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A Human-Pathogen Model for COVID-19 Outbreak

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A Human-Pathogen Model for COVID-19 Outbreak ISSN: 2474-3658 Ojiambo et al. J Infect Dis Epidemiol 2020, 6:184 DOI: 10.23937/2474-3658/1510184 Volume 6 | Issue 6 Journal of Open Access Infectious Diseases and Epidemiology RESEARCH ARTICLE A Human-Pathogen Model for COVID-19 Outbreak: Flattening Epidemic Curve in Kenya Viona Ojiambo1*, Mark Kimathi2, Samuel Mwalili1,3, Duncan Kioi Gathungu1 and Rachel Mbogo3 1Jomo Kenyatta University of Agriculture and Technology, Kenya 2Machakos University, Kenya Check for 3Strathmore University, Kenya updates *Corresponding authors: Viona Ojiambo, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000- 00200, Nairobi, Kenya es globally had exceeded sixteen million [3] with Africa Abstract recording approximately seven hundred thousand of This work introduces a deterministic model of COVID-19 those cases. The 1st case in Kenya was reported on the spread aimed at analyzing non-pharmaceutical interven- th tions in Kenya. The model accounts for symptomatic, as- 13 of March 2020 [4] and currently there are close to ymptomatic and environmental transmission. Using the twenty thousand confirmed cases [3]. SEIR (Susceptible-Exposed- Infected-Recovered) compart- mental model with additional component of the pathogen, The three main types of infectiousness of COVID-19 the dynamics of COVID-19 outbreak is simulated while fo- are asymptomatic, pre-asymptomatic and symptomatic. cussing on the impact of different control measures in the The incubation period for COVID-19 in the pre-asymp- reduction of the basic reproduction number. The resulting tomatic infectious group, which is the time between ex- system of ordinary differential equations (ODEs) is solved using a combination of fourth and fifth-order Runge-Kutta posure to the virus (becoming infected) and symptom methods. Simulation results indicate that non-pharmaceuti- onset, is on average 5-6 days, however it can be up to cal measures such as school closure, social distancing and 14 days. During this period some infected persons can movement restriction emphatically flatten the epidemic peak be contagious. In a symptomatic COVID-19 case, the curve by reducing the basic reproduction number. disease manifests itself with signs and symptoms. As- Keywords ymptomatic transmission refers to transmission of the Basic reproduction number, COVID-19, Intervention, SARS- virus from a person, who does not develop symptoms CoV-2, SEIR-P, Simulation [5]. The basic reproductive number (R0), defined as the average number of secondary infections produced by a Introduction typical case of an infection in a population where every- Corona Virus Disease (COVID-19) is an infectious one is susceptible [6], is affected by the rate of contacts disease that is caused by Severe Acute Respiratory Syn- in the host population, the probability of infection being drome Coronavirus 2 (SARS-CoV-2). The first case was transmitted during contact and the duration of infec- R . reported in main-land China, City of Wuhan, Hubei on tiousness. The COVID-19 0 in Kenya ranges from 1 78 29th December 2019 [1]. Later on, the novel disease to 3.46 [7]. The outbreak is thought to spread through a spread to countries in contact with China resulting in population via direct contact with infectious individuals World Health Organization (WHO) declaring it as a Pub- [8] or cross transmission through pathogen contamina- lic Health Emergency of International Concern (PHEIC) tion in the environment. Specifically, the virus is primar- on 30 January 2020 World Health Organisation [2]. As of ily spread between people during close contact, often 27th July 2020 the number of reported confirmed cas- via small droplets produced by coughing, sneezing, or Citation: Ojiambo V, Kimathi M, Mwalili S, Kioi D, Mbogo R (2020) A Human-Pathogen Model for COVID-19 Outbreak: Flattening Epidemic Curve in Kenya. J Infect Dis Epidemiol 6:184. doi. org/10.23937/2474-3658/1510184 Accepted: December 29, 2020: Published: December 31, 2020 Copyright: © 2020 Ojiambo V, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Ojiambo et al. J Infect Dis Epidemiol 2020, 6:184 • Page 1 of 8 • DOI: 10.23937/2474-3658/1510184 ISSN: 2474-3658 talking [5,6,9,10,11]. After breathing out these droplets sults shall inform health experts in their research of effi- usually fall to the ground or on to surfaces rather than cient and effective ways of flattening the curve. remain in the air over long distances [6,7,9]. People may also become infected by touching a contaminated Materials and Methods surface and then touching their eyes, nose, or mouth In this study a human-pathogen SEIR-P model for [9,10]. The virus can survive on surfaces for up to 72 COVID-19 outbreak is formulated. The model considers hours [11]. a short-time period thereby ignoring the demographics of births and natural death. The total human popula- This kind of transmission has had adverse social and economic impact because all segments of the popu- tion is subdivided into the susceptible human beings S t E t lation are significantly affected. The disease is partic- ( ), the exposed human beings ( ), the asymptomatic infectious population I (t), the symptomatic infectious ularly detrimental to vulnerable social groups such as A population I (t), the critical infectious population I (t), older age-groups, persons with disabilities and people M C with other underlying diseases. If the pandemic is not the hospitalized population H(t) in ordinary wards and properly addressed through policy, the social crisis gen- U(t) in Intensive Care Unit (ICU), the fatalities D(t) and erated from COVID-19 could also increase inequality, the recovered population R(t). The pathogen vector in exclusion, discrimination and global unemployment in the environment is denoted by P(t). the medium and long term United Nations: Department The model in Figure 1 culminates to a ten dimension- of Economic and Social Affairs Inclusion [12]. Inorder al system of ordinary differential equations as follows; to achieve suitable control measures of COVID-19, a dS β SP ββSI(++ I I )() SU + H Susceptible-Exposed-Infected-Recovered (SEIR) mathe- =−−1 23AM C− , matical prediction model is required. Prediction trends dt N N N from the model are largely influenced by policy and so- dE β SP ββSI(++ I I )() SU + H cial responsibility of a given country [13]. These trends =++−1 23AM C ωE, enable us to understand how SARS-CoV-2 could spread dt N N N across a given population and inform control measures dI A =δω Ε−( τ + γ ),I that might mitigate future transmissions [11]. Nonphar- dt A A AA maceutical interventions based on sustained physical distancing have a strong potential to lower and flatten dIM =−−(1δA δω C )EI − τ MM , the epidemic peak in the absence of a vaccine [14]. Pre- dt mature and sudden lifting of such interventions could dI lead to an earlier secondary peak [15]. C =δωEI −+( τ γ ), (2.1) dt C C CC An epidemiological study was done on a hu- man-pathogen model that incorporated individuals dH =τAAI + τ M I M + τ CC IU + φ −+( ξλHH + γ ), H with robust immunity in its compartments; the model dt examined human behavior during the disease outbreak, dU such as ignorance of social distancing so as to advise =ξHU −+( φλU ), health care workers (HCW) [16]. It was established that dt nonpharmaceutical interventions play a major role in dD flattening the epidemic peak. =λHUI ++ λλ, dt H U CC According to the WHO/China Joint Mission Report, 80% of laboratory-confirmed cases in China up to 20 dR =γγHHI + AA, February 2020 have had mild to moderate disease in- dt cluding both non-pneumonia and pneumonia cases; dP whilst 13.8% developed severe disease and 6.1% devel- =η12(IAM ++ I I C )( + ηµ UH +− )P P . oped to a critical stage requiring intensive care [9]. It dt is therefore paramount to come up with an epidemio- with the initial conditions: logical model that factors in hospitalization of the mild S(0) > 0, E(0) > 0, I (0) = 2, I (0) = 2, I (0) = 2, to moderate cases and severity of the disease transmis- A M C sion. H(0) = 0, U(0) = 0, D(0) = 0, R(0) = 0, P(0) = 5000. (2.2) β SP β SI()++ I I β SU()+ H The terms 1 , 2 AM C and 3 de- In this study we explore a human-pathogen model NN N mathematical model with infectious groups, hospital- note the rate at which the susceptible individuals S(t), ized population, human population in intensive care gets infected by the pathogens from environment (P(t)), unit and the fatalities. The study further investigates the infectious humans (IA(t), IM(t) and IC(t)), and hospitalized effects of lifting non-pharmaceutical interventions such humans (U(t),H(t)) respectively. A quarter of reported as school closure, physical distancing and lockdown on cases of COVID-19 are mainly asymptomatic infectious the model. It is envisaged that the dynamics of the re- IA(t), while the larger group is mild infectiousI M(t), with a Ojiambo et al. J Infect Dis Epidemiol 2020, 6:184 • Page 2 of 8 • DOI: 10.23937/2474-3658/1510184 ISSN: 2474-3658 γA IA Human Population IA δAωE τA IA β1 SP β2 S(IA +IM +IC ) β3 S(U+H) N + N + N (1 − δA − δC)ωE τM IM γH H S E IM H R λH H τC IC ξ H φU δCωE D λUU IC U λC IC µp P P η1 (IA + IM + IC) + η2(U + H) Pathogens (SARS-CoV-2) Figure 1: Compartmental model of the SEIR model of transmission of COVID-19.
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