MATHEMATICAL ANALYSIS OF THE GLOBAL COVID-19 SPREAD IN NIGERIA AND SPAIN BASED ON SEIRD MODEL

In this paper, a compartmental model for the transmission dynamics of the new infectious disease referred to as COVID-19 is employed. The model comprises five mutually exclusive compartments (classes) of human population sizes viz: susceptible, exposed, infected, recovered, and death, representing the human dynamics; hence, the name SEIRD model. In the model, the temporal dynamics of the COVID-19 outbreak in Nigeria and Spain are analyzed. The period is between February 15-April 3, 2020 for Spain, and February 27-April 3, 2020, for Nigeria. The analysis of the population data is based on the concerned SEIRD model. Graphical representations of the obtained results are presented. A connection between the contact rate of the infection and the compartmental human population sizes subject to the COVID-19 analysis is revealed. It shows that a decrease in the contact rate of the ‘susceptible and the infected’ classes is a considerable condition leading to a decline in 'the exposed, infected, and death' cases. This decrease is attributed to the control of the possible infecting contacts. The spread patterns for the 2 S.O. EDEKI, I. ADINYA, M.E. ADEOSUN, I.D. EZEKIEL two considered cases are the same. A lot of measures are needed to be put in place to ensure a corresponding increase in the 'recovered class.' The COVID-19 outbreak would remain global and endemic if the infecting contact rate is not well controlled. Thus, adherence to strict public and government policies such as social distancing and isolation is a plausible requirement. For other aspects of epidemiology with related features, this strategy is highly recommended for implementation.


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MATHEMATICAL ANALYSIS OF THE GLOBAL COVID-19 SPREAD mentioned above or cases are valid since the COVID-19 can be contacted via contact with respiratory droplets of infected persons, indicating that the entire population is susceptible to  Owing to the points mentioned earlier, travellers are always at high risk of contracting infectious diseases such as Ebola and the likes. Hence, in [10], it was suggested that international travel authorities should imbibe restriction and proper health screening once an outbreak is noticed. However, these restrictions cannot be enforced for a long time, as many other issues affecting the global economy would come up. For instance, there is no doubt that China had been known to be Nigeria's most leading commercial partners in the whole world, and as a result, there had been substantial inflows and outflows of traffics between these countries.
Unexpectedly, the Nigeria Centre for Disease Control (NCDC) reported that one COVID-19 case, was imported by an Italian, who arrived Nigeria on February 27, 2020, as stated in [11], he further travelled across other states of the country, after which, he was confirmed infected. The latest update by the NCDC shows a rise in the figure of infected persons given as 151 confirmed cases as of April 1, 2020 [12]. Epidemiology is becoming an increasing necessity for the health system of the world today; as such, the situations need good mathematical models to check the spread [13,14]. The recent health situations (hazards) around the world has made it mandatory for scientists to study and determine the causes and best possible way to control, reduce, or eradicate these hazards [15][16][17][18].
Since the beginning of this novel and life-threatening COVID-19, a lot of researches have been conducted across the globe. Many of such are still ongoing, all in a bid to proffer a lasting solution or to control the spread. This paper, therefore, proposed a mathematical model as an extension of the classical SIR model and its modifications for the analysis of the disease spread.
The whole paper is partitioned as follows. In section 1, a detailed introduction is given. Section 2 contains the methodology and the derivation of the model; in section 3, the method of solution is presented, and the model solutions are obtained and discussed. In section 4, the paper is remarkably concluded.

METHODOLOGY
Mathematics has been applied in many studies to proffer solutions to epidemic issues [19][20][21][22][23][24][25][26]. In mathematical epidemic modeling, the main objective is to determine how fast the disease spreads, the number of the population that are being be infected, the effects of migration into and out of affected areas, and possible control measures. The first mathematical epidemic model can be traced to the mathematician, Daniel Bernoulli, in 1766. His work was on how inoculation against smallpox affects the life expectancy of the population [27,28]. Kermack and McKendrick developed the SIR-model, that is, Susceptible-Infectious-Removed (SIR) [29]. They investigated the effects of some factors that aid the spread of an infectious disease right from when an infected person moves to a susceptible population [29]. There have been many extensions of the SIR model, including Susceptible-Infectious-Removed-Susceptible (SIRS), Susceptible-Infectious-Susceptible (SIS) and Susceptible-Exposed-Infectious-Removed (SEIR) [30][31][32][33]. These models have been explored for different health hazards, including the current one, coronavirus disease 2019 (COVID-19). So many researchers had published recent results on various issues relating to epidemiology, and the novel COVID-19 [18,[20][21][22][23][24][25][26][34][35][36][37][38][39][40][41][42]. Zhao et al., [43] considered how inter-city travels affect the spread of the virus using correlation analysis.
They employed the Needleman-Wunsch algorithm, an application of dynamic programming to identify changes in the DNA sequence of the coronavirus. The conventional SIR model or its modified form termed SIRS, claims (by assumption) the disease incubation to be negligible. Such that each susceptible individual (S) becomes infectious once infected and thus move to the infective class (I), who again will move to the recovered class (R) or susceptible class (S), depending on the acquired level of immunity which may be permanent or temporary [44,45].
Whereas, extensions to these models (SIR and SIRS) have been considered on the ground that susceptible individuals, once infected, need to, first of all, go through a latent stage forming exposed class (E) before they become infectious. This scenario can, therefore, lead to either 5 MATHEMATICAL ANALYSIS OF THE GLOBAL COVID-19 SPREAD SEIR or SEIRS model, depending (still) on the acquired level of immunity [46,47]. This present work captures the COVID-19 global issue by adding a compartmental class of death individuals, mainly due to the spread of COVID-19. Hence, the SEIRD model.

Fundamental of the SEIRD Model
In this section, the basic assumptions for the proposed model are presented and the model derivation follows. (ii) Infection is not due to level of education, so the infective class is not sub-divided [45].

Basic assumptions for the SEIRD Model
(iii) Exposure to latency period is permitted.
(v) Infected individuals are quarantined. Though, both sets of dependent variables give the same information regarding the state of the concern epidemic or pandemic.

COVID-19 Spread Model Formulation
During epidemics (disease outbreak within a community) or pandemic (global disease outbreak) situation, some individuals get infected, while some recover after they have been infected.
Meanwhile, some fraction dies due to infectious diseases. It can be captured that some of the susceptible individuals are exposed to the spread before they are infected. Thus, the population at such time can be divided into five compartments resulting in a model herein referred to as SEIRD.
Here, for a time parameter, , In some cases, ( ) RD + , that is, R and D , are referred to as removed individuals. This will be denoted as is as defined earlier, then: The detail partition is shown in the compartmental diagram ( Fig. 1): where the parameter c is the rate of infection (probability of S contracting the disease when in contact with , I  signifies the latency rate of migration,  signifies recovery rate and  is the death rate. By considering Figure 1 via the application of conservation principle, the following dynamics represent a set of a system of differential equations that models the situation: subject to the following initial conditions:

MATHEMATICAL ANALYSIS OF THE GLOBAL COVID-19 SPREAD
The present model (1.2) with its initial data (1.3) extends or complements some vital epidemiological models in literature, such as the researches of [40,45,47,48]. Apart from other parameters considered in [45], the infective class was comprehensively sub-divided into two classes viz: educated and uneducated infected individuals with meaningful results.

METHOD OF SOLUTION
such that: For the sake of simplicity and model analysis, we refer to the following Tables (1 and 2

THE ASSOCIATED MODEL SOLUTIONS
This section presents the solution associated with the proposed model via the method of solution earlier presented. The data in Tables 1 and 2

NUMERICAL AND GRAPHICAL PRESENTATION OF RESULTS
The above solutions are approximate analytical solutions to the proposed SEIRD model. The plots of the resulting solutions are presented in the following figures. Other possible methods of solutions include those of [53][54][55][56][57][58][59][60][61][62]. For the Spain model cases, we consider Fig.2, Fig.3, Fig.4 Similarly, for the Nigeria model cases, we consider Fig.6, Fig.7, Fig.8 and Fig.9,  imply a situation where each infected person makes one (1) possible infecting contact per day on the average of 10 days of observation period (say latency). This for the Spain case model is depicted in Fig.2, Fig. 3, Fig. 4 and Fig. 5.

PARAMETER VALUES AND IMPLICATIONS
Similarly, the parameters 1 were considered to reflect a situation where each infected person makes one (1) possible infecting contact every three (3) days on the average of 10 days of observation period. The results in the two scenarios show a drastic decrease in the 'exposed, infected and death' classes ( Fig.2-Fig.5 compared).
For universality and comparison, the Nigeria case is also considered via the parameters,

CONCLUDING REMARKS
This paper has successfully considered the implementation of a simple mathematical model for the analysis of the global COVID-19 spread. The model compartments partition the population into Susceptible, Exposed, Infected, Recovered, and Deaths individuals; hence SEIRD model.
The pandemic cases analyzed via the SEIRD model were based on data made available by worldometer and WHO between 15/03-03/04/2020 for the case of Spain, and 27/02-03/04/2020 for the case of Nigeria. The reported results of the two countries indicated the same spread patterns for the two considered instances, even though different parameters were used. Due to the considered level of the possible infecting contacts, a reasonable decrease in the 'exposed and infected' likewise the 'infected and death' classes was recorded conditionally, as shown via the graphical representations. Remarkably, more cases would be confirmed at an exponential rate if