key: cord-0787357-v5bmsln8 authors: Mushayabasa, Steady; Ngarakana-Gwasira, Ethel T.; Mushanyu, Josiah title: On the role of governmental action and individual reaction on COVID-19 dynamics in South Africa: A mathematical modelling study date: 2020-07-05 journal: Inform Med Unlocked DOI: 10.1016/j.imu.2020.100387 sha: 7a42a5631b784a94a278b0a30c5ac26dd8ba422f doc_id: 787357 cord_uid: v5bmsln8 Mathematical models proffer a rational basis to epidemiologists and policy makers on how, where and when to control an infectious disease. Through mathematical models one can explore and provide solutions to phenomena which are difficult to measure in the field. In this paper, a mathematical models has been used to explore the role of government and individuals reaction to the recent outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The proposed framework incorporates all the relevant biological factors as well as the effects of individual behavioral reaction and government action such as travel restrictions, social distancing, hospitalization, quarantine and hygiene measures. Understanding the dynamics of this highly contagious SARS-CoV-2, which at present does not have any therapy assist the policy makers on evaluating the effectiveness of the control measures currently being implemented. Moreover, policy makers can have insights on short-and-long term dynamics of the disease. The proposed conceptual framework was combined with data on cases of coronavirus disease (COVID-19) in South Africa, March 2020 to early May 2020. Overall, our work demonstrated optimal conditions necessary for the infection to die out as well as persist. In late December 2019, a novel strand of Coronavirus (SARS-CoV-2) was reported in Wuhan, a 23 central and crowded city of China (Li et al., 2020) . Subsequently, the World Health Organization 24 (WHO) has since officially termed this pandemic the Corona Virus Disease 2019 (COVID-19) [1] . 25 COVID-19 is a rapidly spreading infectious disease and continues to cause several outbreaks in yet infectious, in other words, they are incubating the disease; undetected asymptomatic patients A(t)-these are individuals who would have completed their incubation period and 83 can now transmit the infection. In general, these individuals cannot be recognized if they 84 are not confirmed by RT-PCR or other laboratory testing [14] . The model also includes 85 undetected clinically infected individuals and this population has been further subdivided 86 into two different classes-mild patients I m (t) and severe patients I s (t). In a recent study by 87 Wu and McGoogan [15] , it was noted that approximately 81% of the detected COVID-19 88 patients were of mild symptom and the remainder (about 20%) were severe. In addition, we 89 have also included detected and quarantined patients (both asymptomatic and symptomatic) 90 Q(t); as well as the deceased and successfully recovered individuals and these are respectively vealed that 6% of HCWs at these two institutions were infected with SARS-CoV2 [18] . Thus, 117 the force of infection λ(t), which represents the rate at which susceptible individuals become 118 infected is expressed as: (2) (iv) Susceptible individuals who contract the disease progress to the exposed/latent stage where 135 they will incubate the disease for an average period of σ −1 days. During this period these 136 individuals will not be capable of transmitting the disease. A couple of recent studies on 137 Wuhan COVID-19 dynamics suggest that the average (median) incubation period could be as 138 short as 4 days [4, 20] . (v) Upon the completion of the incubation period, we assume that a fraction f of the exposed 140 individuals move to the asymptomatic stage and the remainder (1 − f ) become symptomatic. Prior studies suggests that of the individuals who become symptomatic, more often there exists two classes, mild patients and severe patients [4, 15] . In particular, in a study on 81% of the cases were mild symptoms (without pneumonia or only mild pneumonia), 14% 145 were severe cases with difficulty breathing, and 5% were critical with respiratory failure, septic 146 shock, and/or multiple organ dysfunction or failure. Based on this assertion, we assume that (vii) Through RT-PCR or other laboratory testing, asymptotic patients are assumed to be detected 156 and quarantined at rate ω. Furthermore, we assume that after an average period of φ −1 days, Based on the aforementioned assumptions we have the following system of nonlinear ordinary differential equations: (3) Figure 1 illustrates the transition of individuals from one epidemiological state to another. denotes the remaining transfer terms are respectively given (at the disease-free equilibrium ) by; It follows that the spectral radius of model (3) which is the reproduction number of the model is 182 given by they are defined as follows From the expression of the reproduction number R eff , we can note the following: (i) Increasing the strength of governmental action to high levels (close to 100%), will lead to a huge 186 reduction in the magnitude of the reproduction number, and the weak actions ( values of α 187 close to zero) will not be effective to reduce the magnitude of the basic reproduction number. (ii) In the absence of governmental actions (α = p 1 = p 2 = ω = q = 0), the number of sec-189 ondary cases that will be generated by each infected individual will be higher compared to 190 when actions are even weak. The expression for the reproduction number in the absence of 191 intervention strategies is given by (iii) We can also observe that if quarantined individuals do not contribute to the generation of new 194 infections, that is, q = R q = 0, then the reproduction number will be less. Table 1 . Other parameters 221 values which were drawn from literature are presented in Table 2 . 222 We fitted the model to cumulative daily new infection data presented in the appendix. The Using baseline values on Table 1 To explore the impact of individual reaction and governmental action on combating COVID-19 245 disease in South Africa, we will simulate model (1) using parameter values in Table 1 Numerical illustration in Figure 8 depicts the effects of different levels of exposed individuals who 279 progress to asymptomatic and infectious stage. day lockdowns with relaxation may not be as effective as an initial 35 day lockdown followed by 296 successive 35 day lockdowns with relaxation. Furthermore, we observed that a detection rate of at 297 least 0.5 per day may lead to a significant reduction of the number of active cases. In addition, we 298 also noted that in the absence of intervention strategies the peak number of cases could be attained 299 around mid-June whereas in the presence of intervention strategies the peak will be attained around 300 the 23rd of August. Hence, we can deduce that the presence of intervention strategies may be 301 responsible for the delay in attaining the peak, thereby prompting policy makers ample time to 302 prepare for various and effective ways of managing the disease. The proposed framework could be of significant importance on understanding the transmission 304 and control of COVID-19. However, we acknowledge that there are several aspects of the disease 305 that are yet to be clearly unraveled, for instance, the duration one remains as an asypmtomatic 306 infectious patient is still debatable. In the event that additional information have been found, it 307 can be used to improve the framework. Here, we provide the dataset that was used in the study. All data are publicly available and can be 310 retrieved on www.worldometer.com. We considered the data ranging from 1 March 2020 to 3 May 311 The authors received no specific funding for this work. to their institution for non-financial support during the time when they were carrying out this study. 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EL 379 PAIS Clinical characteristics of 2019 novel coronavirus infection in China Reproduction number and sub-threshold endemic equi-384 libria for compartment models of disease transmission On the definition and the computation of the basic 387 reproduction ratio R 0 in models for infectious diseases in heterogeneous populations Coronavirus cases The authors are very grateful to the reviewers for their constructive suggestions and comments 316 which helped us to improve the manuscript significantly. In addition, the authors are also grateful The authors are very grateful to the reviewers for their constructive suggestions and comments which helped us to improve the manuscript significantly. In addition, the authors would like to thank their institutions for non-financial support during the time when they were carrying out this study. The authors declare that they have no competing interests.