The house is an integral part of man's daily life and, without it, many factors would be hampered. Different researchers have discussed housing, its classifications, and its importance to man ( Vale et al. 2014; Wahi et al. 2018). It is a global problem, and the government alone cannot provide affordable housing for all its citizens. As such, many housing developers and planners have been actively involved in this sector ( Gayinu et al. 2014; Mohd Noor 2017; Bakhtyar et al. 2013). Many people across the globe suffer from the effect of a lack of better and more affordable sources of accommodation. Different studies have discussed housing shortages in urban and semi-urban areas ( Bhide et al. 2019; Gayinu et al. 2014; Mohd Noor 2017; Bakhtyar et al. 2013; Wahi et al. 2018) and unaffordability due to the high cost of building materials and loans ( Djebarni & Al-Abed 2000). Bhide et al. (2019) argued that housing problems should not be limited to affordable housing, but a robust approach should be adopted to tackle the housing problem. Notable scholars have discussed the difficulties to access affordable houses and the number of occupants that reside in small apartments ( Abiodun 1976).

In some countries, the federal and state government collaborate to provide affordable houses for low-income earners. In recent times, the private sector has played a significant role to complement the efforts of different tiers of government to ensure their citizens have good shelter. In addition, individuals also rent their houses to tenants to complement the housing deficit. The problem with the above is the mode of renting and rent payment. It is often the case that landlords or estate managers require annual or monthly rent payment. In many instances the potential tenants are willing to rent the property, but the rent conditions, such as annual or monthly rent payments or security deposits and the lump-sum payments involved, become major hurdles. These conditions have led many people to be homeless, especially daily income earners.

During the early days of the global system of mobile communication (GSM) in Nigeria, many people were willing to subscribe to a GSM network, but the network carriers configured their system to per-minutes billing, which was very expensive to many intending users and as a result GSM users were relatively few in number. After some time, people began to clamour for per-second billing, which was eventually granted by an indigenous network carrier. With this per-second billing, many people started subscribing for any amount they desired. In several cities, there are numerous unoccupied public and private buildings or houses, and many homeless people are willing to rent but because of the lump-sum payment and high cost of renting they cannot afford decent accommodation. Could this be related to the per-minute billing of GSM networks in Nigeria, in which many wanted to subscribe but because of the high cost of per-minute billing they decided to forgo their intention? For the Nigerian case, the rate of subscription drastically increased when network providers converted to the per-second billing system. The number of sim card registrations for different GSM network providers skyrocketed and many people of different classifications had access to mobile telephones. So, this model can be replicated in the housing sector to accommodate the daily income earners. However, the implementation of this model in the existing conventional public and private housing system may be a herculean task in terms of monitoring tenants' exit times. However, with the technological advancement and transformation of traditional cities into smart cities, it will be possible in the future.

Smart cities and their components have been discussed extensively ( Mohanty, Choppali & Kougianos 2016; Eremia, Toma & Sanduleac 2017; Georgiadis, Christodoulou & Zinonos 2021; Praharaj & Han 2019; Ahvenniemi et al. 2017; Ross & Maynard 2021). We define a smart city as a digitally connected city where all infrastructure is digitally bounded with digital entry permission. With the rapid growth in technological advancement and emergence of smart cities, the existing rent default problem could easily be resolved by the rent subscription model. Smart house concepts are about digitalization, whereby every activity is connected to a digital platform. The smart house implies that everything has a digital relationship with access permission. We define a smart house as a connected platform whereby every facility is connected to a digital framework that requires digital permission. The entry access point requires digital permission, as such subscription for smart houses can be done in a similar way to internet subscriptions. In this case, the subscribed amount is the access key to the house and when this subscription elapses, access to the house is deactivated automatically, thereby denying the tenants/occupants access to the house. An alternative to smart houses is to modify the doors of conventional houses to smart doors, to serve the purpose of the proposed models and thereby solving rental default problems.

In some countries, poor locals, foreigners, and migrant workers with low financial status constitute a large chunk of people with housing difficulties and rental default. Over the years, there have been arguments between landlords and tenants over rent default or rent arrears. This problem also extends to overstay in service apartments or hotels. The problem of rent defaults and refunds has generated many litigations globally. Sometimes the exact amount, based on the length of stay and under-stay, has generated a refund crisis between landlords and tenants.

Certain conditions may alter the plans of either party; the landlord may decide to sell the house, or the tenant may decide to quit the house halfway thereby leading to a refund request. Refund requests are not an easy task for both parties. In some cases, either party will default which may lead to court action, or the case of long-term rental default which the house owner would want to recoup.

Private individuals have resorted to renting part of their houses for economic reasons, while others build for business concerns. The rental policy of such privately owned houses is exclusively decided by the landlords or caretakers. The mode of payments, maintenance, and periodic rental increase is the exclusive preserve of the landlords. These policies, in the form of a tenancy agreement, have been breached by the landlords and tenants due to circumstances that may be beyond their control. The landlord often requests annual or monthly rental. In many instances, rent default and refund have been the subject of discussion and litigation between landlords and tenants. Different studies have discussed rent default in public and private housing management ( Salleh, Johari & Yusof 2008). Salleh et al. (2008) observed that rent default has multiple implications to the parties involved, such as rent refund and rent arrears.

The concept of advance rental payment is a recognized practice and an accepted norm in the society. Rental default and refund are as old as this norm. One difficulty inherent to this concept is the possibility of the tenant requesting a refund midway through the tenancy agreement. In all of these cases, the basic problem is the refund request, which in many situations may not be suitable for the landlord or the tenant. Previous studies have not discussed refund requests, the prevention of rental default problems, or weekly, daily, and hourly housing rent schemes, but have instead focused mainly on affordable houses for low-income earners and rental default. In this paper, we propose rental subscription models whereby rental default is impossible and renting is subscription based. The main contribution of this paper is to solve the rent default problem, thereby making rent arrears impossible in the public and private housing sector and allowing prospective tenants the option to subscribe for house rent based on available resources. Another objective of the study is to provide opportunities for people to have access to a shelter by subscribing weekly or daily, to solve the housing needs of low-income earners and daily income earners.

The rest of this paper is organized as follows. Section 2 describes the recursive annual rental subscription model (R-ARSM). It gives details about the rent subscription models and discussed possible refund request computations based on time. The R-ARSM computes the tenant's annual, monthly, weekly, daily, and hourly subscription balance. Section 3 discusses the adaptive rental subscription model (ARSM) whereby rental default and rental arrears is impossible. Section 4 contains different problems and applications to annual, monthly, daily, and hourly subscription problems, refund requests, discussions, and recommendations. Conclusions follow in Section 5.

The R-ARSM is very flexible and would allow tenants to subscribe annually and provide a flexible refund request framework. The model computes tenancy per month, weeks, days, hours, minutes, and second balance. The five-step R-ARSM is described as follows: ∑ i = 1 n x i = ∏ α ̂ n , (1) where α ̂ is per month rental rate, n denotes the number of months per year and β is a user’s defined boundary constant which accounts for the depreciation of the house and furniture for the period of individual tenancy. Therefore, Equation (1) can be written as ∆ = ∏ α ̂ n (2)

Equation (2) denotes annual rental. Then Equation (3) computes per month rate X = ∆ M + M − 1 β = ∏ α ̂ n M + M − 1 β , (3)

Step 1

From Equation (3), M implies the number of months per year and U is the monthly rate U = ∏ i = 1 M X R i , R = 1 , 2 , . . , M (4) φ = ∏ i = 1 M ∆ ϑ i , (5) where ϑ is a unit constant. Based on the definitions of Equation (4) and Equation (5), we applied the concept of residual discussed in Sharpe (2015), Delucchi (1976), Neter et l. (1996) and Barnett et al. (1975) in a reversed form, and defined the modified residual using Equation (4- 5) to obtain Equation (6) ∆ ∇ = ∏ i = 1 M ∆ ϑ i − ∏ i = 1 M X R i = φ − U (6) where φ is the observed annual rental value and U is the expected sum of monthly rental value. In general, ∆ ∇ denotes per month balance, which is zero at the end of the last month if both landlords and tenants keep to the tenancy agreement and a specified value if the tenancy agreement is breached. Individual occupancy value for multiple occupants can be computed based on Equation (7) ε t = ∆ ∇ α (7) where α denotes the number of occupants per rental period. Equation (7) is a modification of the standardized residual discussed in Haberman (1973) which describes individual cell contribution. The last expression describes multiple occupants’ situations where each occupant contributes to pay for the rent. Equation (6) can be written as ∆ ∇ = 0 if φ = U γ n if φ ≠ U , (8) from Equation (8) ∆ ∇ = 0 means that all parties complied with the agreement while ∆ ∇ = γ n implies a breach of agreement and refund γ n is required. For the multiple occupant’s application, we have ɯ ɯ = 0 , if φ = ∑ ε t and ɯ ɯ = γ n ifφ ≠ ∑ ε t , that is ɯ ɯ = 0 if φ = ∑ ε t γ n if φ ≠ ∑ ε t , (9)

If all occupants comply with the tenancy agreement φ − ∑ ε t = 0 no refund is required while φ ≠ ∑ ε t implies a breach of agreement, hence a refund request would be required.

Step 2

Let X be as defined in step (1) and N denotes the number of days in the given month, then the daily rate X 1 is defined as X 1 = X N (10) γ = ∏ i = 1 N X 1 J i , J = 1 , 2 , … , N (11) w = ∏ i = 1 N X ϑ i (12) ∆ 1 = ∏ i = 1 N X ϑ i − ∏ i = 1 N X 1 J i = w − γ (13)

Where ∆ 1 denotes per month balance which is zero at the end of the last day and individual occupancy value for multiple occupants can be defined as ε q = ∆ 1 α (14) where α denotes the number of tenants per rental period. A similar discussion as in Equations (8- 9) is applicable.

Step 3

Let X 1 be as defined in step 2 and K the number of hours per day, then the hourly rate Y 1 is computed as follows Y 1 = X 1 K (15) γk = ∏ i = 1 K Y 1 e i , e = 1 , 2 , … , K (16) wh = ∏ i = 1 K X 1 ϑ i (17) DH = ∏ i = 1 K X 1 ϑ i − ∏ i = 1 K Y 1 e i = wh − γk (18)

Where DH denotes per day balance which is zero at the end of the last hours and Equation (19) explain per individual rental for multiple occupants ε s = DH α (19) where α denotes the number of occupants per rental period. A similar explanation as in Equations (8- 9) is applied.

Step 4

Let Y 1 be as defined in step 3 and g is the number of minutes per hour, then per minutes rate Y 2 is defined as Y 2 = Y 1 g (20) Td = ∏ i = 1 g Y 2 v i , v = 1 , 2 , … , g (21) sx = ∏ i = 1 g Y 1 ϑ i (22) D Uk = ∏ i = 1 g Y 1 ϑ i − ∏ i = 1 g Y 2 v i = sx − Td (23)

Where D Uk denotes per minute balance which is zero at the end of the last minutes and the individual rate for the period is defined as ε p = D Uk α (24) where α denotes the number of users per rental period. A similar explanation as in Equations (8- 9) is applied.

Step 5

Let Y 2 be as defined in step 4 and l denotes the number of seconds per minute, then per second rate Y 3 is defined as Y 3 = Y 2 l (25) rw = ∏ i = 1 l Y 3 h i , h = 1 , 2 , … , l (26) zs = ∏ i = 1 l Y 2 ϑ i (27) D U = ∏ i = 1 l Y 2 ϑ i − ∏ i = 1 l Y 3 h i = rw − zs (28)

Where D U denotes per second balance which is zero at the end of the last second and the individual rate for multiple occupants for the period is defined as ε w = D U α (29) where α denotes the number of users per rental period. A similar description based on Equations (8- 9) is applied. Therefore, the refund request by either party is based on the following expression

Ʊ = ∆ − ɸ , (*) where ɸ = ∆ ∇ + ∆ 1 + D H + D UK + D U and ∆ is the total annual rental , ɸ denote the total amount of rental for the period, which comprises of the following: ∆ ∇ denotes monthly rent, ∆ 1 number of days (or weekly rental), D H hourly, D UK minute and D U per second rates for the period and Ʊ denotes the total amount to be refunded if the agreement is breached. The description of (*) can be expressed as follows Ʊ = 0 if ∆ = ɸ c if ∆ ≠ ɸ (30)

Ʊ = 0 implies that all parties complied with the tenancy agreement and Ʊ = c implies that a breach of agreement has occurred, therefore the tenant can request for refund.

This subsection discusses the RMRSM. The RMRSM accounts for a period of one month for the tenant, which comprises the number of weeks, days, hours, minutes, and seconds. This model adopts the same process discussed in subsection 2.1. The monthly recursive model is described as follows.

Step 1

For the RMRSM, the penalty constant Ƹ is defined as in subsection 2.1. Let X denote the monthly rate and N number of days per the specific month, therefore the daily rate is defined as X 1 = X N + N − 1 Ƹ , (31)

Hence, we invoke the process discussed in Equations (11- 14) and Equations (7- 9), respectively.

Step 2

Let X 1 denote the definition in step 1 in Equation (31), then Y 1 = X 1 K (32)

Based on Equation (32), Equations (16- 19) and Equations (7- 9) follows.

Step 3

Let Y 1 denote the definition in step 2, Equation (32), and K 1 denotes the number of minutes per hour, then Y 2 = Y 1 K 1 (33)

Based on the definition of Y 1 in Equation (32), the computational procedure described in Equations (20- 24) and Equations (7- 9) is adopted.

Step 4

Let Y 2 be as defined in Equation (33), then Y 3 = Y 2 l (34)

Relying on Y 3 in Equation (34), the computational steps in Equation (23- 26) and Equation (7- 9) follow, respectively.

Therefore,

Ʈ = X − ɸɰ (**) where ɸ ɰ = ∆ 1 + D H + D UK + D U and X is the monthly rate, ɸɰ denote the total rate for the tenancy period which include the following: ∆ 1 number of days (or weeks), D H hourly, D UK minute and D U per second rates and Ʈ denotes the total amount to be refunded. Therefore, the analysis expressed in Equation (30) follows.

The RDRSM finds applications in such areas as hotels, homestays, service apartments, and short rest accommodations. This approach can alleviate some of the associated problems in this sector. This procedure is described as follows.

Step 1

Let X 1 denote daily rate, K corresponds to the number of hours per day and ʠ denotes penalty function, which is user-defined, therefore we have Y 1 = X 1 K + K − 1 ʠ (35)

Using Equation (35) as input, the computation of this procedure follows from Equation (16- 19) and Equations (7- 9), respectively.

Step 2

Let Y 1 be as defined in step 1, Equation (35), and let K 1 denote the number of minutes per hour, then we have Y 2 = Y 1 K 1 (36)

Relying on Equation (36), this procedure follows a similar computational framework in Equations (20- 24) and Equation (7- 9), respectively.

Step 3

Let Y 2 be as defined in step 2, Equation (36), and K 2 denotes the number of seconds per minutes, then Y 3 = Y 2 K 2 (37)

Using Equation (37), Equations (26- 29) and Equations (7- 9) follow immediately.

Therefore, Ʈɼ = X 1 − ɸ ɱɰ (***) where ɸ ɱɰ = D H + D U K + D U

and X 1 is the daily rate, ɸɱɰ total occupancy rate for the period which includes D H hourly, D UK minute and D U per second rates. Therefore Ʈɼ can be defined as in Equation (30) and the discussions follow immediately. The recursive rental subscription model recursively computes the period of tenancy corresponding to the defined rates and allows for an equitable refund if the need arises. This model may not completely solve the problem of rental default, but it affords tenants and landlords a unique opportunity to solve rental refund requests. The discussion in the next section will completely eradicate the rental default problem in the housing sector.

Let W ∆ be the daily subscription rate, K denote the number of hours per day, α be the penalty constant, and β the amount an individual has, then Equation (38- 39) can be written as follows V = W ∆ K + K − 1 α (44) Z = β V (45)

where Equation (44) gives the average rate per hour and Equation (45) determines the number of hours the available resources can sustain. A similar procedure to Equations (40- 42) is applied for this model. We redefined variables in Equations (44- 45) and apply the explanation in Equations (40- 42) to justify the number of minutes the available amount could subscribe. The ADRSM is suitable for daily rental and can be applied to hotels, service apartments, homestay, guest houses, and transit lodges. The uniqueness of the adaptive model is that it can be used for any house with smart doors.

A landlord in a particular city in Nigeria owned an estate of four flats and two self-contained apartments. Over the last four years, he has been managing the estate with little stress due to poor estate conditions or tenant mobility. However, tenant mobility seems to be on the increase due to the coronavirus disease (COVID-19) pandemic. A tenant rented a two-bedroom apartment and paid annual rental of #360,000 and a maintenance fee of #30,000. After six months, two weeks, and four days, the tenant informed the landlord of his intention to relocate to his hometown due to the economic situation occasioned by COVID-19 and his retrenchment from his job. He appealed to the landlord for a refund of the balance of the rent paid, since he is seriously seeking funds to relocate his family to his hometown. Unfortunately, the landlord disagreed, saying the contractual agreement is one year. On compassionate grounds the landlord said he will refund the balance if another tenant is ready to rent the apartment and that is the only option, he could make any refund.

The recursive model can be applied to this problem because it involves periodic refund requests associated with the annual rent paid. To avoid this type of problem in the future, recursive model 2.1 can be incorporated in the future tenancy agreements to enhance smooth refund requests. Let us apply the model to this problem to determine the rental period and refund request. To solve this problem, we input the annual rental rate in the model, the number of months, weeks, days, and if possible, the last hour and minutes of stay so that the refund request would be fair to both parties. The R-ARSM is time-dependent, therefore tenancy period and refund are bounded by time. Table 1 gives the model breakdown of the rate per month, day, hour, minute, and second, respectively.

Table 2 below gives the recursive model breakdown balance for annual rental, per month, per day, hours, minutes, and seconds based on problem 1. For the period of renting, that is, six months, two weeks, four days, and 8 hours 30 minutes. Based on the situation, the landlord decided to compute the total amount for the duration of the tenancy. In this case, we will use Equations (6, 13, 18, 23) and Equations (*, 30), respectively. Applying the above, the tenant refund request would be #175,757.73 (45.1%). This analysis is shown in Table 3 and Figure 1, respectively.

In Table 2, assuming the tenant complied with the tenancy agreement the balance at the end of the last month would be zero and every other computation would be zero. Consider that the tenants breached the tenancy agreement at the end of the 5 ^th month from Table 2, it implies that the tenant balance is #227,500 (58.33%), based on Equations (6, 8). Assume that a tenant that pays monthly rental complies with the terms of the agreement, the balance at the end of the month is zero. On the other hand, if the tenant decides to quit after spending 19 days in the house, the tenant’s balance would be #12,580.645 (38.7%), based on Equation (13). Suppose the tenant paid for one day as in the hotel system and after six hours decided to vacate the room, the balance from the payment would be #786.29 (75%), based on Equation (18). Figure 2 to Figure 4 show the computational framework of the R-ARSM based on Equations (6, 13, 18, 23, 28). These figures imply that the models are time dependent.

Figure 3 and Figure 4 indicate the RMRSM and RDRSM based on Table 1.

Figure 2 to Figure 4 revealed periodic deduction based on the recursive models. So far these illustrations only solve refund requests. How do we solve the rental default and rent arrears problem? Once we solve the rental default problem, we would have also solved the arrears problem.

Assume that Mr. X wanted to rent an apartment in a smart city and the cost of renting is #3,500,000 per annum and he is expected to pay an additional #1,900,000 for utilities and depreciation. At present, he has #3,150,000 and he is not certain that he will complete the annual subscription for the next two months. He is determined to rent the smart house. He approached the smart city manager and explained to him his intention to rent the house and his financial status. He promised to complete the rental at the end of the fourth month. The manager advised him to subscribe with the available resources and that he can resubscribe before the existing subscription elapses, otherwise he may not have access to the house when his subscription elapses. Mr. X agreed and subscribed with the available money. A week before the expiration of the subscribed amount he resubscribed with #305,000 and further subscribed with #10,000. How long would he reside in the house? Table 4 below contains the details.

The adaptive Q in Table 5 indicates that if Mr. X could not continue his subscription based on the available subscriptions in Table 4 his tenancy would elapse. The breakdown analysis of Mr. X’s subscription is shown in Figure 5. The analysis in Table 5 showed that rent arrears and rental default problems could be solved in the future if an adaptive subscription model is adopted for the housing sector by installing smart doors.

Based on Figure 6, if Mr. X does not subscribe after the last day of the 7 ^th month, his tenancy would automatically elapse which implies that the AAC would have been deactivated, but if he immediately resubscribed as in Figure 7 and Figure 8, the AAC will activate and hence his tenancy would be renewed. This process would continue depending on his subscription status. This is summarized in Figure 5.

In Figure 7, he subscribed for 21 days, and in Figure 8 for 16 hours 30 minutes (0.68 day). He is expected to vacate the house at the end of the Q period.

Let assume that Mr. X paid the complete annual rent. Figure 9 is the ARSM for an annual payment and Q duration. After the last day of the last month, Mr. X cannot access the house unless he subscribes again. The adaptive concept is unique in many ways because it can easily be implemented by installing smart doors in existing houses.

With the high rate of rental arrears in the public and private housing sectors, the adaptive model would be the preferred model to solve rental default. On the other hand, the recursive model could be used for the low income and daily income earners since it is very flexible and it updates the rent arrears. In general, the adaptive model is very suitable for the private sector housing developer. The adaptive model is also suitable for shopping mall and other large scale housing developers. These models are recommended to the government for action.

This paper has advanced flexible and rigid rental subscription models. The recursive model can be adopted for ‘traditional’ renting, as they allow for interaction and flexibility with regards to rental default. It also allows for a rent refund. It can be described as a model that could allow housing for people with limited resources. The recursive model is flexible in the sense that the tenant would default and still reside in the house. It is a good model for low-income and daily income earners because it provides space for them to source their rent. The adaptive model is a purely business-concern based model that allows access when subscribed. It is a non-interactive model that helps to solve corporate and high-profile rental default problems. The analysis showed that the application of the recursive model is easy and cheap, however the adaptive model is expensive. The analysis in the tables and figures showed the operational framework of the models. Therefore, the analysis showed that the recursive and adaptive models are suitable to solve rental refund and rental default problems.