key: cord-1024916-gc0szicq authors: Benedict, A. C.; Emmanuel, O.; Tenuche, B.; William-Denteh, W. title: Mathematical Analysis of COVID-19 Transmission Dynamics. A Case Study with Nigeria date: 2020-10-23 journal: nan DOI: 10.1101/2020.10.20.20216473 sha: 6893a1c1cf039d0934bf0526f3fa6844b019e0c8 doc_id: 1024916 cord_uid: gc0szicq Abstract: In this article, we formulated a mathematical model for the spread of the COVID-19 disease and we introduced quarantined and isolated compartments. The next generation matrix method was adopted to compute the basic reproduction number in order to assess the transmission dynamics of the COVID-19 deadly disease. Stability analysis of the disease free equilibrium is investigated based on the basic reproduction number and the result shows that it is locally and asymptotically stable for less than 1. Numerical calculation of the basic reproduction number revealed that which means that the disease can be eradicated from Nigeria. The study shows that isolation, quarantine and other government policies like social distancing and lockdown are the best approaches to control the pernicious nature of COVID-19 pandemic. Key words: mathematical model, basic reproduction number, isolation, quarantine, COVID-19, computer simulation Abstract: In this article, we formulated a mathematical model for the spread of the COVID-19 disease and we introduced quarantined and isolated compartments. The next generation matrix method was adopted to compute the basic reproduction number ሺ ܴ ሻ in order to assess the transmission dynamics of the COVID-19 deadly disease. Stability analysis of the disease free equilibrium is investigated based on the basic reproduction number and the result shows that it is locally and asymptotically stable for ܴ less than 1. Numerical calculation of the basic reproduction number revealed that confirmed cases to 18,827, followed by Abuja (5, 526) , Oyo (3, 226) , Plateau (3, 192 The Covid-19 outbreak is seen as the greatest global threat worldwide because it spreads globally and millions of confirmed cases, accompanied by thousands deaths globally. As of 14 September 2020, WHO reports 28,637,952 confirmed cumulative cases with 917,417 deaths in the world and 18 September 2020, total of 56,956 confirmed cases, total of 48,305 discharged and total deaths of 1,094 occurred in Nigeria [9] . Covid-19 is spread primarily from human to human through direct or indirect close proximity with contaminated surface, objects or with secretion droplets, saliva or respiratory secretions expelled from mouth or nose of an infected individual coughs, sings, sneezes or speaks. The sign and symptoms of Covid-19 usually start two to 14 days after contact with infected person or contaminated surface, this period of time after exposure and before having symptoms is known as incubation period. Common sign and symptoms of Covid-19 include, cough, fever, tiredness, shortness of breath, sore throat etc. Older people with underlying medical challenges like serious heart diseases (heart failure, coronary arterial disease or cardionryopatty), cancer, type 2 diabetes, sickle cell disease, high blood pressure [14] . Mathematical modelling and its computer simulation has attracted the attention of many researchers because of its striking applications in the area of prediction and control of infectious diseases. Many researchers have proposed mathematical models for prediction of infectious disease. [15] , Proposed Mathematical Assessment of the transmission dynamics of HIV/AIDs with treatment effects. Their model comprises of four compartments considering treatment class as the control measure. [12] , proposed a mathematical model for simulating the phasebased transmissibility of a novel Covid-19. In their study, they developed a Bats-Hosts Reservoir-people transmission network model for simulating the potential transmission from source of infection to the human infection. The basic reproduction number from reserviour to person analyzed was estimated to be 2.30 and 3.58 from person to person. Their model revealed that the transmissibility of SARS-COV-2 was higher than the middle east respiratory syndrome within the Middle East countries. Our main contribution in this article is addition of new compartments such as quarantine, isolation andtdeath which are not considered in most existing models, the addition All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Figure 1 gives the schematic diagram of the model. The model is developed under the following mathematical assumptions The recovered individuals do not become susceptible to the disease again. Potion of the infected individuals and isolated individuals can die due to disease. Every susceptible individual has equal chance of being infected. New recruit enters the population through birth and migration. The quarantined individuals can be infected when they come in contact with contaminated surface or infected person. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; Figure 1 Schematic diagram for the model Based on the mathematical assumptions and the schematic diagram, we formulate the following differential equations. In this section we carry out the model analysis such that the equilibrium state, disease free equilibrium, and endemic equilibrium in order to determine stability of the model. At the equilibrium point we have. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; https://doi.org/10.1101/2020.10.20.20216473 doi: medRxiv preprint At the disease free equilibrium The disease free equilibrium of our model equation is The endemic state of our model equations is given below In any disease model, the basic reproduction number ሺ ܴ ሻ is highly significant since it is the most relevant epidemiological distinguishing feature to determine the nature of disease. We therefore, use the next generation matrix approach to evaluate basic reproduction number of the model. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The basic reproduction number is the spectral radius of the matrix which is given by All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; https://doi.org/10.1101/2020.10.20.20216473 doi: medRxiv preprint The Jacobian matrix for the model is given below The characteristic equation is All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; Since all the eigen values are negative, we conclude that the system is locally and asymptotically stable around its disease free equilibrium. We perform numerical simulation of the model in order to compare the results of our model with the real data obtained from published articles by WHO and other researchers, considering a total of 56,956 confirmed cases in Nigeria. Figures 1,2,3 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; https://doi.org/10.1101/2020.10.20.20216473 doi: medRxiv preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; https://doi.org/10.1101/2020.10.20.20216473 doi: medRxiv preprint g r a p h o f e x p o s e d p o p u l a t i o n a g a i n s t t i m e , f r o m t h e g r a p h , t h e e x p o s e d p o p u l a t i o n d e c r e a s e s a s t i m e i n c r e a s e s , t h e e x p o s e d p o p u l a t i o n d e c r e a s e s t o z e r o i n d i c a t i n g t h a t w i t h t i m e t h e C O V I D -1 9 p a n d e m i c c a n b e c o n t r o l l e d i n N i g e r i a s i n c e t h e e x p o s e d p o p u l a t i o n t e n d s t o z e r (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; Fig (8) , is graph of death population against time in Nigeria, the population is at high rate when the outbreak occurs initially but later decreases with time showing that disease induced death tends to zero as time progresses. COVID-19 pandemic is the second world disaster after the World War II, government of Nigeria has taken some policies to control the catastrophic nature of the deadly disease in their own capacity. In this article, we have developed and analyzed a mathematical model which incorporates isolation and quarantine strategies as control measures. Analysis of the model shows that diseases free equilibrium is locally and asymptotic stable. The numerical simulation and analysis of the model revealed that COVID-19 pandemic can be eradicated from the population. [1] First case of corona virus disease confirmed in Nigeria. Nigeria Centre for Disease control. 28 February 2020. Retrieved. 10 March, 2020. [2] Update on COVID -19 Nigeria Centre for Disease and Control (2020) (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 23, 2020. ; https://doi.org/10.1101/2020.10.20.20216473 doi: medRxiv preprint Nigeria Responds to First Coronavirus Case in Sub -Saharan Africa Naming the coronavirus disease (COVID -19) and the virus that causes it. World Health Organization Novel Coronavirus -China World Health Organization (W.H.O) Nigeria records second case of coronavirus Samuel Oyekanmi COVID -19 update in Nigeria Coronavirus disease (COVID -19). Weekly Epidemiological update COVID -19 Dashboard by the Centre for systems science and engineering (CSSC) at John Hopkins University (JHU) Health and family welfare department A mathematical model for simulating the phase -based transmibility of a noval Coronavirus Transmission dynamics of 2019 novel corona virus Mathematical Assessment of the Transmission Dynamics of HIV Report of the WHO-China Joint Mission on Corona virus Disease Mathematical Analysis of Chickenpox Transmission Dynamics with Control Measures No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity