 Original research
 Open Access
 Published:
Mathematical model of the spread of COVID19 in Plateau State, Nigeria
Journal of the Egyptian Mathematical Society volume 30, Article number: 10 (2022)
Abstract
In this research, a mathematical model consisting of nonpharmaceutical control measures is formulated. The developed model helps to examine the transmission of COVID19 infection in Plateau State, Nigeria, using face masks \(c_{f}\) and social distancing \(c_{d}\) as control measures. Data used for the simulation of the developed model were obtained from Nigeria Centre for Disease Control which was fitted to the system of ordinary differential equations using nonlinear least squares method. Results at baseline values \(c_{f} = 0.1\) and \(c_{d} = 0.2\) of control measures indicate 2.3 estimation as basic reproduction number which suggests that COVID19 in Plateau State tends towards endemic state. However, above about 40% in the use of face masks in the population and corresponding above 50% adherence to social distancing could as well bring down the basic reproduction number to a value below 1 necessary for disease eradication. The results at baseline values further indicate that the peak of the COVID19 had been reached in less than 250 days from the first detection date after about 476,455 undetected asymptomatic individuals, 92,168 undetected symptomatic individuals and 83,801 detected quarantined individuals have been fully infectious. Therefore, the policymakers in Plateau State have the possibility of eradicating the disease with further strict nonpharmaceutical control measures provided that the present conditions of analysis remain fairly the same.
Introduction
SARSCoV2 is the virus strain of COVID19 which was first discovered in Wuhan, Hubei China [1]. It is one of the viral infectious diseases which had spread fast across the world. Plateau State, which is one of the thirty six states of Nigeria situated in Africa, detected its first case of the infectious disease on 23 April 2020. However, prior to the detection of the first incidence case, the Plateau State government declared a lockdown on 9 April 2020 which lasted till 15 April 2020 [2, 3].
There has been a considerable spread of COVID19 in Plateau State among the estimated population of 4,200,442 people [4], despite various nonpharmaceutical control measures put in place to curb the spread of the virus. Consequently, five COVID19 isolation centres were provided with a total of 217 bed spaces and provision for homebased care (HBC) was made as an alternative strategy whenever the available bed spaces are filled up. The policymakers of Plateau State also provided preventive materials like face masks and hand sanitizers through their Emergency Operations Centres (EOC) to support patients on HBC—which started in July 2020—in order to stop the spread of the disease to family members. Some measures and materials put in place to reduce the spread of the virus include: use of face masks, hand washing, use of infrared thermometers, hand sanitizers, mass media, social distancing, introduction of mobile testing centres for effective detection, isolation of infected individuals and closure of schools, interstate borders, among others [5,6,7]. While as of 11 June 2020, the state had tested 2032 persons for COVID19, out of which 130 were confirmed positive with 3 deaths, 26 persons were on admission at isolation centres, 99 persons discharged and 35 people were of interest at quarantine centres in Pankshin, Qua’an Pan, Heipang and Mangu [5]. However, as of 28 February 2021, 62,317 samples from people have been tested in Plateau State with 8889 confirmed positive cases [8]. Furthermore, a curfew was imposed from 10 p.m. to 4 a.m.; restriction on gatherings of not more than 50 people under strict observance of COVID19 safety protocols was also announced [9].
The research of Dauda et al. [3] revealed the effect of lockdown, locust of control and state anxiety among residents of Plateau State Nigeria at the time when only three cases of COVID19 were reported with the aid of an online survey using statistical analysis approach. Few researchers have actually considered the spread of COVID19 in Nigeria [3, 10,11,12].
While the research of Iboi et al. [10] focused on transmission of COVID19 in Lagos, Kano and Federal Capital Territory, the study conducted by Okuonghae and Omame [12] focused on Lagos State only. They both investigated the impact of nonpharmaceutical control measures in their areas of coverage. Dauda et al. [3] on the other hand focused on Plateau State with emphasis on statistical analysis of lockdown, locus of control and state anxiety among residents. Furthermore, while some studies focused on hybrid fractional order, fractional optimal control and stochastic models [13,14,15,16], some other recent studies predicted outbreak of coronavirus and assessment of control measures as well as Neyman–Scott point process model for COVID19 both within and across international borders [17,18,19,20]. However, from the foregoing researches and other previously published works [21,22,23,24,25,26,27], there has not been any published research to date which has considered a mathematical model approach for the transmission dynamics of the spread of COVID19 in Plateau State Nigeria.
Different types of numerical methods have been proposed for solving mathematical models. The type of numerical method used for any system of differential equations could largely be dependent on the nature of the model. Alotaibi et al. [28] investigated solution of COVID 19 model, and they concluded that homotopy annoyance and decreased differential change techniques are adequate for solving such model equations. Modified reduced differential transform and homotopy perturbation method have also been used in some other researches [29, 30]. In the research of Mahdy et al. [31], they investigated Rubella ailment disease model using shifted secondorder Chebyshev polynomials type which they claimed had not been used before and concluded that the method is efficient and its simpletouse style makes it suitable for finding convergent solutions to Rubella disease models. Simos [32] solved problems with oscillating solutions with Runge–Kutta–Fehlberg method with phaselag of order infinity and showed its suitability for such problems. Also, Paul et al. [33] investigated behaviour of Lotka–Volterra prey–predator model using Runge–Kutta–Fehlberg method and Laplace–Adomian decomposition method and presented their effectiveness in solving such problems. The research of Handapangoda et al. [34] used Laguerre Runge–Kutta–Fehlberg method for simulating laser pulse propagation in biological tissue. In this study, the proposed model will be solved with existing numerical method based on Fehlberg fourth–fifthorder Runge–Kutta method. For more details on Runge–Kutta–Fehlberg method and some other suitable numerical methods, readers may check [32,33,34,35,36]
The incentive for this study came from high incidence rate of COVID19 in Plateau State according to the data obtained from Nigeria Centre for Disease Control (NCDC) as of 28 February 2021. The high incidence rate of the disease could be due to an increase in detection rate which is directly proportional to an increase in number of laboratory tests conducted in Plateau State. There was a remarkable increase in the number of samples tested in Plateau State compared to other states in Nigeria within the period of this study. The total number of samples tested in the laboratory was 62,317 samples, out of which 8889 samples were confirmed positive for COVID19 as of 28 February 2021. Thus, it is advantageous to measure the impact of nonpharmaceutical interventions on the spread of COVID19 in Plateau State Nigeria.
In this study, we develop a mathematical model to investigate the effect of nonpharmaceutical interventions on the spread of COVID19 in Plateau State, Nigeria. It is essential to use nonpharmaceutical control measures since nonpharmaceutical control measures were used throughout the period of this study.
The organization of the remaining parts of this study is as follows: “Methods” section contains the methods used; “Analysis of model (1)” section considers the analysis of the developed model. Results and discussion are considered in “Results and discussion” section, and conclusion comes up in “Conclusion” section.
Methods
Epidemic data
The epidemic data for this research were obtained from the Nigeria Centre for Disease Control. The centre alerted the Plateau State residents of its first COVID19 incidence case on 23 April 2020 which prompted the Plateau State government to introduce various nonpharmaceutical measures as control strategies. Laboratories were set up to increase the detection of COVID19 incidence rate. There was setting up of quarantine centres, sensitization of the public through the media houses, compulsory use of face masks in public places and enforcement of social distancing among residents in the state. Figure 1 shows reported cumulative daily COVID19 cases from 23 April 2020 to 28 February 2021.
Model formulation
The formulation of the model for this study is based on human to human transmission of the COVID19 at a time t. Let the total human population at the time t be denoted by \(N(t)\) and the population of the susceptible people be \(S(t)\). Let the latent population be \(E(t)\) and infectious asymptomatic and infectious symptomatic population at time \(t\) be \(I_{A} (t)\) and \(I_{S} (t)\), respectively. While \(Q(t)\) represents asymptomatic and symptomatic quarantined population, denote human population that recovered from the COVID19 by \(R(t)\) at time t such that \(N(t) = S(t) + E(t) + I_{A} (t) + I_{S} (t) + Q(t) + R(t)\). It should be noted that the derivation of the model used the assumption that the exposed class \(E(t)\) consists of two sets of people that could transmit the disease and those that could not transmit the disease at latent stage [27]. The schematics representation of the transmission of the COVID19 is shown in Fig. 2.
This schematic representation of transmission of COVID19 infection in Fig. 2 is a tool aiding the formulation of the proposed governing model equations for this study. It should be noted that schematic flow display of infection transmission dynamics is essential to infectious disease modelling whenever some details of model formulations are required in the study of biomathematics. This further suggests that the schematic flow representation could be used to check the correctness or otherwise of the governing model equations. It also means that relationship exists between the schematic flow representation of Fig. 2 and system of ordinary differential equations used for this study. For further details on similar studies where schematic flow representation can be found, readers may check [12, 37].
In this study, the spread of COVID19 in Plateau State Nigeria for the community of people which are distributed homogenously are governed by the following system of differential equations:
where \(\pi\) represents recruitment by birth, \(\mu\) and \(\mu_{i} (i = 1,2)\) are natural deaths and deaths due to COVID19, respectively. The parameters \(c_{f}\) and \(c_{d}\) are the nonpharmaceutical control measures representing the use of face mask and social distancing for \(0 \le c_{f} \le 1,0 \le c_{d} \le 1.\)
Descriptions of the state vectors and other associated parameters of the system of Eqs. (1) are shown in Tables 1 and 2.
Analysis of model (1)
In this section, analysis of equilibria, positivity of solution, basic reproduction number, local stability of the diseasefree equilibrium, local stability of the endemic equilibrium as well as parameter estimation and data fitting of model (1) are presented.
Equilibria of model (1)
The system (1) has the diseasefree equilibrium (DFE) at
and the diseaseendemic equilibrium (DEE) \(\eta_{o} = (S^{*} ,E_{{}}^{*} ,I_{A}^{*} ,I_{S}^{*} ,Q_{{}}^{*} ,R^{*} )\) of (1) is obtained when the righthand side of (1) is set equal to zero, thus
where C is positive constant and \(\Psi_{1} = (1  c_{f} )(1  c_{d} ), \, \Psi_{2} = \alpha_{1} + \mu + r_{1} , \, \Psi_{3} = \alpha_{2} + \mu + \mu_{1} + r_{2} , \,\)\(\Psi_{4} = \gamma + \mu + \mu_{2} , \, \Phi = \pi \Psi_{3} \Psi_{4} \delta r_{1} + \pi \Psi_{3} \alpha_{1} \delta \gamma .\)
Positivity of solutions of model (1)
Theorem 1
For all time t > 0, the solutions of \((S,E,I_{A} ,I_{S} ,Q,R,N)\) of the model (1) are all positive for all \(S(0) \ge 0,E(0) \ge 0,I_{A} (0) \ge 0,I_{S} (0) \ge 0,Q(0) \ge 0,R(0) \ge 0\) and \(N(0) \ge 0\).
Proof
From the governing Eq. (1), we obtain
Hence, the region \(\Omega = \left\{ {(S,E,I_{A} ,I_{s} ,Q,R):S + E + I_{A} + I_{s} + Q + R \le \frac{\pi }{\mu }} \right\}\) is positively invariant which implies positivity of solution.\(\square\)
Basic reproduction number of model (1)
Using nextgeneration matrix approach of Heffernan et al. [38], define square matrices F and V such that \(F = \frac{{\partial F_{i} (x_{o} )}}{{\partial x_{j} }}\) and \(V = \frac{{\partial V_{i} (x_{o} )}}{{\partial x_{j} }}\) where \(F_{i} (x_{o} )\) and \(V_{i} (x_{o} )\) are rates of new infections and different ways to move between the compartments \(i\), respectively. The compartments \(x_{j} (j = 1,2, \ldots ,n \in {\mathbb{N}})\) represent infected compartments for system of Eq. (1), so that
where \(\Psi_{1} = (1  c_{f} )(1  c_{d} )\), \(\Psi_{2} = \alpha_{1} + \mu + r_{1}\), \(\Psi_{3} = \alpha_{2} + \mu + \mu_{1} + r_{2}\), \(\Psi_{4} = \gamma + \mu + \mu_{2}\) and \(g = f\delta + \left( {1  f} \right)\delta + \mu\) and then the dominant eigenvalue of the nextgeneration matrix \(G = FV^{  1}\) is the basic reproduction number \(R_{0}\). Use of Maple software facilitated the process of obtaining inverse of Matrix V through a call to “MatrixInverse” command which is capable of computing inverse of a square matrix and can also be used to obtain Moore–Penrose pseudoinverse of any nonsquare matrix A. The “MatrixInverse” command of Maple adopted for finding inverse of V in this study incorporated LU factorization method based on “LUDecomposition” command in Maple 18 which has the capability to solve a matrix with partial pivoting for any matrix A of the form PA = LU where L and U are lower and upper triangular matrices and P is a permutation matrix. This representation shows that all square matrices can be factorized in this form which is known to be numerically stable and convergent in practice for matrix V being considered in this study [39]. Hence, simple matrix multiplication of F and inverse of V in Maple 18 produced a nextgeneration matrix G whose dominant eigenvalue is the basic reproduction number
for \(\Psi_{1} \,\Psi_{2} \,\delta \,f\beta_{{I_{S} }} \pi + \,\,\Psi_{1} \,\Psi_{2} \,\Psi_{3} \beta_{E} \pi + \,\Psi_{1} \,\Psi_{3} \,\delta \,\beta_{{I_{A} }} \pi > \,\Psi_{1} \,\Psi_{3} \,\delta \,f\beta_{{I_{A} }} \pi\).
Local stability of the diseasefree equilibrium of (1)
Theorem 2
If \(R_{0} < 1\), the diseasefree equilibrium (DFE) \(P_{0}\) of (1) is asymptotically stable but unstable if \(R_{0} > 1\).
Proof
The Jacobian matrix of system (1) for DFE is given by
where \(\Psi_{1} , \, \Psi_{2} , \, \Psi_{3} , \, \Psi_{4}\) and \(g\) are defined as in “Basic reproduction number of model (1)” section. The eigenvalues of the matrix \(J_{Po}\) are
where \(\lambda_{i} (i = 1,2,3) < 0\) indicate negative eigenvalues and \(\lambda_{i} (i = 4) < 0\) whenever \(g\mu > \Psi_{1} \beta_{E} \,\pi\), \(\lambda_{i} (i = 5) < 0\) whenever \(\Psi_{1} \,\delta \,f\beta_{IA} \pi + \Psi_{2} \,g\mu > \Psi_{1} \,\Psi_{2} \beta_{E} \,\pi \, + \,\Psi_{1} \,\delta \,\beta_{IA} \pi\) and \(g\mu > \Psi_{1} \beta_{E} \,\pi\).
Also simplify \(\lambda_{6}\) for \(R_{0}\) to obtain \(\begin{aligned} & \lambda_{6} =  \frac{{\left( {\frac{{\,\Psi_{1} \,\Psi_{2} \,\delta \,f\beta_{{I_{S} }} \pi + \,\,\Psi_{1} \,\Psi_{2} \,\Psi_{3} \beta_{E} \pi + \,\Psi_{1} \,\Psi_{3} \,\delta \,\beta_{{I_{A} }} \pi  \,\Psi_{1} \,\Psi_{3} \,\delta \,f\beta_{{I_{A} }} \pi }}{{\mu \,g\Psi_{2} \,\Psi_{3} }}} \right)\mu \,g\Psi_{2} \,\Psi_{3}  \Psi_{2} \,\Psi_{3} \,g\mu }}{{\Psi_{1} \,\Psi_{2} \beta_{E} \pi + \,\Psi_{1} \,\delta \,\beta_{{I_{A} }} \pi  \,(\Psi_{1} \,\delta \,f\beta_{{I_{A} }} \pi + \Psi_{2} \,g\mu )}}, \\ & \Rightarrow \quad \, \lambda_{6} =  \frac{{R_{0} \mu \,g\Psi_{2} \,\Psi_{3}  \Psi_{2} \,\Psi_{3} \,g\mu }}{{\Psi_{1} \,\Psi_{2} \beta_{E} \pi + \,\Psi_{1} \,\delta \,\beta_{{I_{A} }} \pi  \,(\Psi_{1} \,\delta \,f\beta_{{I_{A} }} \pi + \Psi_{2} \,g\mu )}}, \\ & \Rightarrow \quad \lambda_{6} =  \frac{{\mu \,g\Psi_{2} \,\Psi_{3} (R_{C}  \,1)}}{{\Psi_{1} \,\Psi_{2} \beta_{E} \pi + \,\Psi_{1} \,\delta \,\beta_{{I_{A} }} \pi  \,(\Psi_{1} \,\delta \,f\beta_{{I_{A} }} \pi + \Psi_{2} \,g\mu )}}, \\ \end{aligned}\).and \(\lambda_{i} (i = 6) < 0\) whenever \(R_{0} < 1\) and \(\Psi_{1} \,\delta \,f\beta_{{I_{A} }} \pi + \Psi_{2} \,g\mu > \Psi_{1} \,\Psi_{2} \beta_{E} \pi + \,\Psi_{1} \,\delta \,\beta_{{I_{A} }} \pi\), so that the diseasefree equilibrium of system (1) is locally asymptotically stable for \(R_{0} < 1\), whereas for \(R_{0} > 1\) at least one or more eigenvalues are positive and P_{0} is unstable.\(\square\)
Local stability of the endemic equilibrium of (1)
Theorem 3
The system (1) has endemic equilibrium \(P_{1}\) that is locally asymptotically stable for \(R_{0} > 1\).
Proof
Routh–Hurwitz criterion will be used to prove the local stability of the endemic equilibrium of system (1).
Define the Jacobian matrix \(J_{P1}\) for DEE by
where \( \Psi_{1} = \left( {1  c_{f} } \right)\left( {1  c_{d} } \right),  \Psi_{5} = \beta_{E} E^{*} + \beta_{IA} I_{A}^{*} + \beta_{IS} I_{S}^{*} ,g =  f\delta  \left( {1  f} \right)\delta  \mu\) let \(T =  \Psi_{1} .\Psi_{5}  \mu , \, B =  \Psi_{1} .(\beta_{E} .S^{*} ), \, H =  \Psi_{1} .(\beta_{IA} .S^{*} ), \, D =  \Psi_{1} .(\beta_{IS} .S^{*} ), \, E = \Psi_{1} .\Psi_{5} , \,\)\(F = \Psi_{1} .(\beta_{E} .S^{*} )  \delta  \mu , \, W = (1  f)\delta\) and the characteristics equation for \(J_{P1}\) is
with
\(a_{0} = \left( {\left( {\left( {EB  FT} \right)\Psi_{3} + \left( D \right)\delta \,f\left( {E  T} \right)} \right)\Psi_{2} + \Psi_{3} \,HW\left( {E  T} \right)} \right)\mu \,\Psi_{4}\).
From the foregoing, Hurwitz test of necessary but not sufficient condition for a characteristic equation is considered for stability as all coefficients of the characteristics polynomial (3) exist and can be shown to have the same sign. Though not indicated here due to length of the computation, Routh’s stability criterion of necessary and sufficient condition for stability shows that elements of first column of the Routh’s array have the same sign for the characteristic equation represented by (3) under certain conditions using Maple software; consequently all its roots have negative real parts. Hence, Routh–Hurwitz criterion shows that the endemic equilibrium P_{1} is locally stable for R_{0} > 1.
Parameter estimation and data fitting
Following the approach used by Adedire and Ndam [27], the parameters that indicate the characteristics of the virus are obtained from the literature. Other parameters that represent nonpharmaceutical infections control strategies such as the use of face masks, social distancing as well as those representing circumstantial effects are estimated from the data obtained from NCDC with modest assumptions using parameter estimation process. The choice of the incubation period follows the research of Chen et al. [37] and Rothana and Byrareddy [40] thus \(f = \frac{1}{5.2}\).
The recovery of infected population usually ranges between 3 and 30 days; hence, we made assumption of average recovery period of 15 days for the detected quarantined asymptomatic and symptomatic infectious population with \(\gamma = \frac{1}{15}\).
For recovery rates \(r_{1}\) and \(r_{2}\) from undetected asymptomatic and symptomatic infectious population \(I_{A}\) and \(I_{S}\), we chose \(r_{1} = r_{2}\) representing an average recovery period of 20 days. The assumption of higher recovery rate in the detected quarantined asymptomatic and symptomatic population is based on possibility of special care through administration of certain multivitamins and antibiotics which could significantly boost the immune system of the infected individuals.
Other values of the parameters for nonpharmaceutical control strategies and disease transmission rates together with some initial conditions of model (1) are fitted to the active daily COVID19 cases and cumulative daily COVID19 cases with approach of nonlinear least squares technique using Maple software. The total population of Plateau State is estimated as N(t) = 4,200,442, and the first COVID19 incidence case in Plateau State was reported on 23 April 2020; the initial conditions of model (1) are estimated as \(S(0) \, = \, 4200120\), R(0) = 0, Q(0) = 1 with \(I_{A} (0),I_{S} (0)\) and \(E(0)\) estimated from cumulative daily data of COVID19 obtained from NCDC. It should be noted that the assumption that the data obtained from NCDC are below the actual number of infected cases is adopted due to a few number of tests carried out within the period of this study. Other parameters used in this study are shown in Table 3.
It should also be emphasized here that low initial values are used in parameter estimation process for parameters representing the social distancing and use of face masks. Low parameter assumptions are due to observable low compliance of the population of study to the adoption of strict rules on social distancing and use of face masks. This observable evidence is common among religious gatherings based on their beliefs and also in the market places where rules of social distancing and use of facemasks are not strictly enforced. These initial values representing social distancing and use of facemasks were set low in the fitting of the model Eq. (1) to data using nonlinear least squares method.
Results and discussion
The objective of this section is to present results and discuss investigation of the impact of nonpharmaceutical control measures such as the use of face masks and social distancing on the transmission dynamics of COVID19 infectious disease in Plateau State Nigeria. Simulation of model (1) is also carried out in this section using Maple software. The numerical method used to obtain the solution of the proposed infectious disease model (1) is based on Fehlberg fourth–fifthorder Runge–Kutta method with degree four interpolant. It is a robust numerical method of order O(h^{4}) having an error estimator with order O(h^{5}). The numerical method has the capacity to estimate and control errors using higherorder method which permits automatic adaptive step size h. Further details on the robustness of the numerical method used can be obtained from [42]. The software was used also to obtain equilibrium points of the proposed governing infectious disease model (1) with constraints set for \(S(t) \ge 0,E(t) \ge 0,I_{A} (t) \ge 0,I_{S} (t) \ge 0,Q(t) \ge 0,R(t) \ge 0,t \to \infty\) in line with positivity of solutions of model (1) which has been proved in theorem 1 and seven equilibrium points were obtained.
From the results indicated in Fig. 3, the representation of actual data of the active daily COVID19 infectious individuals (red) is compared with simulations of undetected asymptomatic population \(I_{A}\) (gold), undetected symptomatic population \(I_{S}\) (blue) and detected quarantined asymptomatic and symptomatic infectious population Q (cyan) in Plateau State Nigeria.
The values shown in Fig. 3 mean that the model fits well with the actual data and that the peak of the model is at about 213 days, 211 days and 224 days, respectively, for very actively infectious compartments \(I_{A}\), \(I_{S}\) and Q using the parameters \(c_{f} = 0.1\) and \(c_{d} = 0.2\). The values of the nonpharmaceutical control measures are indicative of low compliance of the Plateau State residents to the use of face masks and social distancing, and Fig. 3 shows that about 476,455 undetected asymptomatic infectious population \(I_{A}\)(gold), 92,168 undetected symptomatic infectious population \(I_{S}\) (blue) and 83,801 detected quarantined asymptomatic and symptomatic population Q(t) (cyan) in Plateau State Nigeria have become infectious at the peak of COVID19.
With nonpharmaceutical control measures \(c_{f} = 0.15\), \(c_{d} = 0.2\) in Fig. 4, results indicate that the peak shifts to the right in range of about 231 days and 244 days from the date of first incidence case when about 421,619 undetected asymptomatic infectious population \(I_{A}\)(gold), 81,243 undetected symptomatic infectious population \(I_{S}\) (blue) and 74,752 detected quarantined asymptomatic and symptomatic population Q(t) (cyan) have been infected at peak of the disease.
Results in Fig. 4 also show that a slight improvement in the attitude of the population towards the use of face masks \(c_{f}\) from a value of 0.1 to 0.15 representing about 5% increase in the use of face masks could have an impact towards reducing the population of infectious individuals to about 364,864, 70,009 and 65,231 against the 421,619, 81,243 and 74,752 people observed at the peaks in Fig. 3 for the compartments \(I_{A} (t)\),\(I_{S} (t)\) and \(Q(t)\), respectively. Further impact of face masks usage is further observed in Fig. 5 with the value of \(c_{f}\) set to 0.2 representing a further 5% increase from the value used in the previous Fig. 4.
Results from Fig. 5 indicate that with \(c_{f} = 0.2\), about 364,864, 70,009 and 65,231 individuals from undetected asymptomatic population, undetected symptomatic population and detected quarantined individuals from fully infectious compartments \(I_{A} (t)\), \(I_{S} (t)\) and Q(t) have contracted the disease and become infectious in the range of about 252 days and 266 days from the date of first case of COVID19 in 23 April 2020.
Effect of social distancing is also investigated as indicated in Figs. 6, 7 and 8. Results from Fig. 6 for social distancing value \(c_{d} = 0.25\) suggest about 414,544 undetected asymptomatic infectious population \(I_{A}\) (gold), 79,844 undetected symptomatic infectious population \(I_{S}\) (blue) and 73,602 detected quarantined asymptomatic and symptomatic population Q(t) (cyan) at the peak of the COVID19 incidence. However, a further increase of about 5% in observing social distancing within the population causes a further reduction in the COVID19 incidence rate to about 350,451 undetected asymptomatic infectious population \(I_{A}\)(gold), 67,174 undetected symptomatic infectious population \(I_{S}\) (blue) and 62,780 detected quarantined asymptomatic and symptomatic population Q(t) (cyan) as indicated in Fig. 7.
Also from Fig. 8, results show a further reduction in the infectious individuals to about 284,516, 54,279 and 51,406 from 350,451, 67,174 and 62,780 obtained in Fig. 7 when the adherence to social distancing among the susceptible population is improved to a value \(c_{d} = 0.35\) for the undetected asymptomatic infectious population \(I_{A}\), undetected symptomatic infectious population \(I_{S}\) and quarantined asymptomatic and symptomatic infectious population Q(t), respectively.
The contour plot of the reproduction number of the model (1) is shown in Fig. 9.
The physical interpretation of basic reproduction number \(R_{0}\) lies in its ability to assess the transmissibility of the COVID19 infections. This means that if \(R_{0} < 1\), the outbreak of the disease will die down and the disease can be eradicated. On the other hand, if \(R_{0} > 1\), the outbreak of the disease will persist and endemic state will be attained. Also, \(R_{0} = 1\) represents the threshold level for determining the persistence level or otherwise of the disease [37, 43].
From Eq. 2, the basic reproduction is 2.3 which is greater than 1 at the baseline values \(c_{f} = 0.1\), \(c_{d} = 0.2\) and this means that the disease is endemic at the baseline values. Observation from Fig. 9 indicates the possibility of disease persistence with values above 1 at the baseline values. It shows that about 40% in the use of face masks and about 50% of social distancing could bring the basic reproduction number to about 0.96 which is just slightly below 1 and it is within the range necessary for eradication of the COVID19 in Plateau State. Figure 9 further indicates that if about 50% of the population use face masks and about 60% adopt social distancing, the basic reproduction number gives about 0.64 which further shows possibility that the disease may die out.
From Fig. 10, the results show \(S \, E \, I_{A} \, I_{S} \, Q \, R\) model together with the cumulative daily infectious individuals from NCDC (red). The population dynamics indicate rise in the rate of recovery and decline in the population of the susceptible community. The peak of the infectious classes is also indicated before equilibria states are attained in each of the classes at baseline values.
Conclusion
In this study, nonpharmaceutical control measures on the spread of COVID19 in Plateau State Nigeria are examined. The emphasis is on the use of face masks and social distancing. The results indicate that COVID19 in Plateau State tends to an endemic state at baseline values of control measures. This means that after the proposed infectious disease model (1) of COVID19 in this study shows prediction of epidemic peak, the level of infection reduces and gradually approaches an endemic state over a period of time. The implication of this is that Plateau State government should be ready to fight the coronavirus for a much longer period than the current transmission wave of the disease.
However, further analysis of results suggests that strict rules on the use of face masks above about 40% of the population and corresponding above 50% adherence to social distancing could as well bring down the basic reproduction number to a value below 1 necessary for disease eradication in Plateau State. Therefore, policymakers in the state need to intensify efforts towards raising the population of users of face masks. Also, compulsory social distancing at public gatherings should be enforced. This study covered a period when nonpharmaceutical control measures were used to curb the spread of COVID19 in Plateau State. However, vaccination of susceptible class of people has just begun, but the quantity of vaccines available is not yet sufficient to vaccinate majority of the population. Consequently, future researches could incorporate vaccine administration into the model when majority of the population of study have been vaccinated.
Availability of data and materials
The data sets used for this study are available at public domain of Nigeria Centre for Disease Control [NCDC] under the title: “An update of COVID19 outbreak in Nigeria”, web link url is at https://ncdc.gov.ng/diseases/sitreps/?cat=14&name=An%20update%20of%20COVID19%20outbreak%20in%20Nigeria. Accessed 22 April 2021.
Abbreviations
 EOC:

Emergency Operations Centres
 HBC:

Homebased care
 COVID19:

Coronavirus disease 2019
 NCDC:

Nigeria Centre for Disease Control
 WHO:

World Health Organization
 S E I_{A} I_{S} Q R:

Susceptible—LatentInfectious Asymptomatic—Infectious Symptomatic—Quarantined asymptomatic and symptomatic—Recovered
References
Shanafelt, T., Ripp, J., Trockel, M.: Understanding and addressing sources of anxiety among health care professionals during the COVID19 pandemic. J. Am. Med. Assoc. 323(21), 2133–2134 (2020). https://doi.org/10.1001/jama.2020.5893
Muhammad, T.S.: Breaking: Lalong declares curfew in Plateau over COVID19” Blueprint Newspaper. https://www.blueprint.ng/breakinglalongdeclarescurfewinplateauovercovid19/ (2020). Accessed 12 Mar 2021.
Dauda, A.S., Ibrahim, D.A., James, A., Teplong, J.I., Jonah, Z.D., Sunday, D., Haa, N.T.: Covid19 lockdown, locus of control and state anxiety among residents of Plateau State Nigeria. Br. J. Psychol. Res. 8(2), 41–48 (2020). (ISSN:20550863)
National Bureau of Statistics: Demographic statistics bulletin, Federal Republic of Nigeria. Vol 3, www.nigerianstat.gov.ng (2017). Accessed 01 Sept 2020.
Theresa, N.: Schools remain shut as Plateau suspends total lockdown. The Vanguard Newspaper. https://vanguardngr.com/2020/06/schoolsremainshutasplateaususpendstotallockdown/amp/ (2020). Accessed 12 Apr 2021.
World Health Organization [WHO]: Demystifying COVID19: WHO supports Plateau State to engage traditional leaders in grass root sensitization. https://www.afro.who.int/news/demystifyingcovid19whosupportsplateaustateengagetraditionalleadersgrassroots (2020). Accessed 21 Mar 2021.
World Health Organization [WHO]: COVID19WHO boosts Plateau State’s home based care strategy.https://www.afro.who.int/photostory/covid19whoboostsplateaustateshomebasedcarestrategy (2020). Accessed 23 Mar 2021.
Nigeria Centre for Disease Control [NCDC]: An update of COVID19 outbreak in Nigeria. https://ncdc.gov.ng/diseases/sitreps/?cat=14&name=An%20update%20of%20COVID19%20outbreak%20in%20Nigeria (2021). Accessed 22 Apr 2021.
Isa, A.A.: Covid19: Governor Lalong lifts lock down imposed on Plateau. The Guardian, Nigeria. https://guardian.ng/news/covid19governorlalongliftslockdownimposedonplateau/amp/ (2020). Accessed 11 Mar 2021.
Iboi, E., Sharomi, O.O., Ngonghala, C., Gumel, A.B.: Mathematical modeling and analysis of COVID19 pandemic in Nigeria (2020). MedRxiv. https://doi.org/10.1101/2020.05.22.20110387v2.articleinfo
Ndam, J.N.: Modelling the impacts of lockdown and isolation on the eradication of COVID19. Biomath, 9, 1–8, 29107 (2020). https://doi.org/10.11145/j.biomath.2020.09.107
Okuonghae, D., Omame, A.: Analysis of a mathematical model for COVID19 population dynamics in Lagos, Nigeria. Chaos Solitons Fractals 139, 110032 (2020). https://doi.org/10.1016/j.chaos.2020.110032
Sweilam, N.H., AlMekhlafi, S.M., Baleanu, D.: A hybrid stochastic fractional order Coronavirus (2019 nCov) mathematical model. Chaos Solitons Fractals 145, 110762 (2021). https://doi.org/10.1016/j.chaos.2021.110762
Sweilam, N.H., AlMekhlafi, S.M., Almutairi, A., Baleanu, D.: A hybrid fractional COVID 19 model with general population mask use: numerical treatments. Alex. Eng. J. 60, 3219–3232 (2021). https://doi.org/10.1016/j.aej.2021.01.057
Sweilam, N.H., AlMekhlafi, S.M., Baleanu, D.: A hybrid fractional optimal control for a novel Coronavirus (2019 nCov) mathematical model. J. Adv. Res. 32, 149–160 (2021). https://doi.org/10.1016/j.jare.2020.08.006
Babaei, A., Jafari, H., Banihashemi, S., Ahmadi, M.: Mathematical analysis of a stochastic model for spread of Coronavirus. Chaos Solitons Fractals 145, 110788 (2021). https://doi.org/10.1016/j.chaos.2021.110788
Shahrear, P., Rahman, S.M.S., Nahid, M.M.H.: Prediction and mathematical analysis of the outbreak of coronavirus (COVID 19) in Bangladesh. RM 10, 100145 (2021). https://doi.org/10.1016/j.rinam.2021.100145
Zhang, L., Ullah, S., Al Alwan, B., Alshehri, A., Sumelka, W.: Mathematical assessment of constant and timedependent control measures on the dynamics of the novel coronavirus: an application of optimal control theory. Results Phys. 31, 104971 (2021). https://doi.org/10.1016/j.rinp.2021.104971
Park, J., Chang, W., Choi, B.: An interaction NeymanScott point process model for coronavirus disease19. Spatial Stat. 47, 100561 (2022). https://doi.org/10.1016/j.spasta.2021.100561
Cai, S., Zhang, T., Robin, C., Sawyer, C., Rice, W., Smith, L.E., Amlot, R., Rubin, G.J., Yardley, L., Hickman, M., Oliver, I., Lambert, H.: Learning about COVID 19 across borders: public health information and adherence among international traveler to the UK. Public Health 203, 9–14 (2022). https://doi.org/10.1016/j.puhe.2021.11.015
Munster, V.J., Koopmans, M., Doremalen, N.V., Riel, D.V., Wit, E.D.: A novel coronavirus emerging in China—key questions for impact assessment. N. Engl. J. Med. 382, 692–694 (2020)
Beesham, A.: Mathematical modelling of COVID19 in South Africa. Asian Pac. J. Trop. Med. 13(5), 235–236 (2020). https://doi.org/10.4103/19957645.283519
Park, J.Y., Han, M.S., Park, K.U., Kim, J.Y., Choi, E.H.: First pediatric case of coronavirus disease 2019 in Korea. J. Korean Med. Sci. 35(11), e124 (2020).
Sahin, A.R., Erdogan, A., Agaoglu, P.M., Dineri, Y., Cakirci, A.Y., Senel, M.E., Okyay, R.A., Tasdogan, A.M.: 2019 novel coronavirus (COVID19) outbreak: a review of the current literature. Eurasian J. Med. Oncol. 4(1), 1–7 (2020). https://doi.org/10.14744/ejmo.2020.12220
Liang, K.: Mathematical model of infection kinetics and its analysis for COVID19, SARS and MERS. J. Infect. Genet. Evol. 82, 104306 (2020). https://doi.org/10.1016/j.meegid.2020.104306
Ambrosio, B., AzizAlaoui, M.: On a coupled timedependent SIR models fitting with New York and NewJersey States COVID19 Data. Biology MDPI 9(6), 135 (2020). https://doi.org/10.3390/biology9060135,hal02883239
Adedire, O., Ndam, J.N.: A model of dual latency compartments for the transmission dynamics of COVID19 in Oyo State Nigeria. Eng. Appl. Sci. Lett. 4(1), 1–13 (2021). https://doi.org/10.30538/psrpeasl2021.0056
Alotaibi, H., Gepreel, K.A., Mohamed, M.S., Mahdy, A.M.S.: An approximate numerical method for mathematical and physical studies for covid19 models. Comput. Syst. Sci. Eng. 42(3), 1147–1163 (2022). https://doi.org/10.32604/csse.2022.020869
Gepreel, K.A., Mohamed, M.S., Alotaibi, H., Mahdy, A.M.S.: Dynamical behaviors of nonlinear coronavirus (COVID19) model with numerical studies. Comput. Mater. Contin. 67(1), 675–686 (2021). https://doi.org/10.32604/cmc.2021.012200
Gepreel, K.A., Mahdy, A.M.S., Mohamed, M.S., AlAmiri, A.: Reduced differential transform method for solving nonlinear biomathematics models. Comput. Mater. Contin. 61(3), 979–994 (2021). https://doi.org/10.32604/cmc.2019.07701
Mahdy, A.M.S., Mohamed, M.S., Lotfy, K., Alhazmic, M., ElBary, A.A., Raddadi, J.N.: Numerical solution and dynamical behaviors for solving fractional nonlinear Rubella ailment disease model. Results Phys. 24, 1–10, 104091 (2021)
Simos, T.E.: A Runge–Kutta Fehlberg method with phaselag of order infinity for initialvalue problems with oscillating solution. Comput. Math. Appl. 25(6), 95–101 (1993)
Paul, S., Mondal, S.P., Bhattacharya, P.: Numerical solution of Lotka Volterra prey predator model by using Runge–Kutta–Fehlberg method and Laplace Adomian decomposition method. Alex. Eng. J. 55(1), 613–617 (2016)
Handapangoda, C.C., Premaratne, M., Yeo, L., Friend, J.: Laguerre Runge–Kutta–Fehlberg method for simulating laser pulse propagation in biological tissue. IEEE J. Sel. Top. Quantum Electron. 14(1), 105–112 (2008)
Amer, Y.A., Mahdy, A.M.S., Shwayaa, R.T., Youssef, E.S.M.: Laplace transform method for solving nonlinear biochemical reaction model and nonlinear EmdenFowler systems. J. Eng. Appl. Sci. 13(17), 7388–7394 (2018)
Mahdy, A.M.S.: Numerical solutions for solving model timefractional Fokker–Planck equation. Numer. Methods Partial Differ. Equ. 37(2), 1120–1135 (2020)
Chen, T.M., Rui, J., Wang, Q.P., Zhao, Z.Y., Cui, J.A., Yin, L.: A mathematical model for simulating the phasebased transmissibility of a novel coronavirus. Infect. Dis. Poverty 9(24) (2020). https://doi.org/10.1186/s40249020006403
Heffernan, J., Smith, R., Wahl, L.: Perspectives on basic reproduction ratio. J. R. Soc. Interface 2, 281–293 (2005)
Trefethen, L.N., Bau, D.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (1997). ISBN 9780898713619
Rothana H.A., Byrareddy, S.N.: The epidemiology and pathogenesis of coronavirus diseases (COVID19) outbreak. Journal of Autoimmunne 109:102433 (2020).
Eikenberry, S.E., Mancuso, M., Iboi, E., Phan, T., Kostelich, E., Kuang, Y., Gumel, A.B.: To mask or not to mask: modeling the potential for face mask use by the general public to curtail the covid19 pandemic. Infect. Dis. Model. 5, 293–308 (2020)
Hairer, E., Norsett,.S., Wanner, G .: Solving ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993). ISBN 3540566708.
Chen, S., Yang, D., Liu, R., Zhao, J., Yang, K., Chen, T.: Estimating the transmissibility of hand, foot, and mouth disease by a dynamic model. Public Health 174, 42–48 (2019)
Acknowledgements
Not applicable.
Funding
The authors did not apply for any funding; consequently, no funding was received for this study.
Author information
Authors and Affiliations
Contributions
OA and JNN gathered the data and constructed the model equations. Both authors interpreted the results obtained from the model equations and approved the final manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Adedire, O., Ndam, J.N. Mathematical model of the spread of COVID19 in Plateau State, Nigeria. J Egypt Math Soc 30, 10 (2022). https://doi.org/10.1186/s4278702200144z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s4278702200144z
Keywords
 COVID19
 Coronavirus
 Disease
 Epidemic
 Plateau State
 Nigeria
Mathematics Subject Classification
 97M10
 93A30
 81T80
 34C60
 03C30
 00A71