Dynamic model of infected population due to spreading of pandemic COVID-19 considering both intra and inter zone mobilization factors with rate of detection

Highlights

• •A dynamic model of infected population to estimate the propagation pattern of COVID -19 pandemics.
• •Provision of incorporating several influencing factors in proposed model such as inter and intra zone mobilization, detection rate etc.
• •Different remedial steps are applied as operating procedures to the model in both standalone and hybridized mode.
• •Flexibility of obtaining probable infected pattern of a particular region or country by proper tuning of the different coefficient of proposed model.
• •Validation of the model for three geographical locations within India with real world data.

Abstract

Most of the widely populated countries across the globe have been observing vicious spread and detrimental effects of pandemic COVID-19 since its inception on December 19. Therefore to restrict the spreading of pandemic COVID-19, various researches are going on in both medical and administrative sectors. The focus has been given in this research keeping an administrative point of view in mind. In this paper a dynamic model of infected population due to spreading of pandemic COVID-19 considering both intra and inter zone mobilization factors with rate of detection has been proposed. Few factors related to intra zone mobilization; inter zone mobilization and rate of detection are the key points in the proposed model. Various remedial steps are taken into consideration in the form of operating procedures. Further such operating procedures are applied over the model in standalone or hybridized mode and responses are reported in this paper in a case-studies manner. Further zone-wise increase in infected population due to the spreading of pandemic COVID-19 has been studied and reported in this paper. Also the proposed model has been applied over the real world data considering three states of India and the predicted responses are compared with real data and reported with bar chart representation in this paper.

1. Introduction

Recent outbreak of Corona virus disease or COVID-19 global pandemic turns out to be one of the most severe threats to the mankind since first inception of the disease in 2019 at Wuhan, China due to a novel virus whose specific source of origin is not yet identified [1], [2], [3]. Also, after the inception, this virus spreads rapidly, first in China and then in more than 80 countries across the globe [4]. One of the preliminary reasons behind such rapid spreading of the disease has been identified as the contagious nature of the alleged virus which enables cumulative increase in the number of infections through daily anthropologic activities that require social interactions [5]. Also the stability property of the disease free equilibrium of COVID-19 indicates that proper vaccination for cure from this virus is not yet developed [6]. Therefore, social distancing and rapid detection test have been evolved as the most acceptable preventive measures in recent time [7,8]. Social distancing is attempted to achieve by different administrative bodies through local and global lockdown. Such lockdowns are imposed through the restriction on daily human activities as well as population mobilization in both local and global level [9,10]. On the other hand, rapid tests have been performed to detect the presence of the virus among the potential victims and if detected, sterner social distancing is being imposed on the infected persons through quarantine and isolation with proper medication and observatory procedure [11]. But such social lockdown and restriction on human mobilization can bring some severe and sweeping impact on economic sectors which in turn cause some detrimental effects on both social life and mental health of the human being [1], [2], [3], [4]. Therefore, the prediction of probable duration of lockdown is absolute necessity and needs to be addressed as top priority which requires continuous monitoring of the spreading pattern and timeline of the COVID-19 both locally and globally as well as the recovery rate and pattern of infected population [11]. But the continuously changing genetic structure of the respective virus make prior prediction of the disease difficult and ambiguous [5,11] which in turn brings delay in devising plan on omitting lockdown fully or partially.

Under such scenario, the demand for precise model to predict the exact nature of the spread of COVID-19 is ever increasing to envisage a proper protective strategy of preventing the aforementioned pandemic. All such predictive models can be broadly divided in two categories. In first category, samples are collected from one or more certain population considering different pandemic parameters such as doubling rate [12], basic reproduction factor [13], serial intermission [14] etc and then perform statistical analysis on collected samples to make required prediction of aforementioned pandemic. Also based on such analysis, several statistical models have been proposed to detect actual inter country infected cases [15] as well as to trace unidentified cases [16], to determine the effects of local and global migration of people [17,18] etc. Also different advanced statistical techniques have been used to predict the outbreak of corona virus in [19], [20], [21], [22], [23].

In second category, dynamic modelling has been used to assess the nature of COVID-19 pandemic [24] more accurately. Initially the final size and timeline of COVID-19 pandemic was predicted based on dynamic SIR model [25]. More advanced SIER model was brought in use to predict different factors associated with the disease and possible measures [26,27]. In such dynamic models, several factors like transmission process and risk [28], effects of isolation and quarantine [29] etc., are also included to make prediction more accurate. An advanced version of SIER model namely e-ISHR model has also been proposed to introduce the effects of time delay in the existing models [30]. Also, as the dynamics of COVID-19 pandemic is inherently nonlinear in nature like other epidemics [24], [25], [26], [27], [28], [29], [30], there will always be the provision of implementing some state of the art nonlinear dynamical methods proposed in recent times [31], [32], [33], [34], [35], [36] to make different forecast of spreading dynamics of COVID-19.

Various recent studies in the field of pandemic COVID-19 reflect the need of research in relation to determine spreading pattern of the pandemic and influence of different factors on it more accurately, which motivates the current studies performed in this paper. In view of these, a dynamic model to predict the pattern and volume of infected population due to the spread of COVID-19 has been proposed in the present paper considering several real life factors such as intra and inter zone mobilization, lockdown on local and global activities before detection, rate of detection and the effects of quarantine after detection. Also the zone-wise increase in infected population due to spreading of pandemic COVID-19 has been given special emphasis in this paper. Various remedial steps are taken into consideration in the form of operating procedures. Further such operating procedures are applied over the model in standalone or hybridized mode and corresponding responses are reported considering several case studies to indicate that imposing restriction on intra and inter zone mobilization as well as proper quarantine leads to the flattening of pandemic curve.

Finally the proposed model is applied over the real data of few states of India and the predicted responses are reported in the Appendix section of this paper compared with real data. Also the proposed model has the provision of simulating various operating procedures as remedial steps in both standalone as well as hybridized mode to reduce the propagation or spreading strength of concerned pandemic in a particular geographical region which is useful in determining possible measure to be implemented based on the demographic properties of that region to counter the spreading of COVID- 19 pandemic.

2. Proposed model

The proposed dynamic model of infected population due to spreading of pandemic COVID-19 is represented in Fig. 1 . This model considers three major factors such as intra zone mobilization; inter zone mobilization and rate of detection. In this model a country/state/territory is divided into N number of zones. At any point of time when it has been realized that such pandemic viral infection is spreading out and the time (day) has been taken as the initial time and total non-detected (implies Non-quarantined) infected alive population on the day has been taken with zone wise distributions.

Fig. 1.

Block diagram of the proposed model.

The parameters which are marked in the proposed model are detailed as follows.

$P_i\left(t\right)=$Number of alive non-detected infected population till time $t\left(day\right)$in Zone $i$(Excluding death / detected with quarantined / cured).

$P_{d{q}_i}\left(t\right)=$Number of detected with quarantined infected population till time $t\left(day\right)$in Zone $i$(Including death after detection / detected with quarantined / cured after detection).

${\lambda }_i\left(t\right)=\text{Distribution}\text{factor}\text{of}{i}^{th}\text{Zone}\text{at}{t}^{th}\text{day},$

$$\sum _{j=1}^N{\lambda }_j=1$$ (1)

$D_i\left(t\right)=$Number of death of infected population but not detected till time $t\left(\text{day}\right)$in Zone $i$

$$\delta D_i\left(t\right)=\gamma P_i\left(t-T_d\right)$$ (2)

where, $\gamma$is the death factor and $T_d$is the average death time delay in days.

$C_{d{t}_i}\left(t\right)=$Number of cured infected population belongs to detected with quarantined population till time $t\left(\text{day}\right)$in Zone $i$.

$C_{nd{t}_i}\left(t\right)=$Number of cured infected population belongs to non-detected population till time $t\left(day\right)$in Zone $i$.

$$\delta C_{d{t}_i}\left(t\right)={\beta }_{dt}\delta P_{d{q}_i}\left(t-T_c\right)$$ (3)

$$\delta C_{nd{t}_i}\left(t\right)={\beta }_{ndt}\left\{P_i\left(t-T_c\right)-\delta D_i\left(t-T_c\right)-\delta P_{d{q}_i}\left(t-T_c\right)\right\}$$ (4)

where, ${\beta }_{dt}$and ${\beta }_{ndt}$are the factors to become cure and $T_c$is the average time delay to become cure in days.

$$\delta P_{d{q}_i}\left(t\right)=r{d}_i\left(t\right)\left\{P_i\left(t\right)-\delta D_i\left(t\right)\right\}$$ (5)

where, $r{d}_i\left(t\right)$represents rate of detection of $i^{th}$Zone at time $t$.

$$\begin{array}{ccc}& & P_i\left(t+1\right)=P_i\left(t\right)+{{\alpha }_{mw}}_i\left(t\right)P_i\left(t\right)+\sum _{\stackrel{k=1}{k\ne i}}^N\left\{{{\alpha }_{mo}}_{ki}\left(t\right)P_k\left(t\right)\right\}\hfill \\ & & -\delta D_i\left(t\right)-\delta P_{d{q}_i}\left(t\right)-\delta C_{nd{t}_i}\left(t\right)\hfill \end{array}$$ (6)

where,

${{\alpha }_{mw}}_i\left(t\right)=$Enhancement factor of $P_i\left(t\right)$due to intra zone mobilization in Zone $i$at time $t$.

${{\alpha }_{mo}}_{ki}\left(t\right)=$Enhancement factor of $P_k\left(t\right)$due to inter zone mobilization from $k^{th}$to $i^{th}$Zone at time $t$and ${{\alpha }_{mo}}_{ki}\left(t\right)={{\alpha }_{mo}}_{ik}\left(t\right)$has been considered.

$${\lambda }_i\left(t+1\right)=\frac{P_i\left(t+1\right)}{P\left(t+1\right)}$$ (7)

$P_{total}\left(t\right)=$Total infected population (with/without detected including death) at the end of $t^{th}$day.

$$P_{total}\left(t\right)=\sum _{j=1}^N{P}_j\left(t\right)+\sum _{j=1}^N\underset{0}{\overset{t}{\int }}\delta D_j\left(t\right)+\sum _{j=1}^N\underset{0}{\overset{t}{\int }}\delta {P_{dq}}_j\left(t\right)$$ (8)

In this model the update of alive non-detected infected population is done by Eq. (6) and further the distribution factors can be updated by Eq. (7). If mobilization of any zone is stopped then in this model the corresponding factors have to be zero. Also in this model death factor, average death time delay, factors to become cure and average time delay to become cure have also been considered. The average death delay time indicates the delay between infected and death, whereas average time delay to become cure indicates the factor associated will be applied over the delayed infected population and the resulted population does not infect further. In this model the rate of detection also has been considered as a function of time. Further clustering of zones has been considered based on the geographical locations to classify the zones where direct inter zone mobilization may happen. Two adjacent clusters may have some common zones in the region of intersection. Further various operating procedures have been presented in Table 1 , which can be applied over the proposed model at any point of time to impose damping over the infected population response. Considering the operating procedures various case studies are simulated and have been reported in the subsequent section.

Table 1.

Operating procedure.

Label	Description
OP-1	Standard lock-down but mobilization happens
OP-2	Inter zone mobilization of few zones are stopped but intra zone mobilization happens
OP-3	Both intra and inter zone mobilization of few zones are fully stopped
OP-4	Rate of detection with quarantined increases

It is very much obvious that the community transmission of COVID-19 is mostly due to the alive infected population which is not detected or quarantined till date. Further how such infected population in each zone changes due the crucial realistic cause of spreading such as intra and inter zone mobilization factors with rate of detection has been estimated. Also day wise death and cured/recovered are considered in the proposed model with various realistic coefficients such as death factor, average death time delay, factors to become cure, average time delay to become cure etc. This model is reported with Eqs. (1)–(8) and also with block diagram form as illustrated in Fig. 1.

3. Simulation results and discussions

The clustering of zones for the simulation is represented in Fig. 2 and random initializations of population are reported in Table 2 . The proposed dynamic model of infected population due to spreading of pandemic COVID-19 has been simulated with Case-1 parameters and the response of number of alive non-detected infected population $\sum P_i\left(t\right)$has been represented in Fig. 3 . Also the total infected population (with/without detected including death) $P_{total}\left(t\right)$is represented in the Fig. 3. The responses indicate that the patterns are very much similar to the patterns of infected population of various countries.

Fig. 2.

Clustering of zones considering geographical location.

Table 2.

Initialization of population for simulation.

Zone#	$P_i$/${P_{dq}}_i$	Zone#	$P_i$/${P_{dq}}_i$	Zone#	$P_i$/${P_{dq}}_i$
1	7/2	41	8/1	72	8/1
4	10/1	42	20/3	74	8/2
8	9/2	43	6/0	90	8/1
13	6/1	46	5/1	98	10/2
37	6/4	61	9/9	Other zones	0/0
$t$ = 20, $D_i$= 0 for all $i$

Fig. 3.

Total infected population (with/without detected including death) $P_{total}\left(t\right)$and infected but not detected alive population $P\left(t\right)$with operating procedure OP-1 (Case-1).

Further, simulations of various case studies as per Table 3 have been carried out and reported in Fig. 4 . It is found that the rate of change of infected population are slowing down for Case-2 to 6 compared to Case-1 and in few cases the non-detected infected population are reducing. Again simulations are carried out by applying OP-4 with incremental rate of detection per day basis starting from a certain day and the responses are reported in Fig. 5 . It seems that the responses are quite satisfactory compared to Case-1.

Table 3.

Descriptions of case studies.

Label	Description
Case-1	OP-1 with $\gamma =$0.1%, ${\beta }_{dt}={\beta }_{ndt}=$2%, $r{d}_i=$1% (for all $i$), ${{\alpha }_{mw}}_i=$8%, $T_c$= 5, $T_d$= 5, ${{\alpha }_{mo}}_{ki}=\{\begin{array}{cc}0.1\%\hfill & ;\text{if}\text{Zone}i,k\notin \text{same}\text{cluster}\\ 0\hfill & ;\text{else}\end{array}$For all $t$
Case-2	Case-1 for $t<110$day, Case-1 with OP-2 for $t\ge 110$day for zones= 1 to 20 and 61 to 80
Case-3	Case-1 for $t<110$day, Case-1 with OP-3 for $t\ge 110$day for zones= 1 to 20 and 61 to 80
Case-4	Case-1 for $t<110$day, Case-1 with OP-3 & OP-4 for $t\ge 110$day for zones= 1 to 20 and 61 to 80, $r{d}_i=$10% ($i$=1 to 20 and 61 to 80)
Case-5	Case-1 for $t<110$day, Case-1 with OP-3 for $t\ge 110$day for all zones
Case-6	Case-1 for $t<110$day, Case-1 with OP-4 for $t\ge 110$day for all zones $r{d}_i$(for all $i$) increases by 0.5% per day

Fig. 4.

Comparative analysis of infected but not detected alive population $P\left(t\right)$with various remedial steps initiated from day=110.

Fig. 5.

Comparative analysis of infected but not detected alive population $P\left(t\right)$with various incremental rate of detection initiated from day=110.

Furthermore, for in-depth studies of the proposed model, the zone-wise surface maps have been reported in Fig. 6 . When the model is operated with the parameters as per Case-1, the zone-wise non-detected infected population are increasing day by day (Fig. 6(a)). But when OP-3 applied to the model from day 110 for the zones 1 to 20 and 61 to 80 as per Case-3, the zone-wise non-detected infected population are reducing for the said zones (Fig. 6(b)).

Fig. 6.

Zone-wise surface map of infected but not detected alive population $P_i\left(t\right)$, (a) operating procedure OP-1 (Case-1), (b) operating procedure OP-3 (Case-3).

The proposed model is simulated with some initial population as per Table 2. Various operating procedures have been applied as remedial steps in standalone or hybridized mode and the responses indicate the effectiveness. Fig. 4 illustrates that if mobilization of population is reduces or test for detection increases or both then the increase in infected population will slowdown or may reduce. Further Fig. 5 clearly describes that increase in test for detection day by day with quarantine will slowdown or reduces the enhancement of non-detected infected population. Further zone-wise surface map (Fig. 6) patterns illustrate that how infected population profile changes in each zone if certain operating procedures have been imposed in few of the zones. It is obvious that remedial course of action with same intensity may not be possible to be imposed in all zones at same point of time. In view of these if certain zones are handled with rigid course of action then how the patterns may improve compared to other zones are studied and reported in Fig. 6.

In view of the reported simulation responses it has to be admired that hybridization of various operating procedures may improve the situation by slowing down propagation of infected population.

Further to validate the proposed model, various real world data [37] have been taken into consideration and responses are reported in the Appendix-A section. In this section three numbers of states of India such as West Bengal, Tamilnadu and Gujarat, have been taken into consideration as zones for prediction of spreading of infected population. Considering the data available in [37] upto 31-July-2020 the day-wise prediction of the entire Aug-2020 has been simulated by using the proposed model. Fig. A1 (a) represents the factor ${\alpha }_{mw}.rd$to find the linear trend line which will be used to get the intra zone mobilization factor with rate of detection for the month of Aug-2020. Since in [37], migrated population is taken zero, the inter zone (in this case state to state) mobilization factor is assumed to be zero. A crucial factor behind the spreading of infection is the non-detected infected population roaming throughout the zone. In order to this it may be assumed that the infected population to be detected in next few days (here taken 07 days) are already infected at present but not detected or quarantined. This consideration is employed to find the trend line as reported in Fig. A1(a). Further, factor $\left({\beta }_{dt}\right)$to find the cured/recovered population from detected infected population has been determined taking $T_c$= 5 with moving average basis (Fig. A1(b)).

Fig. A1.

Proposed model based predicted data of West Bengal, India. (a) Factor (${\alpha }_{mw}.rd)$with trend line, (b) factor${\beta }_{dt}$, (c) comparative analysis of cumulative predicted infected population with real world data available at [37], $T_c$= 5, $T_d$= 5, (d) comparative analysis of cumulative recovered population (detected) with real world data available at [37].

In Fig. A1(c), the day-wise predicted cumulative infected population for the month of Aug-2020 has been reported with bar chart representation in comparison with real data [37] of West Bengal, India. Furthermore, in Fig. A1(d), the day-wise predicted cumulative cured/recovered population for the month of Aug-2020 has been reported with bar chart representation in comparison with real data [37] of West Bengal, India. In similar manner, predicted cumulative population both infected and recovered, compared with real data for the zones Tamilnadu, India and Gujarat, India, have been illustrated in Figs. A2 and A3 .

Fig. A2.

Proposed model based predicted data of Tamilnadu, India. (a) Factor (${\alpha }_{mw}.rd)$with trend line, (b) factor${\beta }_{dt}$, (c) comparative analysis of cumulative predicted infected population with real world data available at [37], $T_c$= 5, $T_d$= 5, (d) comparative analysis of cumulative recovered population (detected) with real world data available at [37].

Fig. A3.

Proposed model based predicted data of Gujarat, India. (a) Factor (${\alpha }_{mw}.rd)$with trend line, (b) factor${\beta }_{dt}$, (c) comparative analysis of cumulative predicted infected population with real world data available at [37], $T_c$= 5, $T_d$= 5, (d) comparative analysis of cumulative recovered population (detected) with real world data available at [37].

4. Conclusion

In this paper a dynamic model of infected population due to spreading of pandemic COVID-19 considering both intra and inter zone mobilization factors with rate of detection, have been proposed with various operating procedures. Considering the operating procedures as followed in this paper, various case studies have been simulated and reported with adequate responses. The responses obtained from the simulation of the proposed model are seen to have considerable similarities with the patterns of infected population of various countries as reported in the literatures which indicates the predictability of the proposed model. The coefficients or factors associated with the proposed model are needed to be tuned to get the pattern of infected population of any particular country. Various operating procedures have been applied as remedial steps in standalone or hybridized mode after a certain day and the responses indicate the effectiveness. Further this study also investigates that imposing various strict administrative protocols in certain zones by improving such factors may be achieved improved responses. Further the proposed model is having various provisions to fed external inputs to realize the effects of imposing different remedial steps. In addition to this the proposed model has been applied over the real world data considering three states of India and the predicted responses are compared with real data and reported with bar chart representation in the Appendix-A section.

Future scope of research

This proposed model can be tuned to validate the prediction for others states or country.

CRediT authorship contribution statement

Mousam Ghosh: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization. Swarnankur Ghosh: Formal analysis, Writing - original draft, Writing - review & editing, Visualization. Suman Ghosh: Resources. Goutam Kumar Panda: Writing - original draft, Writing - review & editing. Pradip Kumar Saha: Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Appendix

Fig. A1, Fig. A2, Fig. A3