundary conditionsand initial conditionswhere is a bounded domain in with a smooth boundary .
, , and , respectively, are the normal derivatives of , and on the boundary of the domain . Moreover, , , and .
We set with .
Setting a map , the system of (11) is written aswhere is a parameter. Thus, F satisfies the Lipschitz condition in the subspace of , and the system of (11) can be written in the Banach space , where .where , , , 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 data , there exists a unique nonnegative solution of the system of equations (8) defined on with . Moreover, for and , , is a classical solution of the system of equations (8).
Proof 1. In this proof, the Euclidean norm is used to achieve the results. Setting and for any sufficiently small , we obtainAlso, we can see thatThis implies that is bounded above by . System (8) admits a unique nonnegative solution on with and for .
We haveBy solving and , using the method of integrating factors yieldsAt , it implies, for .To establish the uniformly boundedness of and , we haveThen,where is a finite positive constant. Also,Solving the equation yieldswhere is a finite positive constant.
We can be deduced that there exist positive constants and that depend, respectively, on and and on and such thatFrom the above, it has been proved that , , and are uniform bounded on . Therefore, it follows from the standard theory of semilinear parabolic systems that is the maximal existence time for the solution of the system of equations (11). This completes 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 , and leads in the qualitative solution of the system of (7) without the influx of foreigners. As a result, equation system (4) is as follows:
The eigenvalues of the linearized system of (27) are obtained aswhere, , , and .
We observe that , , and , 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 . 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
We also have .
Here, J is a Jacobian matrix of the linear terms, is the Laplacian operator in one dimension, and is a matrix that contains the coefficients of diffusion, that is,together with , and .
The solution of the system of (29) is of the form
For the time-independent component, satisfies the following equation with the same boundary conditions as the system of (29):
By the Hartman–Grobman theorem, the system of (7) at is unstable. At ,where,
Using the Routh–Hurwitz criterion for determining the sign of the roots of the characteristic polynomial in (35), the system of (7) at is unstable if at least one of the following conditions is not met: , , and . Since all the parameters are positive real numbers if , then the characteristic polynomial in (34) has at least one positive root. This implies that the system of (7) is unstable at . We obsvered that 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 Theorem 3.
Theorem 3 (Lyapunov function for the system of ODEs). Suppose a system of ordinary differential equations,is an increasing function on . Then,where is the fixed point of , is the Lyapunov function for constructing the stability of the systemof ODEs (19).
Proof 2. We can see that is a monotone increasing function on for . At , we observe thatThis implies that has a minimum value at . Therefore, is a fixed point of .
Differentiating with respect to t, we haveSince the gradient of is decreasing on [0, ], it implies that is a Lyapunov function for the system of (3). This completes the proof.
This implies that has a minimum value at .
2.3.1. Application of Theorem 2 in Determining the Global Stability of the System of ODEs
We can see that
Lastly, we can see that
Since , , , , and ,
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.
Setting , , and where , is the constant wave speed of the foreigners into the country; the first equation in the system of equation (4) becomes
, since the movement of a quarantined person is restricted so he or she cannot travel to another country.
Similarly, we have
Setting , we have
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.
We use the HPM to find the solution of the linearized system of (7) as follows.
Setting into (52) yields
The complementary function of (51) iswith and .
Setting the particular solution , finding the first and second derivative and substituting the results into (51) yield the particular solution . Thus, the general solution of the proportion of susceptible is given by
Similarly, the proportion for exposed people and infected people is given byrespectively.
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.
Settingyields the following system of ODEs:
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
Linearizing the system of ODEs (30) yields the following results: