A mathematical model is presented which is designed to be applicable to Coronavirus case data from any cou ntry and which can serve as a tool to carry out predictive analysis and sensitivity analysis with the goal of minimizing virus related deaths. The model has been successfully applied to Covid- 19 case data in five countries – China, South Korea, Iran, Ital y and the USA. The model is versatile and can be applied to other countries and regions as well. Key words: Coronavirus, Covid 19, modelling, mathematical , infectious disease , data analysis INTRODUCTION Mathematical modelling of spread ing of infectious diseases is a well -established field [1,2]. Many infectio us diseases such as Hepatitis C, HIV and Lyme disease have been extensively studied using mathematical models. There have been a surge of recent efforts in modelling Coronavirus spreading as r eport ed in the literature [3,4 ]. In this paper we present a mathematical model which analyzes Covid -19 case data (daily values of number of cases, deaths and recovered) from five countries with very different case outcomes. The model fits the Covid -19 case data for these countries very well. The model also enables the reader to make near term predictions and sensitivity analysis for any country. The Covid -19 virus has spread rapidly through many countries in the world in the first three months of 2020 causing widespread deaths. The same virus with the same infectious and fatality properties however had different number of casualties in different countries. It is important to have a mathematical model that can explain the case data from various countries and that uses parameters which can be measured and which can be used to do sensitivity and what if analyses. This is the motivation behind the model presented in this paper. METHODOLOGY In this section we study the propagation of a new infectious disea se through an isolated country’s population. With some simple modifications, the procedure can be applied to recurring outbreak of the same virus and also to the normal state in which there is population movement among countries . Let n be the number of days since the onset of virus infection . Figure 1 is a schematic of our model. © Santanu Basu 2 of 12 In incubation (#days = j Number = In)With mild symptoms (#days = k Number = Sn)With acute conditions (#days = m Number = An)Deaths (Number = Dn)Recovered (Number = Rn) Daily infection rate = Tn Symbols: daily recovery rate = γ , daily death rate = d , daily testing rate = bSnapshot on day n since virus outbreakTn(In-1+ Qn-1 Sn-1+ An-1) Asymptomatic (#days = k Number = Qn)γ1 d1 d2γ2Patients survived after m daysCountry’s healthy population = P1stday of virus infection on n=1 1stday of symptoms on n=j+1 1stday of acute condition on n=j+k+1 Figure 1. Schematic of the mathematical model for the virus infected cases The entire population of virus infect ed people is classified in six categories: I – people in whom the virus is in incubation Q – people who are infected with the virus but who do not exhibit any symptoms S – patients w ho exhibit mild symptoms of the viral infection A – patients who are in acute condition D – people who have died as a resul t of viral infection R – people who have recovered from the viral infection On any day n, the total number of people who have been infected with the virus is N n, where Nn = In + Qn + Sn + An + D n + R n (1) We denote the incubation period for the virus in number of days by j. The model assumes each infected person spends j days in incubation. We assume that after th e incubation period of j days, each infected person either is asymptomatic but carries the virus for another k number of days, or shows mild symptoms of the disease for k number of days . For each patient during each day in this mild symptom phase, the probabilities of recovery and death are denoted by γ 1 and d1 per day. At the end of k number of days, the patients who have not recovered or died are assumed to enter the acute condition phase as shown in figure 1 . We assume that the acute condition period las ts for m number of days . For each patient du ring each day in the acute condition phase, the probabilities of recovery and death are denoted by γ2 and d2 per day. The patients who survive m days in acute condition are assumed to be fully recovered. © Santanu Basu 3 of 12 Let I n be the total population of infected people who are still in the incubation period on day n. Let I ni be the population of those infected people in the incubation period on day n who have been infected for i number of days (i ranges from 1 to j) . In = In1 + In2 + …..+I nj = ∑i Ini (2) Let Sn be the total population of infected people who are showing mild symptoms of the disease on day n. Let S ni be the population on day n of those infected patients who have been showing mild symptoms for i numb er of days (i ranges from 1 to k) . Sn = Sn1 + Sn2 + …..+ Snk = ∑i Sni (3) Let Qn be the total population of infected people who have gone through the incubation period and in whom the virus still lives for k number of days, but who are asymptomatic of the disease. Let Qni be the population on day n of those asym ptomatic people who have been carrying the virus for j+i number of days (i ranges from 1 to k) . Qn = Qn1 + Qn2 + …..+ Qnk = ∑i Qni (4) Let An be the tot al population of patients who are in acute condition on day n. Let Ani be the population on day n of those infected patients who have been in acute condition for i number of days (i ranges from 1 to m) . An = An1 + An2 + …..+ Anm = ∑i Ani (5) The total number of people who are currently infected on day n and who are able to transmit the virus is (I n + S n + Qn +An). Out of this population, only (S n + A n) number of people show outward signs of the disease. Ironically in the beginning of the outbreak, when (S n + A n) is small, the problem is sometimes underestimate d without realizing that the disease is spreading exponentially everyday by the entire infected population (I n + S n + Q n +A n). The time to take swift and decisive preventive measures is when (S n + A n) is small . We denote the rate of transmission of the virus from an infected person to a hea lthy person per day by Τn. Let N 0 be the number of infected people on the first day. For example , the number s of infected people on the first two days of the outbreak are: I1 = N0 (6) I2 = N0 (1+Τ1) (7) In general, the number of newly infected people on day n is given by In1 = Τn-1 (In-1 + S n-1 + Qn-1 + An-1) (8) A fraction, f a, of the newly infected people is asymptomatic of the disease. An estimate of f a = 30% has been given in reference [9 ], which has been used in this model calculations . The nu mber of new patients with mild symptoms on day n is given by Sn1 = (1 - fa) In-1j (9) Similarly the number of new asymptomatic patients on day n is given by Qn1 = fa In-1j (10) The number of new patients with acute condition on day n is given by © Santanu Basu 4 of 12 An1 = (1 – γ1 - d1) Sn-1k (11) The number s of patients who recover and die from mild symptoms on day n are given by γ 1 Sn and d1 Sn respectively. Similarly t he number s of patients who recover and die from acute conditions on day n are given by γ2 An and d2 An. Sni+1 = (1 – γ1 - d1) Sn-1i for 0 ≤ i ≤ k-1, n>1 (12) Qni+1 = Q n-1i for 0 ≤ i ≤ k-1, n>1 (13) Ani+1 = (1 – γ2 - d2) A n-1i for 0 ≤ i ≤ k-1, n>1 (14) The number of deaths caused by the virus on day n is given by Dn = d1 Sn + d2 An (15) The number of pati ents who recover from the disease on day n is given by Rn = γ1 Sn + γ2 An + (1 – d2) Anm (16) Before the first person showing symptoms of the disease caused by the new virus and shortly thereafter up to day n 1, the transmission rate Τ n is assumed to be c onstant at an initial value Τ 0: Τn = Τ0 for 1 ≤ n ≤ n1 (17) Typically the authorities take steps to prevent the spreading of the disease such as patient isolation, testing, contact tracing and social distancing. As a result Τ n decreases with time. The model is capable of including actual time dependent transmission data, however at this time such data for the Coronavirus case is not available. A reasonable approximation is exponential decay of the rate of transmission Τn from infected persons to the g eneral population. Let G be the number of days it takes for Τ n to decrease by 63% of Τ0, and let Τ f be the long term steady state value of Τ n. Τn = Τf + (Τ0 - Τf) exp ( - (n – n1) / G) for n1 < n (18) Three parameter s n1, G and Τ f were found to significa ntly affect the number of cases from the virus. The data analyses carried out in this paper clearly showed that the actual case data for countries with smaller number of cases could be fit well with smaller G and n 1. In this paper, we will limit the analysis to the first outbreak of the viru s. It may also happen that social distancing measures are lifted due to economic necessity before the virus is completely extinguished within the population, which would cause Τ n to rise again and a possible recurrence. The recurrence can be modelled in future using the methodology given in this paper. Let σn per day per person be the rate of testing of the entire population, P of the country including the infected people with out symptoms (I n + Q n) on any day n. Let bn per day per person be the testing rate of patients with symptoms. T he active number of infected cases which are confirmed by testing and reported can be expressed as : Bn = σn (In + Qn) + b n (Sn + A n) (18) Total cumulative number of reported cases, C n is given by Cn = Bn + ∑i bi (Di + R i) for 1 ≤ i ≤ n (19) In the model, the testing rates σ n and bn are considered time dependent as follows : © Santanu Basu 5 of 12 bn = 0 for 1 ≤ n ≤ j (20) bn = b0 for j+1 ≤ n ≤ n 2 (21) bn = b0 + ((1 - b0) (n – n2) / nb) for n2 ≤ n ≤ n 2 + n b (22) bn = 1 for n > n2 + n b (23) and, σn = 0 for 1 ≤ n ≤ n2 + n b (24) σn = σf ((n – (n2 + n b)) / H) for n2 + n b ≤ n ≤ n2 + n b + H (25) σn = σf for n > n 2 + n b + H (26) The rationale for choosing differen t testing rates is as follows. During the first j number of incubation days of the outbreak there is no testing and bn = 0. During the first outbreak of the virus, it takes several days to identify the new virus and to develop an initial test , during whi ch time , the testing rate of the patients showing symptoms is low, denoted by b 0. Let nb be the number of days to ramp up the testing of the patients with symptoms to 100% final testing rate . The testing rate σ n for the general population including thos e who are infected and not showing symptoms is assumed to be zero until n = n 2 + n b during which time testing rate is being ramped up for those showing symptoms. Let H be number of days it takes to build up the testing capability to test the general popul ation at a final rate of σ f . This recognizes that large scale production of testing supplies and development of infrastructure and resources for administering of tests in large numbers take time. It is to be noted that there is daily variation and pro bability distribution function for each of these model parameters. The time dependent data on these parameters for the new Coronavirus in each country are not available yet. When the values of these parameters become available on a later date, these can be inserted in the model. Fo r near real time analysis of the case data, we make realistic initial assumptions as described in detail in this section and then numerically iterate on them to fit the published data on Covid -19 infect ed cases for each country . Model parameters We test this model to explain the number of Covid -19 cases in a small number of countries at first – China, South Korea, Iran, Italy and USA . We collected the data on number of cases, deaths and number of recovered people from reliable sources [5 -8] for th e period Jan uary 1st, 2020 to March 31st, 2020. The actual data used in this paper is included in table I in the appendix. There are seven teen variable parameters in the model description. These are j, k, m, N o, To, Tf, n1, G, b0, n2, nb, γ1, γ2, d1 and d2, σf and H. We start with reasonable assumptions for each parameter and then use the published data on total number of cases, total number of deaths and total number of recovered patients for each country to iteratively optimi ze the parameter values using the mathematical model . We also make sure that the values for the parameter are reasonable. For example, for j, k and m, we make use of the information presented on the WHO website [5 ], which states “Most estimates of the in cubation period for COVID -19 range from 1- 14 days, most commonly around five days. These estimates will be updated as more data become available.” Another such source of information is a published study done at the World Health Organization, “Using available preliminary data, the median time from onset to clinical recovery for mild cases is © Santanu Basu 6 of 12 approximately 2 weeks and is 3- 6 weeks for patients with severe or critical disease. Preliminary data suggests that the time period from onset to the development of sev ere disease, including hypoxia, is 1 week. Among patients who have died, the time from symptom onset to outcome ranges from 2 -8 weeks.” [6]. We carry out multivariate optimization to fit the data for each cou ntry between Jan 1 and Mar 31 of 2020. Starting with reasonable initial conditions in the model, iterations are made in the algorithm until the residual errors in fitting t he published data during Jan 1 and Mar 31, 2020 for each country are minimized. At that point the final values of all parameters f or this predictive model for each country become available. This can be considered as the training set. Once the optimization is completed, the country -specific parameters are determined and the model becomes powerful to yield information such as (1) pre dicted number of future cases in a country or (2) sensitivity analysis such as the effect on number of casualties on days to implement social distancing. The model can also be used to quantify if there is a recurrence of the disease once social distancing measures are lifted. We discuss th e results in the next section. RESULTS AND DISCUSSION We used the data on number of cases, deaths and number of recovered people from reliable sources [5-8] for the period January 1st, 2020 to March 31st, 2020. The actu al data used in this paper is included in table I in the appendix. The results are shown for China, South Korea, Iran, Italy and USA in figures 2 -6. The same model fits the data for five countries in three continents remarkably well as shown in the figur es. As stated before, w e have done multivariable optimization and iterated on seventeen parameters to obtain the best fit between the model prediction and the actual data for each country. Figure 2 shows the comparison of the actual data and the model p redictions for China in five categories : (a) total cases, (b) total deaths, (c) total recovered, (d) daily new cases and (e) daily deaths. In the literature [5,6,8 ], the daily case data for China became available since Jan 22, 2020. Before that time, the data has been scarce. On Dec 19, 40 cases were reported in China, 59 cases were reported on Jan 9, 200 cases were reported on Jan 20 and 571 cases were reported on Jan 22. The 1st death was reported on Jan 11, a total of 2 deaths were reported on Jan 17 which increased to 3 on Jan 20 and to 17 on Jan 22. We used the mathematical model to best fit the case data for China between Jan 1 and Mar 31, 2020 and to obtain the values for parameters used in the model for China. For example, t he best fit for th e data with minimum least squares error was obtained by using initial number of infected people, N 0 = 10, initial transmission factor T 0 = 0.54, Jan 15 start date for transmission factor decrease and a transmission decay rate of 1/G, where G= 6 days . The spike in the actual daily case on Feb 12 could not be fitted with model, as it was a one- time event in the actual data set . It should also be mentioned that the case data fr om China was revised on April 17th which is outside the data set used in this pape r. © Santanu Basu 7 of 12 a. Total Cases b. Total Deaths c. Total Recovered d. Daily New Cases e. Daily DeathsCountry: China Data Model Prediction020000400006000080000100000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20China Total Cases S. Basu 200404●Data __ Model 05000100001500020000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20China Daily Cases S. Basu 200404●Data __ Model010002000300040005000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20China Deaths S. Basu 200404●Data __ Model 020000400006000080000100000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20China Recovered Cases S. Basu 200404●Data __ Model 0100200 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20China Daily Deaths S. Basu 200404●Data __ Model Figure 2. Data and model predictions for Covid -19 cases in China Figure 3 shows the comparison of the actual data and the model predictions for South Korea. The daily case data for South Korea has been avai lable in literature [5,6,8 ] since Feb 15. There were 28 total cases on Feb 15 and the first Covid -19 death was on Feb 20. The model fitted the South Korea case data with minimum error with N0 = 3, T0 = 0.47 , Feb 12 start date for transmission factor decrease and a transmission decay rate of 1/G, where G= 6 days. Swift action to reduce the transmission facto r when the infected population was small is what we believe to be the primary reason for very low Covid -19 deaths in South Korea and which is supported by the model . a. Total Cases b. Total Deaths c. Total Recovered d. Daily New Cases e. Daily DeathsCountry : South Korea Data Model Prediction050001000015000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20South Korea Total Cases S. Basu 200404●Data __ Model 050010001500 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20South Korea Daily Cases S. Basu 200404●Data __ Model0100200300 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20South Korea Deaths S. Basu 200404●Data __ Model 01020 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20South Korea Daily Deaths S. Basu 200404●Data __ Model0500010000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20South Korea Recovered Cases S. Basu 200404●Data __ Model Figure 3. Data and mode l predictions for Covid -19 cases in South Korea Figure 4 shows the comparison between the model prediction and the actual data for Iran . The daily case data for Iran has been available in [5,6,8 ] since Feb 19. There were 2 total cases and equal number of deaths on Feb 19 . The model fitted the Iran Covid -19 case data with T 0 = 0.31, Feb 20 start date for transmission factor decrease and a transmission decay rate of 1/G, where G= 15. For Iran with higher number of casualties than South Korea, the transm ission rate decay factor, © Santanu Basu 8 of 12 G is higher (15 as compared to 6) . Figure 6 shows the model predicting (solid curve in figure 4(d)) the number of daily new cases to level off at the end of March 2020 timeframe. a. Total Cases b. Total Deaths c. Total Recovered d. Daily New Cases e. Daily DeathsCountry : Iran Data Model Prediction0200004000060000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Iran Total Cases S. Basu 200404●Data __ Model 025005000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Iran Daily Cases S. Basu 200404●Data __ Model020004000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Iran Deaths S. Basu 200404●Data __ Model 0100200300 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Iran Daily Deaths S. Basu 200404●Data __ Model0100002000030000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Iran Recovered Cases S. Basu 200404●Data __ Model Figure 4. Data and model predictions for Covid -19 cases in Iran Figure 5 shows the comparison between the Covid -19 case data and the model prediction for Italy. The first reported Covid -19 case in Italy was on Feb 15, at nearly the same time as South Korea and Iran. There were 3 total cases on Feb 19 and the first death was on Feb 21 . The model fitted the Italy Coronavirus case data with T 0 = 0.38, Feb 16 start date for transmission factor decrease and a transmission decay rate of 1/G, where G= 16. Comparing case data for Italy and South Korea, we find that the model parameter, G is 16 for Italy as compared to G=6 for South Korea. Transmission containment actions were swifter in South Korea. As shown in figure 7, the model predicts the number of daily new cases reaching a maximum at the end of March 2020 timeframe (solid curve in figure 5(d)) . a. Total Cases b. Total Deaths c. Total Recovered d. Daily New Cases e. Daily DeathsCountry : Italy Data Model Prediction050000100000150000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Italy Total Cases S. Basu 200404●Data __ Model 0500010000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Italy Daily Cases S. Basu 200404●Data __ Model050001000015000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Italy Deaths S. Basu 200404●Data __ Model 050010001500 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Italy Daily Deaths S. Basu 200404●Data __ Model0100002000030000 1-Jan-20 31-Jan-20 1-Mar-20 31-Mar-20 30-Apr-20Italy Recovered Cases S. Basu 200404●Data __ Model Figure 5. Data and model predictions for Covid -19 cases in Italy Figure 6 shows the comparison of the actual data and the model predictions for USA. The daily case data for USA was obtained from