Modeling Covid-19 Pandemic Responses in Malaysia for the First 145 Days Duration (Jan. 25 – June 17

Abstract


Introduction
As of June 17, 2020, the total global covid-19 cases were 8,295,151 cases with recovery and fatality rates of 52.4 and 5.4%, respectively [1].The World Health Organization (WHO) had characterized covid-19 as pandemic on March 11, 2020 [2], and as of June 17, 2020, the pandemic has been affecting of more than 200 countries worldwide [1].With the initial name of 2109-nCoV, the virus is a positive, enveloped, and single-strand RNA [3].It has many similarities with the MERS-CoV (Middle East Respiratory Syndrome) and SARS-CoV (Severe Acute Respiratory Syndrome).The first suspected case in Wuhan, China was on December 8, 2019 [3].Wuhan of Hubei is the epicenter of the outbreak of 2019-nCoV or 2019 Novel Coronavirus , and specifically the disease is called Covid19.It was reported on Jan25, 2020 ( date coincided with the first covid-19 wave in Malaysia) that the epidemic had caused at least 41 fatalities ,and the virus had spread to other countries, including Singapore, Japan, the US, France, and Australia [4].
In Malaysia, the first wave occurred during the period of Jan 24 -Feb.26, 2020 with 22 cases including the daily zero case for the period of 11 days (Feb.16 -26, 2020) [5].In this wave, there were 12, 8, and 2 cases of those had a travel history to affected countries , close contacts , from a humanitarian mission, respectively, and by the end of wave 1, all of them were fully recovered and discharged [ 6].The second wave started from Feb., 27, 2020 [7].The covid-19 cases that initially low ( first wave) in number and principally sourced from importation had aggressively increased due to the localized infection starting in the late February to early March that partly contributed by the Tabligh gathering in Sri Petaling Mosque with 14,500 participants .From this point, the cases had reached to the figure of more than 2,500 cases at the end of March.Additionally, by March 16, 2020, all states and Federal territories had recorded Covid-19 pandemic infection [8,9].
The above observed exponential and alarming increasing trend of covid-19 cases, and if it is not well and appropriately controlled and mitigated, the nation economy and wellbeing of the people would be negatively affected.Thus, the interventions in containing and mitigating the covid-19 spread are being made by the Malaysian government through the enforcement of the movement control order (MCO) starting from March 18, 2020.The MCO has been enforced in phases, i.e., MCO1 ( March 18 -31, 2020), MCO2 ( April 1 -14, 2020), MCO3( April 15 -28, 2020), MCO4 ( April 29 -May 12, 2020), MCO5 ( May 13 -June 9,2020 : conditional MCO), and currently recovery MCO (RMCO: 10 June -August 31, 2020) [9].In MCO1, there are orders of prohibiting mass gathering; requirements of undergoing health check and 14-day self-quarantine for Malaysians returning from abroad; restricting the entry of tourists and foreign visitors; closing all kindergartens, schools, and pre-university institutions; closure of higher education institution; and the closure of all government and private premises with the exception of those providing essential services to the public.All above are fundamentally enforcing social distancing and the practices of 'new normal' ways of life.The MCO extensions are being enforced due to significant increase of cases over time and the possibility of cases to peak at Mid-April as predicted by the WHO.In the CMCO and RMCO, due to convincing lower daily cases, some relaxations of orders for reopening the business, and allowing more movements and activities with new normal ways [ 9,10], but these relaxations are still bound to the previous or updated standard operating procedures that principally and basically based on the social distancing and self-hygiene or personal-hygiene practices.
There must be a social and scientific basis, and professional opinion to be used in forming the form of intervention and relevant time duration or period in controlling and mitigating the pandemic.A part from qualitative tools, mathematical modeling could be used as guideline or tools in the planning and making decision related to any intervention to be made in mitigating certain disease epidemic for the benefit of the public health [11].In epidemic or pandemic of disease that especially caused by viral infection, the cases would be normally trended with nonlinear pattern and chaotic in nature.These can be seen from the scientific reports and publications in the infection of covid-19 [3], SARS [12], H1N1 [13], and Ebola [14].The viruses need hosts to grow (or replicate) in a given conducive environment where the hosts would be ranging from humans, domestic animals, and to natural ecosystem [15].As generally termed as microbial, the growth rate of the population size in the unlimited and limited resources would be proportional and a function of population size, respectively [16].Thus, it is assumed that the enforcements of serial MCO phases in Malaysia would lead into environment of limiting resources for the spread of covid-19 virus.Thus, by taking the equivalent of Covid-19 virus population size to covid-19 infected persons, the cumulative cases over days model is basically represented by the growth logistic function.Several workers have used this function or other exponential related models in describing epidemic/pandemic of viral diseases including covid-19 virus [3, 17, 18, 19, 20, 21, and 22].The applicability of logistic function is generally used in a quite high number of modeling works.In neural network study that implemented in artificial neural networks (NAA) with artificial neurons, the logistic function is generally used as an activation function in the NAA [23].
As of June 17, 2020 (145 days after the first positive diagnosed covid on Jan. 24, 2020), the Malaysia's recoveries and fatalities were 92% and 1.4 %, respectively; and the respective global figures were 52% and 5.4% [1].With the assumption of no additional significant new covid wave occurrence, the daily cases' curve would be seen as on the down-trend that observed within the enforced rehabilitation MCO phases towards reaching the covid-free or non-significant covid infestation condition.Thus, based on data of Jan. 25 -June 17(145 days), i.e., considering the pandemic has already been entering into the 'maturity stage', the study was firstly aimed at the applicability of the proposed serial layered-logistic growth function in determining mathematically the trends of covid-19 pandemic on cumulative cases, recoveries, fatalities, and active cases over days; and its related dynamic model.Secondly, the aim was to identify the information and parameter values that could be obtained or extracted through modeling technique in understanding the observed dynamic of the pandemic.With these identifications, specifically, the model would be used to determine that parameter value that linking all above models, and to determine the possibility of the occurrences of several cycles of wave during the study period.The information could also be used as guidelines for reallocation and remobilization of resources by the relevant authorities at postdate days of duration where the data were used for modeling.Thirdly, the aim was to forecast the quantitative information for days beyond the duration of data used in the study.
In overall, the study was aimed at observing the correctness and appropriateness of the enforced MCO phases in the covid-19 mitigation by overlying the model trends on the MCO zonings or phases.It was noted that the zero daily cases in wave 1 were 11 days [5] as opposed to the covid-19 virus incubation period of 14 days [24] and the required zero daily cases for 28 days [25].Additionally, the early phase of the cumulative cases trend in China [3] was quite resembling to that in Malaysia.Thus, it would be logical, this study had grouped waves 1 and 2 in Malaysia to be as one wave only.

Microbial population growth.
Theoretically, the rate of microbial population growth that hypothetically including viruses in certain culture/environment over time (   ) is proportional to and a function of population size under an unlimited and limited resources or environment, respectively [16].The resultant function is shown in Eq. (1).
Where, t=time, n=number of microbial or colony size,  and  = constant and population size, respectively.
The  and (1 -1  ) represent the quantitative intrinsic property and the sum total of environmental restriction/limitation, respectively [16].In this discussion, we would replace '' with ' ', i.e., it is a generally used for mathematical elaboration.If we define the microbial growth rate as

𝑑𝑦 𝑑𝑡
, then the relative growth rate (R) can be written as shown in Eq. ( 2).
By mathematical manipulation of Eq. ( 2) that based on the definition of relative growth rate (R), the resulted function is written in Eq. ( 3).
Where, R = k (1 - BY solving differential equation of Eq. ( 3), the growth size ( ) over time at instant time is given by Eq. ( 4) as shown below: When b=  , then  is the constant of integration [16], and the time constant () could be computed by , and it is basically a very small or minute number.The curve of Eq. ( 4) is usually called as the logistic or autocatalytic model [16].

(a) Model of cumulative cases over days
In a situation of covid-19 virus pandemic in Malaysia, it is of almost impossible in counting the virus.Thus, the numbers of infected patients (responses) or the covid-19 cases were used in the analysis.The data on cumulative covid-infected cases, recovery, fatality, and active cases; and daily cases over days as sourced from the daily updates by The Ministry of Health, Malaysia were used [26].The above Eq.( 4) was then used in modeling the responses (Covid-19 infected cases), i.e., by using y as the cumulative cases.
There are several non-linear models in describing growth, but in this analysis, Eq. ( 4) is used since it is the most popularly used model in describing numerically the growth of organisms.In this study, the data collected for the period of Jan 25 -Jun 17, 2020 were used.This data duration that equivalent to 145 days were observed to be in condition of the 'matured' stage of the pandemic, i.e., up to the region of the RMCO phase enforcement [26].
In making the model functioning for catering possible several number of serial-time-wise infestation cycles [or subsequent smaller cycle(s)], the model of Eq. ( 4) was extended in having the so called the -logistic function.For example, the bi-logistic model was used by Meyer in his work in the field of diffusion of social phenomena [27].The -logistic function used in predicting cumulative number of cases (   ) is shown in Eq. (5).
The   value was the number of days that showed the beginning of observed significant jth wave.The pj value is similar to 'IF' statement in mathematical operation but with the exception of without the presence of physical branching in one continuous mathematical function [28].Thus, the effective day durations for cycle 1 was t=( 1 +1 to  2 -1) days , and for cycle 2 was t=(  2 to  3 -1) days, for cycle 3, t= ( 3 to  4 -1) days, and finally for cycle j was t=(   to ∞ ) days.At cycle 1,  1 =0 since the analysis started from day=1.For cycles 2, 3 ,and j , the nets of relative days ( t -  ) would be 0 , and consequently the cyclej would have 'extremely' small effect (  ) of cyclej-1.In the analysis, mathematically Eq.( 5) would be in operation within the range of day=1 to days=∞.
The innovation of analytical method was made in having that at j=1 (first infection cycle), the yi would progress logistically from the point of   = 1 + 1 and it would be approaching the asymptotic value ( asymptotic_1 ) or  1 (→  1 ) when   approaching and beyond  2 days as computed by using  1 ,  1 ,  1 and  1 constants.Subsequently, within  2 [second cycle), the computation of   would be using values of constants of  2 ,  2 ,  2 , and  2 .Within the  2 cycle, if   < or = 2 , the yi → is  2 /(1 +  2 ) at  1 -elevated y-axis.Starting from point of   = 2 , the   would move in the 'new' logistic function and it would be approaching to  2 asymptotic value ( asymptotic_2 ) as   approaching and beyond  3 days.This  2 was an additional or partial asymptotic value, and thus the new asymptotic value or cases from these two cycles of infestation would be  1 + 2 .The above systematic computation would be similarly applied for the possible additional cycle occurrences.

(b) Daily cases vs. days model
By taking the derivative of Eq. ( 5), the predicted daily increment of cases [(   )  ] at  ℎ day was computed by model of Eq. ( 6).The subscripts were as shown in Eq. (5).
With the anticipation of several additional or cycles of occurrences of covid-19 pandemic in Malaysia, approaches of discussion in the cumulative cases vs. days were similarly applicable to the discussion of progressive movement of the daily cases versus days.One cycle would represent one 'bell shape' distribution of daily cases versus days.The predicted curve of Eq. ( 6) with  =  would have  serial 'bell shape' curves indicating the presence of  additional cycles of Covid-19 pandemic.The curve that had highest peak with widest tail-spread of density distribution would be the primary or main cycle among other smaller cycles.

(c) Inflection point and time constant
By considering on only non-layered logistic model [Eq.( 4)], the inflection point ( ) and time of inflection occurrence (   ) could be computed by functions of  = /2 and   = loge / , respectively.The inflection point is the point of the occurrence of changes between the increasing and decreasing daily rates in the cumulative cases over days.The daily rate (
In an exponential decay curve of y=yoe -bt , where yo is the maximum value, and y and t are the dependent variable and time, respectively, and thus, the reciprocal of constant k, i.e.,  = 1  is the time constant [29].At point when t =  (unit: t -1 ), then y=yoe -1 and yielding y=yo*0.368since e -1 =0.368.This illustrates the needed t value for reducing  value to 36.8% of   , and thus, this time constant could be used in measuring the speed of the decay.By mathematical manipulation, Eq. ( 4) could be rewritten as is time constant in reducing (   − 1) to 36.8% of  or relatively speaking of increasing y.In this study, the time constant relative to the  value would also be used descriptively in comparing the aggressiveness of infection of primary pandemic cycle to the subsequent serial additional cycle(s).

(d) Models of recovery and fatality.
The fractions of cumulative recoveries and fatalities over cumulative cases were used in modeling the recovery and fatality versus days.Due to the complementarity between recovery and fatality, and for having positive incremental percentages over days, the fraction were computed based on the total of partial asymptotic cases (  ) that sourced from the j cycles as shown in Eq. (7).
The models used for predicting the fractions of recovery (   ) and fatality (  ) over days were similar to the form and associated conditions of Eq. ( 5).The respective constants would then be   ,   ,   , and   ; and   ,   ,   , and   for recovery and fatality, respectively.The functions used in computing predicted accumulative recovery (   ) and fatality (   ) at  ℎ day are indicated in Eqs. ( 8) and ( 9), respectively.
(e) Model of active cases.
The cumulative active or remaining cases at  ℎ day was computed by deducting the cumulative recovery and fatality from the cumulative cases.The active cases are cases that still having covid-19 infection and they are probably contagious to other susceptible individuals.Thus, the predicted active cases at  ℎ day (  ) could be computed by Eq. ( 5) -Eq.( 8) -Eq.( 9).In full, the model could be rewritten as shown in Eq. (10). )} The respective subscripts in the model are as above indicated.

(f) Model fitting and additional analyses
In this study, the model fitness was analyzed by computing coefficient of determination (R 2 ).For models of Eqs. ( 5), (8), and ( 9), the R 2 values were computed by using the model and total sum of squares of the non-liner model.The regression technique of observed data vs. predicted data was used in computing R 2 for models of Eqs. ( 6) and (10).The one-way analysis of variance (ANOVA) and the LSD test were carried out for daily cases that sourced from importation, local Malaysian citizens, and local foreigners of the May 18 -June 17, 2020 data.This was carried out in determining the main causes of the additional cycle occurrence.
Data collection and statistical analysis tools.
Data (Jan.25 -Jun.17, 2020) that sourced from the Ministry of Health, Malaysia were used in this analysis [26].The data on cumulative cases, recovery, and fatality were used in computing the daily cases, recoveries, and fatalities.Additionally, the daily cases that based on importation, local Malaysian citizens, and local foreigners during May 18 -June 17, 2020 were also recorded.The statistical analyses (inclusive of the graphic presentations) were executed by writing SAS programming of SAS9.4 version statistical package [30].

Results and Discussions
Cumulative and daily cases As earlier stated, this analysis were made based on the positive covid-19 infection cases of cumulative cases, recoveries, and fatalities over days; and the associated daily cases.It was not based on the number of tests or screening made on the target groups or communities.Based on the collected data of Jan. 25 -Jun.17, 2020, Figure 1 shows the curve of cumulative cases over days that resulted from the model shown in Eq. ( 5) with its associated values of constants indicated in Table 1.The respective observed and predicted data over days are given in Table 2.The model was significantly fitted to the observed cases with R 2 = 99.9%(p <0.0001).
As up to Jun. 17, 2020, Figure 1 shows that the curve contained additional serial smaller cycles of infestation.At this point, we could say that the analysis shows the covid-19 pandemic in Malaysia was made up from primary cycle and two additional serial cycles of pandemic as the predicted outputs were highly fitted to the observed values.Based on the dynamic curve (daily cases curve), the left and right tails of cycle 2 are mixed with the cycle1 right-tail and cycle3 left-tail, respectively.In general, the primary cycle mainly occurred for the period of Jan. 25 -Apr.19, 2020, while the other two serial two additional cycles occurred during Apr.20 -May 17, 2020 and May 18 ,2020 and onwards, respectively.In this study, it was felt logical (due to the shown curvature) that the first or primary cycle ( j=1) was basically made up of initially declared waves that occurring during Jan. 25 -Feb.28 and Feb. 29 -Apr.19, 2020 with its partial asymptotic cases (  1 ) of 5,889 (Table 1).The respective partial   values for the second (j=2) and third (j=3) cycles were 968 and 1,643 cases, respectively.Thus, the   value of model (Eq.( 7)) was 8,500 cases.By inserting values in Table 1 into Eq.( 5), the tri-logistic model is explicitly written as shown in Eq. (11).The presence of these smaller additional cycles would be evidently seen in the following discussion that based on the predicted daily cases as the first derivative of Eq. ( 5) [Figure 2 and Eqs. ( 6and 12)] since the derivative function would be more sensitive in showing the dynamic of the pandemic [3].
where   = predicted cumulative cases at ith day, As shown by Figure 1 and Table 2, the observed cases were little bit higher at days=141 -145 (Jun.13 -17) than the predicted cumulative cases by 0.07% -0.49%.This would probably suggest that, in spite of in this rehabilitation MCO (RCMO), there are possibly still 'small 'amount of resources (environmental factors and hosts) available for Covid-19 infection and additionally with small import cases.Also, at this higher region of days, the number of cases of the local foreigner workers is quite 'significant'.Thus, the study suggests that the actions and intervention of MCO (RCMO) should be appropriately maintained during the pandemic.The enforcement of RMCO until August 31, 2020 is then appropriate.With the current effective covid mitigating (under RMCO), it is expected the cumulative cases over days would not much higher over the model   (8,500 cases) in the postdate days of the study period.As based on the respective y-axes in the common curve, the inflection points of the primary cycle (2,945 cases) was higher than those of the others two additional cycles (484 and 821 cases for cycles 2 and 3, respectively) (Table1).Based on the starting points the jth cycles, the above trend was also observed in the predicted days for the occurrence of inflection point (), where the respective values were 68.1 days (68.1 days from Jan. 25), 100.9 days (13.9 days from Apr. 28), and 125.8 days (5.9 days from May 15).At these points of days, the predicted maximum daily cases for cycles 1, 2, and 3 were 194.3, 78.7, and 89.1 cases day -1 , respectively (Figure 2 and Table 1).By using Table 1 and Eq. ( 6) that yielded from the first derivatives of Eq. ( 5), the tri-logistic related function of the predicted daily rate is explicitly written as shown in Eq. ( 12).The fitness of the model was significant with R 2 = 70.4% (n=144, p < 0.04).The fluctuation of the observed daily cases moved around along the predicted curve.The occurrences of one primary and two additional cycles in the pandemic dynamic were clearly indicated in (+.−.(  −) ) 2 (12) where (

𝑑𝑦 𝑑𝑡
)  = predicted daily cases at ith day, By taking the reciprocal of   , the day constant was computed (   = 1/  ).The   values for first, second, and third cycles were 7.58, 3.60, and 4.64 day -1 , respectively (Table 1).As based on the relation of   and   /  value, it shows that the cycles 2 and 3 were faster in reaching the respective   values that that of cycle 1, but the   values for cycles 2 and 3 were much lower than that of cycle 1.By considering simultaneously the values of   ,   ,   , and the respective predicted maximum daily cases, the primary cycle or cycle 1 was more aggressive in covid-19 pandemic than the other two cycles (Figures 1 and 2).
The first derivative was sufficient in indicating the sensitivity of the pandemic during the study period, and thus the second derivative in computing the dynamic acceleration and deceleration were not computed in this study.The middle cycle (cycle 2) was affected by the Cycles 1 and 2. The actions taken by the government (several stages of movement control order (MCO) and followed by the conditional) imposed nationwide had resulted encouraging positive results in managing the pandemic that only yielding two subsequent additional smaller or weaker cycles.The Ministry have indicated that the extension of MCO into phase 4 would enable the Ministry to work towards completely eradicating the transmission of Covid-19 [31].The MCO listed among others were 'stay at home', 'no interstate travelling', 'social distancing', 'self-hygiene awareness and practices' , ' controlling and restricting international travelling', and actively upgrading and operating the covid-19 screenings or tests and health related facilities and pandemic treatment in designated public and private hospitals.As of today (June 17, 2020), the government is enforcing the rehabilitation MCO due the down trending of daily cases, cumulative active cases , and reaching the mode of cumulative cases plateauing.
Being a tropical country (annual mean temperature and humidity of 27 o C and > 80%, respectively) [32], Malaysia's high temperature and humidity had probably made the MCO effective in controlling and mitigating the covid-19 pandemic.Even though it is still debatable, it is probably that the virus spread and infection would be lower by higher temperature and humidity.As indicated in Hong Kong, the risk of severe acute respiratory syndrome (SARS) infection would be at 18 folds higher in daily cases at lower temperature versus higher temperature [33].In term of SARS viability, it will rapidly lost at higher temperature and humidity indicating the virus transmission is more favorable in sub-tropical versus tropical countries [34].As of SARS-CoV-2 (covid-19 virus), warmer temperatures are associated with lower rate of Covid-19, but with modest effect [35].In other developments, the results (not yet peerreviewed) show that the weather of cold and dry apparently will boost the spread of covid-19 virus [36].
The medical expert from Taiwan warns of the covid-19 reoccurrence during the coming winter, and if the prevention intervention is not made, the pandemic would be getting worst [37].
The covid-19 infection in Malaysia is probably and highly due to the person-to-person contact rather than the contacts with physical surfaces that has virus droplets within an environment, locality or residential/business premises.By using data of May 18 -June 17, 2020 (coincided with the cycle 3 pandemic), the mean + standard error of daily cases of local foreigners, imports, and local citizen were 38.1 + 10.9, 7.2 + 1.2, and 6.9 + 0.8 cases, respectively.The value for local foreigners was significantly higher than those of the imports and local citizen (LSD test at p=0.05).Thus, in cycle 3, the local foreigners were the main source for cycle 3 occurrence.The imported cases were mainly sourced from those of citizens or permanent residents that returning home from various countries with covid-19 pandemic.These non-citizen locals are mainly workers in the manufacturing and construction industries, and in the supply and retail business.They are mainly staying in houses and hostels that mainly provided by their employers under 'crowded' occupancy condition that conducive for person-to-person contact for covid-19 infection.
Figure 2: The observed and predicted daily covid-19 cases over days in six movement control order (MCO) phases during the period of Jan. 25 -Jun.17, 2020 and Jan. 25 -Jun.30, 2020, respectively.The vertical solid lines referenced to time of inflections that associated with the predicted daily maximum of the primary cycle (cycle 1), cycle 2, and cycle 3.

Cumulative recovery and fatality cases
Figure 3 shows the observed and predicted accumulative recovery and fatality cases over days that computed by Eqs.(8 and 9) and the associated data are indicated in Table 2.The respective constant values of the models are given in Table 3.Based on the constants and Eqs.(8 and 9), the equations that related to the tri-logistic function are explicitly written in Eqs.(13 and 14).The related R 2 values (model fitness) for both recovery and fatality models were > 99% (p <0.0001).The trends of these two models were in tandem with that of the cumulative cases vs. days.The day constant values (measuring speed of the dynamic) of recovery model for cycles 1, 2, and 3 were 6.2 , 3.7 , and 8.2 day -1 , respectively, while the respective values for the fatality were 6.1, 3.6, and 13.5 day -1 .These values were generally close to those of cumulative cases model.Specifically in cycle 3, the   value of recovery was lower than that of the fatality indicating recovery rate was faster than the fatality during the CMCO and RMCO phases.This could be interpreted as the results of the effective control interventions and efficient medical facilities and services made and provided by the government and her related agencies, especially the Ministry of Health and the enforcement bodies.
Where   = predicted cumulative recoveries at ith day,   = predicted cumulative fatalities at ith day,   = ith day.
The predicted percentages of recovery and fatality were computed by (predicted cumulative recovery/cumulative cases)*100% and (predicted cumulative fatality/cumulative cases)*100%, respectively.Based on the trends shown in the analysis, the final percentages of recovery and fatality were estimated at 98.4% and 1.6%, respectively.As at the end of this study period (June 17, 2020), the predicted recovery and fatality percentages were 89.4% and 1.43%, respectively, while the respective observed percentages were 92.5% and 1.42%.At the same date, as based on the reports of dated June 17, 2020, the global recovery and fatality were 52.4% and 5.38%, respectively [ 1].
The respective figures for North America, Europe, and Asia were 43.9% and 5.81%; 53.3% and 8.20%; and  and lower fatalities.The models of high and significant fitting to the recorded data had indicated that the trends of the increasing and decreasing of recoveries and fatalities, respectively were well synchronized and fitted to the enforcement of the MCO phases (Figure 3).In actual fact, the fatality rate should be lower since 80% of the fatalities were those simultaneously of having the chronic diseases (hypertension, diabetes, etc.) [38].

Active cases
Cumulative active cases are the net or resultant of covid-19 infected cases after subtracting the cumulative recoveries and fatalities from the cumulative cases.These cases could be contiguous to other susceptible persons or communities.Figure 5 shows the curve of cumulative cases computed based on Eq. ( 15) that generally and originally in the form of Eq. (10).The associated predicted and observed data are shown in Table 2.
Where,   = cumulative active cases at ith day,   = ith day.
The predicted active cases were significantly fitted to the observed values with R 2 =97% (p <0.0001).The predicted cumulative value on April 14, 2020 (day=115) (beginning of MCO phase 3) estimated by the Ministry of Health, Malaysia was 2,033 cases (45% of cumulative cases).This value was lower than the observed value (2400 cases: 50% of cumulative cases) and the predicted cases of Eq. ( 15) (2,333 cases: 48% of cumulative cases).This would indicate that the tri-logistic model was successfully developed in modeling covid-19 responses in term of active cases over days.The highest predicted cumulative active cases occurred in the MCO phase 2 (Cycle 1).This then down trended that subsequently followed by an increase that resulted small peak as related to the occurrence of an additional serial smaller cycle (cycle 2).It then continued for the down trend that subsequently once again experiencing a quite substantial amount of peaking occurred in the CMCO phase that associated to the occurrence of the third cycle of the pandemic.As discussed earlier, the cycle 3 was mainly caused by the cases of the local foreigners (workers).Thus, apart from the predicted daily model, the dynamic of the pandemic could also be seen or detected from the model of active cases.The changes of curve trend were affected by the occurrences of the smaller serial cycles (cycles 2 and 3) over days.On June 17, 2020, the predicted percentages of active cases was 8.6% of the predicted cumulative cases, while the observed record was 6.1 %.At this date, the percentage globally was 42.2%, and the figures for Singapore and Indonesia were 24.3% and 55.3%, respectively.During the rehabilitation MCO phase, the forecast cumulative active cases on June 30, 2020 would be 202 cases (2.4% of predicted cumulative cases).

Model linkages and tri-logistic model applicability
The tri-logistic models of the recoveries and fatalities were linked to the cumulative cases function by parameter of   [Eq.( 5) vs. Eqs.(7 and 8)].On the other hand, the fractions of cumulative recoveries and fatalities had common denominator, i.e.,   , and these two fractions were complement to each other since mathematically at the end season of the model ( at t  ∞), the sum of them would be 1.0 or unity.Subsequently, active cases would be the resultant of Eq. ( 5) -Eq.( 7) -Eq.( 8).The accumulative ceases, recoveries, and fatalities models were highly fitted to the observed values with R 2 > 99% at p < 0.0001.The interconnected model as shown by the cumulative active cases was equally and highly fitted to the observed records with R 2 =97% (p<0.0001).The models were developed based by the approach of data driven that in contingent with the microbial growth model.The basic logistic function is used widely in the diverse fields including in the field of artificial neural network as its activation function.Thus, it could be concluded that the model used in this study could be used elsewhere currently or in the future in modeling the responses of covid- This partly shows the applicability of the function in modeling the responses of the pandemic in Malaysia.
Since the data duration used in this modeling study was considered as 'matured' (cumulative cases at about plateauing level), the forecasts on the daily cases and the cumulative figures of cases, recoveries, fatalities, and active cases at postdate days of the study period could be made.The predicted daily cases; the cumulative cases, recoveries, fatalities, and active cases at the end of the study period (June 17, 2020: 145 days) were 6 cases day -1 ; 8473, 7615, 122, and 737 cases, respectively.The respective observed figures were 10 cases day -1 ; 8515, 7873, 121, and 521 cases.The respective forecast values at June 30, 2020 (158 days) were 1 case day -I ; 8500, 8168, 128, and 202 cases.With the assumption of 'matured model' , there would be a minimal more resources left for pandemic spread as indicated by the slight higher observed vs. predicted cumulative cases in the RMCO zone.This would caution the authority to stay alert in maintaining the MCO enforcement.If no any further odd-extraordinary cycle occurrence, the country would probably be enjoying a non-significant covid or 'covid-free' pandemic by the July end or mid-August, 2020.
The purposes of the enforcements of the movement control order (MCO) in several phases (phases 1 -4, conditional MCO , enhanced MCO, and rehabilitation MCO) are generally to control and mitigate the spread of the pandemic, and specifically to reduce drastically the daily cases, the fatalities, and the active cases; and to increase the recoveries.The social distancing, self-hygiene, and having the 'new normal' ways of life are basically the centrals of the listed orders in the above MCOs.Even though it is still debatable, some researchers and workers have indicated that the activity and growth (replication) of viruses would be lower in the condition or environment of higher temperature and humidity.Being a tropical country, it is hypothesized or even 'predicted' that Malaysia would have an additive effect of MCO enforcement in mitigating covid-19 than those of the cooler countries.This could be seen as Malaysia is among the countries currently of having high recoveries and low fatalities.The predicted of all responses were tallied and in tandem to the objectives of MCO phases' enforcements.As in the RMCO, the observed and predicted values of daily cases and cumulative active cases were low, while the cumulative cases and recoveries were seemed at and about plateauing levels, respectively.Thus, this modeling study had indicated the validity of the correctness and appropriateness of the MCO phases' enforcements by the government.
The models were developed for the cumulative cases, recoveries, fatalities, and active cases vs. days; and daily cases vs. days.Based on cumulative records, the four models were related to each other by the key parameter of atotal (asymptotic total), and the resultant model was indicated in the cumulative active cases model.Thus, basically, it was only one model in representing the variable of cases, recoveries, fatalities, and active cases.As indicated from the analyses, even though with high number of observations ( n=145), yet each of the models of cumulative cases, recoveries, and fatalities vs. days had fitness of the R2 > 99% at p <0.0001, and the resultant model had R2 of 97% at p<0.0001.From the above exercise, it could be seen that the model involving the tri-logistic growth function (or even higher layered-logistic model) is unique in modeling responses of covid or covid-related pandemic/pandemic that involving several serial cycle of infections, and thus, this approach of the non-linear mathematical modeling could be used elsewhere.Additionally, the basic logistic growth function is used in the diverse fields of studies and applications.By knowing the valid estimates of respective parameters or constants, the model could be used for simulation purposes of the covid or covid-related epidemic or pandemic.Historically, the outputs of response model could probably used as guidelines of making action in mitigating future covid or covid-related infestation with the assumption of having quite 'similar' pattern of infestation even with the expected mutation of virus would be occurring in the future.
The  The observed and predicted daily covid-19 cases over days in six movement control order (MCO) phases during the period of Jan. 25 -Jun.17, 2020 and Jan. 25 -Jun.30, 2020, respectively.The vertical solid lines referenced to time of in ections that associated with the predicted daily maximum of the primary cycle (cycle 1), cycle 2, and cycle 3.

Figure 1 :
Figure 1: The observed and predicted of cumulative covid-19 cases over days in six movement control order (MCO) phases during the period of Jan. 25 -Jun.17, 2020 and Jan. 25 -Jun.30, 2020, respectively.The vertical solid lines are referenced to inflection points of the primary cycle (cycle 1), cycle 2, and cycle 3 as indicated in the Figure.

Figure 2
Figure 2 that showing three 'bell shape' curves ( with three amplitudes) of predicted daily cases versus
observed and predicted of cumulative covid-19 cases over days in six movement control order (MCO) phases during the period of Jan. 25 -Jun.17, 2020 and Jan. 25 -Jun.30, 2020, respectively.The vertical solid lines are referenced to in ection points of the primary cycle (cycle 1), cycle 2, and cycle 3 as indicated in the Figure.

Table 3 :
The constants and day constant of recovery and fatality models that based on fraction of cumulative recoveries and fatality to the total asymptotic value of the cumulative cases vs. days.
19 or covid-related epidemic/pandemic.All the curves of responses were in tandem with the phases of four MCOs, CMCO, and RMCO, and thus indicating the validity of correctness and appropriateness of MCO enforcements in mitigating Covid-19 in Malaysia.The intervention (MCO enforcement) and health/medical/safety services and facilities should be synchronized and coordinated well in the objective of achieving stable   , since achieving the plateauing of cases is vital to systematically controlling, mitigating, and finally eradicating covid-19 pandemic of the Nation.They are the cycle 1(1 o cycle) [Jan.25 -Apr.27],cycle 2 [Apr.28 -May 14] and cycle 3 [May 15 and onwards].Above is evidenced by the following discussion.There were three 'bell shape' curves in the dynamic model (derivative of cumulative cases vs. days) related to the pandemic cycles 1, 2, and 3 with the maximum predicted daily cases of 194.3, 78.7, and 89.1 cases day -1 , respectively.The respective days of occurrences of these maximums were 68.1, 100.9, and 125.8 days within the region of MCO1/ MCO2, MCO4, and CMCO, respectively.These trends were also observed in the predicted cumulative active cases.Thus, the above had indicated the presence of the 3-cycle covid-19 pandemic in Malaysia.By integrating the above discussion with the day constant values ( cycles 1, 2, and 3 : 7.6, 3.6, and 4.6 day -1 , respectively) relative to partial asymptotic cumulative cases ( cycles 1, 2, and 3 : 5889, 968, and 1,643 cases, respectively), the aggressiveness among cycles were in the order of cycle1 > cycle 3 > cycle 2.As indicated in the findings, the occurrence of cycle 3 was mainly contributed by the high cases among local foreigners (workers).Thus, it is suggested the reallocation and remobilization of resources should be targeted in containing the covid-19 infection among the local foreigners while making surveillance activities continually among local citizens, especially in the phase of RCMO enforcement.Thus, the study has shown that by employing the proposed serial trinonlinear logistic growth model that developed based on data of Jan. 25 -Jun.17, 2020 (145 days) and simultaneously in contingent with the consideration of the quantitative behavior of microbial population growth had been successful mathematically in identifying of the 3-cycle covid-19 pandemic in Malaysia.
ConclusionAs indicated in the section of Introduction of this paper, the current wave is considered as only one wave that initially are consisted of wave 1( Jan 25-Feb.26)and wave 2 ( Feb. 27 and onwards).Within this reclassified one wave, the study has indicated that covid-19 in Malaysia has three cycles of the wave.