key: cord-0882463-w77w6x5z authors: Mbabazi, F. K.; Yahaya, G.; Awichi, R.; Olupot, P. O.; Rwahwire, S.; Biira, S.; Luboobi, L. S. title: A Mathematical Model Approach for Prevention and Intervention Measures of the COVID-19 Pandemic in Uganda date: 2020-05-11 journal: nan DOI: 10.1101/2020.05.08.20095067 sha: 38e8343c1894b79471d3c61b976fbc96dfa78aeb doc_id: 882463 cord_uid: w77w6x5z The human-infecting corona virus disease (COVID-19) caused by the novel severe acute respiratory syndrome corona virus 2 (SARS-CoV-2) was declared a global pandemic on March 11th, 2020. Current human deaths due to the infection have raised the threat globally with only 1 African country free of Virus (Lesotho) as of May 6th, 2020. Different countries have adopted different interventions at different stages of the outbreak, with social distancing being the first option while lock down the preferred option for flattening the curve at the peak of the pandemic. Lock down is aimed at adherence to social distancing, preserve the health system and improve survival. We propose a Susceptible-Exposed-Infected-Expected recoveries (SEIR) mathematical model to study the impact of a variety of prevention and control strategies Uganda has applied since the eruption of the pandemic in the country. We analyze the model using available data to find the infection-free, endemic/infection steady states and the basic reproduction number. In addition, a sensitivity analysis done shows that the transmission rate and the rate at which persons acquire the virus, have a positive influence on the basic reproduction number. On other hand the rate of evacuation by rescue ambulance greatly reduces the reproduction number. The results have potential to inform the impact and effect of early strict interventions including lock down in resource limited settings and social distancing Corona Virus Disease 2019 (COVID- 19) , first discovered in Wuhan City, Hubei 2 Province, China on December 31st 2019 [7] , has established itself as the most 3 devastating global pandemic todate. The disease has not respected borders, 4 socio-economic developments of countries or states, and personal status. COVID-19 5 caused by Severe Acute Respiratory Syndrome Corona Virus 2 (SARS-CoV-2) has 6 spread worldwide [1] . Even in countries and states with very high levels of emergency 7 response preparedness, COVID-19 has had early and projected ramifications in all areas 8 of life including health, economy, education, travel, development and security. Globally, 9 the number of cases confirmed to the disease has surpassed 3.5 million people with over 10 248,313 deaths (CFR: 7% ) and 1,157,014 recoveries [5] . The evolution of the current 11 pandemic has dis-proportionally affected various countries. The top most affected 12 countries both in confirmed cases and mortality burden include: USA, Spain, Italy, 13 France, UK and Germany. With USA showing the highest death toll 68,602 persons 14 (with 5.8% Case fatality rate), as of May 4th, 2020 [5] . In Africa, an estimated 44,483 15 confirmed cases; 1,801 death, with case fatality rate of 4% and 14,921 recoveries have 16 occurred, as of May 4th, 2020 [23] . Countries that have not reported COVID-19 cases 17 as of, May 4th, 2020 include: Kiribati, Lesotho, Marshall Islands, Micronesia, Nauru, 18 North Korea, Palau, Samoa, Solomon Islands, Tonga, Turkmenistan, Tuvalu, 19 Vanuatu [25] . The pandemic however, has both direct and indirect effects and 20 ramifications in all sectors of life including health, economy, trade, travel, education and 21 governance. 22 In Uganda, the index case was confirmed on March 21, 2020 [4] . Despite immediate 23 lock down and intense public health interventions including contact tracing, the country 24 has registered eighty-nine (89) cases as of May 4th, 2020 [5] . Majority of these cases are 25 imported cases including recent truck drivers from the East African region. The total 26 number of foreign truck drivers who have tested positive for COVID-19 is thirty (30), of 27 these nineteen have returned to their respective countries whereas eleven are admitted 28 at different hospitals in Uganda [6] . The community transmission through contacts has 29 emerged and the extent of which remains unknown since many of those who traveled updates as of May 4th, 2020 indicate that a total of thirty nine thousand, two hundred 36 thirty two (39,232) persons have been tested, of whom eighty nine (89) are confirmed 37 cases, eight hundred sixty one (861) have been discharged from institutional quarantine, 38 four hundred forty six (446) are under institutional quarantine, one thousand three 39 hundred two (1,302) are contacts listed, eight hundred eight (808) are under follow up, 40 141 are under self quarantine, eighteen (18) are active cases and fifty two (52) have fully 41 recovered following successful treatment [3, 6] . The distribution of COVID-19 confirmed 42 cases by residence is shown in [6] . and Newspapers are not affordable to the majority. This calls for the need to have 53 awareness programs extended to the rural areas in order to educate the population 54 about the spread of the COVID-19 virus, its prevention and control measures. This 55 would limit the number of exposed and infected persons in the country. The country is implementing a model of early lock down and fractional testing of 57 high risk populations including travelers and contacts of confirmed cases. This is as 58 opposed to models in China, Europe and USA where the lock down followed 59 unprecedented number of cases and deaths. The only similar model of the developed 60 world to that employed by the Uganda government is the Greece model. Most African 61 countries including those in the East African Region have also followed early lock down. 62 The biggest percentage of Uganda population is rural population and may be at the risk 63 of contracting the disease. Awareness by mass media, are limited by existing resources 64 and other socioeconomic factors, and it is generally difficult for these awareness 65 programs to be disseminated to the whole host population. 66 Mathematical models have been used during the outbreak of COVID-19 in China, Italy and other countries to give direction to policy and decision makers in government 68 institutions. The commonly used model is the SEIR and include works of [12, 13, 18, 19] . 69 A study done by Rovetta [20] has used the SEIR model to predict and inform 70 governments of different countries about the COVID-19 pandemic. The models have 71 been modified by Hang et al. [14] , Zhu and Zhu [15] , Wan et al [16] and Cao et al. [17] 72 to include the asymptomatic classes, symptomatic classes, quarantine population, self 73 isolation and death classes in order to assess the impact of the disease and predict the 74 epidemic in the populations. In our model [8] designed for the early in-country outbreak of COVID-19, we 76 predicted how the rate at which COVID-19 would spread in the country without 77 prevention and intervention measures. Approximately one hundred twenty seven (126) 78 persons were predicted to have contracted the disease in two weeks and four thousand, 79 three hundred sixty nine (4, 370) persons to have contracted the disease in a month if no 80 prevention and intervention measures are put in place (see Fig 2) . We have since 81 developed a hypothesis based on these data and model. We hypothesize that with . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 11, 2020 sub-population divided into aware persons S a (t) and unaware persons S u (t) at time t, 99 the exposed/quarantined persons (individuals with a travel history) E(t) and infected 100 persons (infectious with disease symptoms) I(t), the infected persons on treatment 101 expected to recover R(t) at time t. If the tracing of contact is considered, a fraction p of 102 persons exposed to COVID-19, is quarantined. The quarantined persons can either 103 transfer to the infected compartment or to aware susceptible compartment S a (t) 104 depending on whether infectious or not. 105 We assume Uganda to be a closed system with a constant population 106 N = 45, 395, 554 during the epidemic and the exposed population initially to consist all 107 returnees E = 18, 128. The unaware population is increased through aware persons 108 loosing memory about information on prevention and intervention measures. The 109 unaware persons get infected either in contact with infected individuals or with infected 110 objects and transfer to the exposed population at a mass incidence rate bS u I, where . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . https://doi.org/10.1101/2020.05.08.20095067 doi: medRxiv preprint The aware sub-population is increased by persons who implement the existing 116 prevention and control measures learned from media. The aware population may get in 117 contact with infected objects and persons because human tend to forget due to some 118 social factors and transfer to the exposed population. The exposed population initially 119 consists of persons with a travel history that were checked on arrival, quarantined for 120 fourteen days (incubation period), however some persons never went through the 121 process of checking and mixed with the community. The quarantined persons in the 122 exposed population are tested after the incubation period of 14 days, if tested negative 123 a proportion (1 − p)δE transfer to the aware population and a proportion pδE transfer 124 to the infected class, with p the proportion of persons that acquire infection and δ the 125 incubation rate. The infected population is assumed to decrease at a rate r + q i + q s , where r is the 127 rate of evacuation by rescue ambulances, q s is the rate at which individuals with mild 128 symptoms isolate themselves from the population, q i is the rate at which infected . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . In this study, upon giving a transition diagram, assumptions and description of 145 model parameters, the ordinary differential equations for the population change of each 146 sub-population are stated as Parameter estimation Basic data used in this study were obtained from the daily epidemic announcements by 151 media and Ministry of Health. Release of cumulative data about COVID-19 in terms of 152 confirmed cases of infected, critical, total deaths, recovered and cumulative tested 153 cases [5] . We assume persons that had a travel history to be quarantined in institutional 154 centers for 14 days. Information from official websites and previous studies done as of 155 April 27, 2020. The initial population conditions with regard to system (1) for Uganda are set to: latency period is assumed to be Erlang distributed with mean 5.2 days (SD 3.7) [11] . 159 We estimate the removal rate of infected persons on treatment expected to recover = 160 φ = 1 nr k i=1 p2 Nc , 161 with k i=1 p 2 = cumulative recoveries, N c = total of confirmed cases, n r = number 162 of days for which an individual takes to recover, φ = 55 21×97 = 0.027. The average disease duration for COVID-19 is 21 days, hence the rate of recovery is 164 given by γ = 1 21 = 0.04761 per day. The steady states and the reproduction number, R n0 The endemic steady state 172 The endemic steady state is got by setting the R.H.S of Eq (1) to zero. That is: Hence the endemic steady state E * q = (S * u , S * a , E * , I * , R * ) on a set Ω is given as, where 174 With 175 e = β(d + q + l + h), a 2 = β(d + k)I * We note that, S * u is biologically feasible provided 176 (ζM + e)(ρ + β(d + k)) > ρ, β(d + k)I * < 1 − ρ. Whereas 177 S * a is biologically feasible provided The biological meaning of this endemic state is that the disease establishes itself in the 179 population and persists for a long period. . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . https://doi.org/10.1101/2020.05.08.20095067 doi: medRxiv preprint infection is likely to propagate and persist in the population. The basic reproduction 186 number is a vital threshold in modeling infectious diseases that show the threat of an 187 infectious pathogen with respect to the epidemic spread. The magnitude of R n0 is 188 significant in determining the severity of the disease, and help to draw plans and design 189 control strategies. Since COVID-19 epidemic is in it's early stage of spread, we assume 190 S 0 u (0), S 0 a (0) ∈ S 0 to be near the infection-free steady state value S 0 u (0), S 0 a (0), and 191 approximating differential equations of the exposed and infected classes to a linear 192 system: with Eq (3), the linearization has been separated into two parts, i.e. first matrix 195 represents infection rates and the second matrix represents a combination of transition 196 rates and growth rate. Let G =the matrix of infection rates and U =the matrix combination of transition 198 rates and growth rate. Such that We find the next generation matrix GU −1 . Then the spectral radius of the product of 201 G and U −1 to be the reproduction number [9] . From Eqs (4) . . Eq (5) Using initial parameter values in Table 2 , the reproduction number is estimated to be 209 R n0 = 0.468. The reason of this geometric mean during the early stages of . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . https://doi.org/10.1101/2020.05.08.20095067 doi: medRxiv preprint The sign of elasticity index explains whether R n0 increases (positive sign) or reduces 225 (negative sign) with the parameter while the magnitude establishes the relative 226 significance of the parameter (see Table 3 ). Such indices give direction to decision 227 /policy makers on important parameters to be targeted for cost effectiveness and 228 practical control strategies. . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . https://doi.org/10.1101/2020.05.08.20095067 doi: medRxiv preprint Numerical results and discussion 245 We apply the Runge Kutta fourth and fifth order to solve system (1) with the help of 246 MATLAB. . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. . CC-BY-ND 4.0 International license It is made available under a 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 May 11, 2020. Regarding COVID-19 pandemic situation in Uganda, we proposed an SEIR epidemic 282 model that incorporated prevention and intervention measures. This research illustrates 283 capabilities of the SEIR model in predicting and therefore informing the general public 284 about the impact of COVID-19 using a mathematical approach. The results obtained 285 will be used to predict, inform and monitor the progress, timing and magnitude of the 286 COVID-19 pandemic in Uganda. We computed the reproductive number and it worked out as R n0 = 0.468. We note 288 that R n0 is less than unity, thus forecast that several strategies in combination There is need for collaborative effort from citizens especially truck drivers and 295 neighbours from Eastern African region in order to combat COVID-19 pandemic. In 296 addition, there should be focus on strict inland mediation and awareness at borders 297 including nearby villages in order to reduce on exogenous imported cases. It is important to ensure fast detection, awareness, treatment and sufficient medical 299 supplies are maintained. It is also important that persons with mild symptoms are 300 maintained in institutional facilities. 301 We recommend that the Sub-Saharan countries including East Africa should adopt 302 the model used to construct reliable intervention strategies to eliminate COVID-19 303 pandemic. The question why Severe Acute Respiratory Syndrome Corona Virus 2 305 (SARS-CoV-2) re-emerges after 1 year and 2 months, shall be answered by the model 306 we intend to embark on soon. The research team intends to further conduct empirical studies in our local 308 communities in order to inform the public about the impact of COVID-19 especially on 309 prevention and intervention measures in Uganda. The stigma faced by recovered 310 persons calls for special attention. There is need to inform and sensitize the community 311 on how to cope and live with the victims. The authors declare no conflicts of interest. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 11, 2020. 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