One of the key factors to control the spread of any infectious disease is the health care facilities, especially the number of hospital beds. To assess the impact of number of hospital beds and control of an emerged infectious disease, we have formulated a mathematical model by considering population (susceptible, infected, hospitalized) and newly created hospital beds as dynamic variables. In formulating the model, we have assumed that the number of hospital beds increases proportionally to the number of infected individuals. It is shown that on a slight change in parameter values, the model enters to different kinds of bifurcations, e.g., saddle-node, transcritical (backward and forward), and Hopf bifurcation. Also, the explicit conditions for these bifurcations are obtained. We have also shown the occurrence of Bogdanov-Takens (BT) bifurcation using the Normal form. To set up a new hospital bed takes time, and so we have also analyzed our proposed model by incorporating time delay in the increment of newly created hospital beds. It is observed that the incorporation of time delay destabilizes the system, and multiple stability switches arise through Hopf-bifurcation. To validate the results of the analytical analysis, we have carried out some numerical simulations.
Citation: A. K. Misra, Jyoti Maurya, Mohammad Sajid. Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11628-11656. doi: 10.3934/mbe.2022541
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Abstract
One of the key factors to control the spread of any infectious disease is the health care facilities, especially the number of hospital beds. To assess the impact of number of hospital beds and control of an emerged infectious disease, we have formulated a mathematical model by considering population (susceptible, infected, hospitalized) and newly created hospital beds as dynamic variables. In formulating the model, we have assumed that the number of hospital beds increases proportionally to the number of infected individuals. It is shown that on a slight change in parameter values, the model enters to different kinds of bifurcations, e.g., saddle-node, transcritical (backward and forward), and Hopf bifurcation. Also, the explicit conditions for these bifurcations are obtained. We have also shown the occurrence of Bogdanov-Takens (BT) bifurcation using the Normal form. To set up a new hospital bed takes time, and so we have also analyzed our proposed model by incorporating time delay in the increment of newly created hospital beds. It is observed that the incorporation of time delay destabilizes the system, and multiple stability switches arise through Hopf-bifurcation. To validate the results of the analytical analysis, we have carried out some numerical simulations.
1.
Introduction
Several large-scale epidemic outbreaks have occurred in recent decades, including Ebola, SARS, Zika virus, swine flu, and COVID-19, contributing to low socio-economic status and inadequate healthcare access. The relative shortage of healthcare facilities, especially hospital beds, is a growing issue that can influence the subsequent care provided to the patients. Consequently, it is observed that, in order to ensure the welfare of its population, a city must have an adequate number of hospital beds. The city incurs unnecessary costs if the number of hospital beds is too large, but there is the risk of disease outbreaks if it is too small. In 2017, there were only 5 hospital beds for every 10,000 persons (i.e., Hospital-bed-population ratio) residents in India, while this value is 22, 28, and 128 for Saudi Arabia, the United States, and Japan, respectively [1,2]. This Hospital-bed-population ratio is very small in low-income countries [1]. According to WHO statistics, the data on in-patient bed density is one of the significant tools to assess the level of health service delivery [3]. Thus, it is more crucial to discern the impact of limited hospital beds on future dynamics of the disease and determine the number of hospital beds that can be appropriate to control infection.
There has been a considerable effort put into developing some realistic mathematical models for a better understanding of transmission dynamics of existing and emerging infectious diseases [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Such mathematical models focus on comprehending the epidemiological transmission pattern and signifying the impact of public health services to prevent the possible outbreak and transmission of the disease. For instance, Asif et al. [18] in their study, explain the transmission of influenza by using a diffusive epidemic model. Bozkurt et al. [17] study a mathematical model for COVID-19 transmission with fear in the population. In these models, the total human population has been split into different categories depending on the status of individuals concerning infection. The simplest epidemic models to study and to tackle with are SIR and SIS type models [14,19]. In these models total human population is either subcategorized into two classes (i.e., susceptible (S) and infected (I)) or three classes (i.e., susceptible(I) and recovered (R)). Umar et al. [19] establish an SIR epidemic model for dengue fever and study it using the artificial neural networks along with the Levenberg-Marquardt backpropagation technique, i.e., ANNs-LMB. Apart from this, only a few mathematical modelers have incorporated the hospital beds and other medical resources in their proposed model to address the impact of healthcare resources on the control of disease outbreaks [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. In this regard, Karnon et al. [27] describe several case studies that illustrate the role of mathematical modeling in improving healthcare delivery with various mathematical techniques. In each case, they found that mathematical research has been able to add to the expertise of the clinicians and policymakers. Further, considering the WHO hospital-beds-population ratio (HBPR) rationale, Shan and Zhu [32] have studied an SIR epidemic model by considering standard incidence rate and nonlinear recovery rate that depends on the number of hospital beds. Also, Saha and Ghosh [31] consider a deterministic model with logistic growth in susceptible population and a nonlinear treatment rate with limited hospital beds and found that the dynamical behavior of the proposed model not only depends on the basic reproduction number but also affected by the number of hospital beds, vaccination, and other medical resources. Moreover, Njankou and Nyabadza [24] have presented an epidemiological model to investigate the future course of Ebola virus disease when hospital beds are limited. Their mathematical findings suggest that the supply of hospital beds in Ebola treatment units has a decisive role in reducing the number of Ebola patients. To analyze the impact of hospital beds and spatial heterogeneity, Zhang et al. [35] has proposed a diffusive SIS model in the heterogeneous environment for prevention and control of an infectious disease. During the COVID-19 pandemic in 2021, Booton et al. [22] established a regional mathematical model to estimate the number of infected individuals, death caused by the infection, and the required number of hospital beds (acute and intensive care). Also, Area et al. [23] developed a compartmental mathematical mode model for COVID-19 to predict the number of beds required in intensive care unit. Their study is helpful to manage the number of beds in intensive care unit.
Further, Misra and Jyoti [29] have discussed a mathematical model with a limited number of regular and temporary hospital beds. In their study, they assumed that a fixed number of temporary hospital beds are introduced during an epidemic outbreak to reduce the surge in hospitals. From this, it is found that the introduction of temporary hospital beds can lessen the number of infected individuals and the number of deaths caused by the infection. However, the medical resources used to control the disease are always limited, and the policies that intervene vary depending upon the number of infected individuals [30]. Thus, the common wisdom is that the number of hospital beds should be increased with the increasing number of infected population.
From the above, it is shown that maintaining enough beds in the hospital is decisive to restrain the spread of infection in the population. Motivated from the prior discussion, in this paper, we study the disease dynamics when the number of hospital beds is being incremented in proportion to the increasing number of infected individuals. Also, the process to construct new hospital beds require a lot of effort, time, and money, which may lead to some time lag between the construction of new hospital beds and the current data of infected individuals. Therefore, we also introduce a delay in the increment of new hospital beds to intervene in a more realistic situation.
2.
The mathematical model
It is a well-established fact that new beds are created in hospitals every day, and this has considerable implications for future patient dynamics of infectious diseases. Thus, to assess this effect on a disease outbreak, we have classified the total human population N(t) into three categories: S(t), I(t), and H(t) sequentially represent the population of susceptible, infected, and hospitalized at time t. The allocation of hospital beds in any hospital depends on the demand. Thus, we assume that Hb is the number of newly created hospital beds that are increased with an increasing number of infected population at time t that are created at a rate ϕ. We also consider that some hospital beds are not functioning properly and thus decreased with a rate ϕ0.
The term (Ha+Hb−H) (here H≤Ha+Hb) in model system (2.1) represents number of available hospital beds at any instant of time t, where Ha is the total number of pre-existing hospital beds. The basic notion of the proposed model is that healthy people come in contact with infected individuals, contract infection, and become infective. Here, we assume that the contact rate of susceptible population with infected individuals is proportional to the total population size N; therefore, we consider the simple mass action incidence rate βSI for the proposed model. This term βSI, represents the transmission of individuals from the susceptible class to the infected class at time t, where β is the average number of adequate contacts per unit time with infected individuals (also called transmission rate). The constants d, α, ν, and ν1 sequentially represent the natural mortality, disease-induced mortality, self-recovery and hospital recovery rates. The parameters A and k1 are immigration and hospital bed occupancy rates, respectively. We assume that some hospitalized population may die in the hospital because of infection, thus experience some extra mortality due to the emerged infectious disease with proportionality constant θ.
From the above consideration, the rate of change of each class of human population (i.e., susceptible, infected, and hospitalized) and newly created hospital beds is given by the following system:
Here, S(0)=S0>0, I(0)=I0≥0, H(0)=H0≥0 and Hb(0)=Hb0≥0.
3.
Basic properties
3.1. Model equilibria
The model system (2.1) always has equilibrium E0(Ad,0,0,0) at which the infected population is zero (i.e., I=0). Thus, the basic reproduction number is obtained as
R0=βAd(ν+α+d+k1Ha).
For I≠0, the components S, I, H of any endemic equilibrium satisfies,
We set, D0=A22+4βϕ0k1d(R0−1)(ν+α+d+k1Ha)(d+θα+ν1){ϕ0(α+d)+ϕ(d+θα)}.
Further, for R0>1 the equation F(I)=0 always has a real positive root and has two real positive roots if and only if R0<1, A2<0, and D0>0, that can be written as:
I∗1=−A2+√D02A1,andI∗2=−A2−√D02A1, here I∗1>I∗2.
Moreover, for A2<0 and D0=0, we obtained a threshold value Rc of R0, that can be written as
Rc=1−A224A1ϕ0d(ν+α+d+k1Ha)(d+θα+ν1).
Therefore, the results regarding existence of equilibrium of model system (2.1) are concluded in the theorem below:
Theorem 1.The model system (2.1) has
(i) A disease-free equilibrium E0(Ad,0,0,0), which always exists.
(ii) A unique endemic equilibrium E∗1, it exists whenever R0>1.
(iii) A unique endemic equilibrium E∗1, it exists when R0=1, provided A2<0, otherwise no endemic equilibrium exists.
(iv) Two endemic equilibria E∗1 and E∗2, exist whenever R0<1, provided A2<0 and D0>0; otherwise no endemic equilibrium exists. These two equilibria (E∗1 and E∗2) collide and unite into a unique endemic equilibrium E∗3 when D0=0 and A2<0 (or equivalently R0=Rc).
3.2. Local stability analysis
At equilibrium E0 the Jacobian matrix for model system (2.1) is given by
The above matrix yield four eigenvalues (i.e., −d, −(d+θα+ν1), −ϕ0, and (R0−1)(ν+α+d+k1Ha)), from which three are always negative and (R0−1)(ν+α+d+k1Ha) depends on the value of R0. Thus, results regarding nature of equilibrium E0 are compiled in the theorem below:
Theorem 2.The equilibrium E0, is locally asymptotically stable until R0<1, as R0 exceed unity, it becomes unstable.
Now, the Jacobian matrix for model system (2.1), calculated at any endemic equilibrium is written as
Therefore, based on the above expression, we can say that I∗=I∗1 gives, B4>0. Further, I∗=I∗2 gives B4<0 when R0<1. Therefore, using the Routh-Hurwitz criterion, we can state that the equilibrium E∗1 is locally asymptotically stable iff
B3>0andB3(B1B2−B3)−B21B4>0,
(3.3)
whereas E∗2 is unstable whenever exists. Thus, the results regarding local stability behavior of endemic equilibria are compiled in theorem below:
Theorem 3.If system (2.1), has simply a unique endemic equilibrium E∗1, then it is locally asymptotically stable iff B3>0andB3(B1B2−B3)−B21B4>0. If there exists exactly two endemic equilibria, then the endemic equilibrium with high endemicity (i.e., E∗1) is locally asymptotically stable iff B3>0andB3(B1B2−B3)−B21B4>0, whereas the equilibrium with low endemicity (i.e., E∗2) is unstable, whenever exists.
Thus, the matrix J0(β∗) simply has one zero eigenvalue at R0=1, and the rest three eigenvalues (i.e., −d, −(d+θα+ν1), and −ϕ0) are negative. Therefore, to identify the direction of transcritical bifurcation, we first obtain the right eigenvector (˜U=(˜u1,˜u2,˜u3,˜u4)T) and left eigenvector (˜V=(˜v1,˜v2,˜v3,˜v4)) of the matrix J0(β∗) corresponding to the zero eigenvalue, which are given as follows:
Now, the coefficients ˜a and ˜b, in [39] for model system (2.1) can be calculated as ˜a=4∑i,j,k=1˜vk˜ui˜uk∂2˜fk∂˜xi∂˜xj(E0,β∗), and ˜b=4∑i,k=1˜vk˜ui∂2˜fk∂˜xi∂β(E0,β∗) (here fk, k=1,…,4 are right hand side of d˜xidt, i=1,…,4 of model system (2.1)). Thus, we have
From the above expression, it is clear that ˜b is always greater than zero, thus applying the condition (i) and (iv) of Theorem 4.1 of [39]. Therefore, the following theorem emerges:
Theorem 4.The direction of transcritical bifurcation for model system (2.1) is
(i) backward if [β∗(β∗Ad−ν)+k1dϕϕ0]<k1Ha(k1d+β∗ν1d+θα+ν1), and
(ii) forward if [β∗(β∗Ad−ν)+k1dϕϕ0]>k1Ha(k1d+β∗ν1d+θα+ν1).
4.2. Saddle-node bifurcation
Here, we demonstrate the saddle-node bifurcation phenomenon through the application of Sotomayor's theorem. Keeping this aim in mind, we first select β as a bifurcation parameter. According to Theorem 1 system (2.1) possess exactly one endemic equilibrium E∗3 at R0=Rc (or β=βc) at which the Jacobian matrix J∗(E∗3,βc) has a simple zero eigenvalue. Let ˆUT=(ˆu1,ˆu2,ˆu3,ˆu4) and ˆV=(ˆv1,ˆv2,ˆv3,ˆv4) are respectively, the right and left eigenvector of matrix J∗(E∗3,βc), with respect to the zero eigenvalue. Thus, we have
Now, we consider that ˆF=(ˆf1,ˆf2,ˆf3,ˆf4), here ˆf1, ˆf2, ˆf3, and ˆf4 are respectively, the right hand side of dS/dt, dI/dt, dH/dt, and dHb/dt in model system (2.1). Thus, we have
From Eqs (4.1) and (4.2), Sotomayor's theorem demonstrates that saddle-node bifurcation exists at β=βc (or equivalently R0=Rc). Therefore, the resulting theorem is as follows:
Theorem 5.The equilibrium E∗3 is saddle in nature, iff Hb∗3≠H∗3, and k1≠βc(d+θα)d.
4.3. Hopf-bifurcation
We select transmission rate β as a bifurcation parameter to establish existence of Hopf-bifurcation at equilibrium E∗1. As a result, we can write each coefficient of the characteristic Eq (3.2) as a function of β, which gives
ξ4+B1(β)ξ3+B2(β)ξ2+B3(β)ξ+B4(β)=0.
(4.3)
We consider that, β=βr gives Bi(βr)>0 (i=3,4), and B3(βr)(B1(βr)B2(βr)−B3(βr))−B21(βr)B4(βr)=0. Therefore, the Eq (3.2) can be written as
(ξ2+B3B1)(ξ2+B1ξ+B1B4B3)=0.
(4.4)
The Eq (4.4) has four roots ξi (i=1,…,4), with a pair of purely imaginary roots ξ1,2=±iψ0, where ψ0=(B3/B1)1/2. The nature of remaining two roots (i.e., ξ3 and ξ4) can be identified with help of following set of equations:
{ξ3+ξ4=−B1,ψ20+ξ3ξ4=B2,ψ20(ξ3+ξ4)=−B3,ψ20ξ3ξ4=B4.
Therefore, we can say that ξ3 and ξ4 are always lies in the left half plane. Further, we choose any point in the ϵ−neighborhood of β for which ξ1,2=ω(β)±iκ(β). When we substitute this in Eq (4.3) and writing separately real and imaginary components, we obtain
ω4+B1ω3+B2ω2+B3ω+B4+κ4−6ω2κ2−3B1ωκ2−B2κ2=0,
(4.5)
and
4ωκ(ω2−κ2)−B1κ3+3B1ω2κ+2B2ωκ+B3κ=0.
(4.6)
As we know that κ(β)≠0, therefore from Eq (4.6), we have −(4ω+B1)κ2+4ω3+3B1ω2+2B2ω+B3=0. Substituting this in Eq (4.5), we obtain
Here, we show the existence of BT-bifurcation of co-dimension 2 in a small neighborhood of endemic equilibrium E∗3. Now, one can easily note that B4|R0=Rc=0, therefore if we assume B3|R0=Rc=0, and B2|R0=Rc>0, Equation (3.2) produces a simple zero eigenvalue with algebraic multiplicity 2, and two eigenvalues (i.e., σ3 and σ4) are negative or contain negative real parts.
Now, we shift the endemic equilibrium E∗3 to origin using the transformation ˜y1=S−S∗3, ˜x2=I−I∗3, ˜y3=H−H∗3 and ˜y4=Hb−Hb3∗. Thus, the model system (4.6) can be written as
where Q20=Q20 and Q11=Q11+2P20. Thus, we have the following theorem:
Theorem 7.If Q20≠0 and Q11≠0, then model system (2.1) manifests Bogdanov-Takens singularity of co-dimension 2 around the endemic equilibrium E∗3.
5.
Model with delay (τ>0)
It is well-known that the proposal for the creation of new hospital beds is based on the assessment of current data on infected individuals, but, this process of establishing new hospital beds requires a lot of effort, time, and money. This requirement may lead to some time lag between the construction of new hospital beds and the current data of infected individuals. Thus for the proper arrangement of new hospital beds, it is more reasonable to consider time lag in the increment of newly created hospital beds. In this regard, we have considered that at time t, the increment in new hospital beds is in accordance with the number of infected individuals reported at time t−τ (for some τ>0). Taking this into account, the dynamics of model in presence of delay can be expressed by the following system:
Here, S(0)=S0>0, I(ε)=I0≥0 for ε∈[−τ,0), H(0)=H0≥0 and Hb(0)=Hb0≥0.
5.1. Stability analysis
This section aims to study the stability behavior of endemic equilibrium E∗1 of model system (5.1) in presence of time delay and also explore the possibility of Hopf-bifurcation at the equilibrium E∗1 by taking τ as a bifurcation parameter [40]. For this, we first linearize the model system (5.1) about the endemic equilibrium E∗1(S∗1,I∗1,H∗1,H∗b1) by using the following transformations: S(t)=S∗1+s(t), I(t)=I∗1+i(t), H(t)=H∗1+h(t), and Hb(t)=H∗b1+hb(t). Thus, the linear system of model system (5.1) around the equilibrium E∗1 is given as follows:
dϑdt=Q1ϑ(t)+Q2ϑ(t−τ),
(5.2)
where ϑ=[s(t),i(t),h(t),hb(t)]⊤, Q1=[−(βI∗1+d)−(βS∗1−ν)ν10βI∗10k1I∗1−k1I∗10k1(Ha+H∗b1−H∗1)−(d+θα+ν1+k1I∗1)k1I∗1000−ϕ0], and Q2=[0000000000000ϕ00]. Therefore, the linearized system (5.2) has the characteristic equation as follows:
Now, to show the occurrence of Hopf-bifurcation, the Eq (5.3) must have a pair of purely imaginary roots. Thus, we substitute, φ=iη(η>0) in Eq (5.3) and separate real and imaginary parts, we have the following equations:
η4−ϱ2η2+ϱ4=−ς2ηsinητ−(ς3−ς1η2)cos(ητ),
(5.4)
ϱ1η3−ϱ3η=ς2ηcos(ητ)−(ς3−ς1η2)sin(ητ).
(5.5)
Substituting η2=ω in the sum of square of Eqs (5.4) and (5.5), we obtain fourth degree polynomial equation in ω as follows:
P(ω)=ω4+G1ω3+G2ω2+G3ω+G4=0,
(5.6)
here G1=ϱ21−2ϱ2, G2=ϱ22−2ϱ1ϱ3+2ϱ4−ς21, G3=ϱ23−2ϱ2ϱ4−ς22+2ς1ς3, G4=ϱ24−ς23.
Now, the nature of solutions of Eq (5.6) is discussed in the following cases:
P1: If all Gi's(i=1,…,4) in P(ω) are positive then by Descarte's rule of sign, Equation (5.6) posses no real positive root and therefore the Eq (5.3) has no purely imaginary roots for τ>0. Thus, roots of Eq (5.3) are either negative or contain the negative real part, when τ>0. Therefore, the following theorem emerges:
Theorem 8.The equilibrium E∗1, is locally asymptotically stable for all τ>0 if E∗1 is feasible, whenever the condition (P1) is fulfilled, whereas it is stable in presence of delay.
P2: If none of the coefficients Gi's(i=1,…,4) in Eq (5.6) are positive, then by using the Descarte's rule of sign, we can say that the Eq (5.6) has exactly one real positive solution if one of the following conditions holds:
(b1):G1>0, G2>0, G3>0, and G4<0.
(b2):G1>0, G2>0, G3<0, and G4<0.
(b3):G1>0, G2<0, G3<0, and G4<0.
(b4):G1<0, G2<0, G3<0, and G4<0.
If either of the conditions bi's (i=1,…,4) is fulfilled, then the Eq (5.3) posses a pair of purely imaginary roots ±iη0. Further, from Eqs (5.4) and (5.5) corresponding to positive value of η0, we have
tan(η0τ)=Θ1Θ2,
where, Θ1=ς2η0(η40−ϱ2η20+ϱ4)+(ς3−ς1η20)(ϱ1η30−ϱ3η0) and Θ2=(ς3−ς1η20)(η40−ϱ2η20+ϱ4)−ς2η0(ϱ1η30−ϱ3η0). Therefore, the value of τm with respect to the positive value of η0 can be obtained as follows:
τm=mπη0+1η0tan−1(Θ1Θ2),form=0,1,2,…
By taking the advantage of Butler's Lemma [41], one can easily note that the stability of equilibrium E∗1 of model system (2.1) is maintained for τ<τ0. Now, to identify, whether the Hopf-bifurcation exists or not as τ increases through τ0, we prove the following lemma:
Lemma 1.If condition (P2) is fulfilled, thus the following condition holds:
sgn[d(ℜ(φ))dτ]τ=τ0>0.
Proof. Differentiating Eq (5.3), with respect to τ, gives
Thus, it can be noted that P′(η20)>0 if one of the conditions (bi)'s (i=1,…,4) is satisfied. This proves the Lemma 1.
Thus, the following result emerges:
Theorem 9.If conditions (3.3) holds and any one of the conditions (bi)'s (i=1,…,4) is fulfilled, then equilibrium E∗1 is locally asymptotically stable for all τ∈[0,τ0) and trn into unstable for τ>τ0. The model system (5.1) manifests a supercritical Hopf-bifurcation when τ=τ0, generating a family of periodic solutions bifurcating from equilibrium E∗1 as τ crosses τ0[42].
Remark 1.Further, if none of the conditions (bi)'s (i=1,…,4) is satisfied, then it is possible that Eq (5.6) posses more than one positive roots. In this case, Equation (5.3) exhibits pair of two purely imaginary roots. Therefore, we numerically deliberate the result for which Eq (5.6) posses two real positive roots.
P3: Analytically, the condition for which Eq (5.6) posses two real positive real roots is not easy to obtain, i.e., Equation (5.3) has two pair of purely imaginary roots. So, we discuss the result numerically in which Eq (5.6) has two positive real roots. Numerically, it is obtained that the Eq (5.6) has two positive real roots ω+ (corresponding to η2+) and ω− (corresponding to η2−) where ω+>ω−, i.e., the characteristic Eq (5.3) has two pairs of purely imaginary roots ±iη±. For these positive values of η±, from Eqs (5.4) and (5.5) we can obtain the positive value of τ±m as follows:
Hence, if Eq (5.6) contains two real positive roots then P′(ω2+)>0 and P′(ω2−)<0 and therefore the transversality conditions hold. This concludes the proof of Lemma 2.
Thus, the following result emerges regarding the Hopf-bifurcation theorem of functional differential equation [42], which is stated as follows:
Theorem 10.If Eq (5.6) contains two real positive solutions and condition (3.3) is fulfilled, then a positive integer ˜q exists, for which ˜q stability switches occur from stability to instability and later the system becomes unstable. More specifically, when τ∈[0,τ+0),(τ−0,τ+1),…,(τ−˜q−1,τ+˜q), the equilibrium E∗1 is stable while it is unstable when τ∈(τ+0,τ−0),(τ+1,τ−1),…,(τ+˜q−1,τ−˜q−1).
6.
Numerical simulations
The qualitative behavior of the system (2.1) around equilibrium has been discussed in previous sections to comprehend the disease dynamics and get insights into the feasibility of equilibria, their local stability, and Hopf-bifurcation in absence as well as in the presence of delay. We have also identified the existence of transcritical, saddle-node, and BT-bifurcation in the absence of delay. To simulate the model system (2.1), we choose a set of hypothetical parameter values given in Table 1.
Table 1.
Parameter description of model system (2.1) and their considered values used for numerical simulation.
Parameter
Description
Value
A
Immigration rate
20 day−1
β
Transmission rate of individuals from susceptible class to infected class
2×10−3 person−1day−1
k1
Hospital bed occupancy rate
5×10−3 day−1
d
Natural mortality rate
0.01 day−1
ν
Self recovery rate
2.5×10−3 day−1
ν1
Hospital recovery rate
2×10−3 day−1
Ha
Total number of hospital beds
100
α
Disease induced mortality rate
0.2 day−1
ϕ
increment rate of new hospital beds
0.02 day−1
ϕ0
the rate at which hospital beds reduces
0.01 day−1
θ
Disease induced mortality coefficient of hospitalized individuals
For the hypothetically chosen set of parameter values in Table 1, the existence conditions for equilibrium E∗1 are fulfilled and basic reproduction number is R0=5.614>1. Also, the components of endemic equilibrium E∗1 is obtained as
First, to see the effect of newly created hospital beds on infected individuals' death due to infection, we generate a bar graph with k1=0.00002 and varying the increment rate of new hospital beds (ϕ) from 0.01 to 0.046 (Figure 1). This figure shows the change in percentage of susceptible (blue bar), infected (green), hospitalized (yellow), and deaths due to infection for different values of ϕ. One can easily observe that as we increase the increment rate of new hospital beds, will increase the percentage of the hospitalized population as well as decrease the percentage of infected individuals and deaths due to infection.
Figure 1.
The graphical representation of the change in percentage of disease-induced deaths (blue), hospitalized (yellow), infected (green), and susceptible (green) individuals for different increment rates of newly created hospital beds with k1=0.00002.
Further, we plot equilibrium curve in R0−I plane to demonstrate the different dynamics of model system (2.1). First, when we choose k1=0.0007, the model system (2.1) displays the transcritical bifurcation in forward direction, Figure 2(a). This figure reveals that when basic reproduction number R0>1, the disease will persist in the population and eventually go extinct when R0<1. Further, when we increase the hospital occupancy rate to k1=0.0011, the direction of transcritical bifurcation changes in backward, which fledges a complex situation, Figure 2(b). From this figure, we can see that two equilibrium points (one stable and one unstable) will collide and unite into one equilibrium point when the equilibrium curve reaches its turning point and disappear thereafter, which shows that the model system (2.1) exhibits saddle-node bifurcation at that turning point (marked with SN in Figure 2(b)). It can also be seen from this figure that the system exhibits a bistable case when R0∈(Rc,1), which means the disease will persist or not in the population is entirely dependent on the initial population of infected individuals. Also, keeping R0 less than the Rc will be necessary to eradicate the disease from the population.
Figure 2.
Bifurcation plot in R0−I plane for model system (2.1). (a) forward bifurcation occurs when k1=0.0007, (b) backward bifurcation occurs for k1=0.0011. The green curve indicates equilibrium points that are stable, and the brown curve indicates equilibrium points that are unstable. The violet curves represent the time series plot of infected individuals generated for the value of basic reproduction number indicated with red dots.
Furthermore, for the chosen set of parameter values except β=0.0016, we generate the surface plot of basic reproduction number R0 and its threshold quantity Rc by varying the hospital occupancy rate k1 (0–6×10−3) and total number of hospital beds Ha (0–300), Figure 3. From this figure, we can observe that with the increment in hospital occupancy rate, an increment in the number of hospital beds becomes necessary to eradicate the disease.
Figure 3.
Surface plot of basic reproduction number (R0) and its threshold (Rc) with β=0.00016. The green surface represents basic reproduction number and blue surface represents its critical value Rc.
When we generate the equilibrium curve in R0−I plane with hospital occupancy rate k1=0.001212, model system (2.1) exhibits two Hopf-points H1 (at R0≈0.9771) and H2 (at R0≈0.9956), Figure 4(a). From this figure, we observe that if R0<0.97134 (SN point), the disease will die out from the population, on the other hand, if R0∈(0.97137,0.977083) or R0∈(0.995607,1) the disease will persist or extinct from the population will depend on the initial size of the infected individuals. Further, between Hopf-points H1 and H2 model system (2.1) enters into limitcycle oscillation by forward supercritical Hopf-bifurcation and disappears via backward supercritical Hopf-bifurcation. This dramatic behavioral change of the model system will make the situation unpredictable between H1 and H2; thus, to make any decision to control the prevalence of disease, we have to reduce R0 below 0.97710 or increase above 1. Furthermore, when we increase the value of hospital occupancy rate to 0.00125, the model system (2.1), exhibits supercritical Hopf-bifurcation at R0=1.01585 and disappear via Homoclinic bifurcation, Figure 4(b).
Figure 4.
Bifurcation plot in R0−I plane for model system (2.1). (a) k1=0.001212, (b) k1=0.00125. The green curve indicates equilibrium points that are stable, and the brown curve indicates equilibrium points that are unstable. The violet curves represent the time series plot of infected individuals, generated for the value of basic reproduction number indicated with red dots.
Further, for k1=0.0028, the model system exhibits limitcycle bifurcation, Figure 5(a). This figure evinces that when R0<1.0402 the endemic equilibrium E∗1 violates the condition of stability and surrounds itself with a stable limitcycle. Also, for R0>1.0402, the stable endemic equilibrium E∗1 surrounds itself with two limit cycles, from which one is stable, and another is unstable, and these two limit cycles collide, unite and disappear via limitcycle bifurcation when R0≈1.1388. To get the clear idea about limitcycle bifurcation, we generate the bifurcation plot in R0−S−I space, Figure 5(b).
Figure 5.
Bifurcation plot for model system (2.1) when k1=0.0028 in (a) R0−I plane and (b) R0−S−I space. The green curve indicates equilibrium points (limitcycles) that are stable, and the brown curve indicates equilibrium points (limitcycles) that are unstable.
For k1=0.0085 model system (2.1) showcase the subcritical bifurcation and disappear through homoclinic bifurcation, Figure 6(a). This figure reveals that for R0>0.6271 the stable endemic equilibrium E∗1 surrounds with the unstable limitcycles, but for R0<0.6271, endemic equilibria E∗1 and E∗2 both become unstable. The bifurcation plot in R0−S−I space is shown in Figure 6(b).
Figure 6.
Bifurcation plot for model system (2.1) when k1=0.0085 in (a) R0−I plane and (b) R0−S−I space. The green curve indicates equilibrium points (limitcycles) that are stable, and the brown curve indicates equilibrium points (limitcycles) that are unstable.
To show the existence of BT-bifurcation of co-dimension 2, we have plot the bifurcation diagram in R0−ϕ, Figure 7(a). This algebraic curve in R0−ϕ represents the bifurcation plot near the BT-point (0.987376, 0.100291) at which the components of endemic equilibrium are (1920.168,2.2247,33.0449,22.3125). From Figure 7(a) construct with three curves i.e., saddle-node (SN), Hopf (H) and Homoclinic (Hom) that are represented with red, violet and green color, respectively. These three curve divide the R0−ϕ plane into four regions (Regions Ⅰ–Ⅳ). The dynamics of model system (2.1) is different in these four regions. The model system (2.1) exhibits a unique endemic equilibrium at the BT point (R0,ϕ)=(0.987376,0.100291), which is saddle in nature and a disease-free equilibrium, Figure 7(b). Further, there exists only disease-free equilibrium, when we choose the value of R0 and ϕ from Region Ⅰ, i.e., (R0,ϕ)≈(0.992,0.112), Figure 7(c). The model system (2.1) exhibits a disease-free equilibrium and two endemic equilibria from which one is stable focus, and another is saddle-node when we consider (R0,ϕ)≈(0.997,0.112) (Region Ⅱ), Figure 7(d). Again two endemic equilibria occurs when we choose the values of R0 and ϕ from Region Ⅲ, i.e., (R0,ϕ)≈(0.997,0.106), Figure 7(e). From which, one endemic equilibrium is saddle in nature and another is unstable, which surrounds itself with and stable limit cycle. Furthermore, two endemic equilibria, for the values taken from Region Ⅳ, i.e., (R0,ϕ)≈(0.996,0.102). The endemic equilibrium with high endemicity is unstable and another is saddle-node, Figure 7(f).
Figure 7.
(a) Bifurcation plot in R0−ϕ Plane, (b)–(f) Phase Portrait in S−I−H space.
Also, to demonstrate the effect of hospital occupancy rate on the disease dynamics, we have plot the bifurcation diagram in R0−k1 plane. This algebraic plane again consists of three curve saddle-node, Hopf and homoclinic, which unfold the BT-point, Figure 8(a). The saddle-node (SN), Hopf (H) and Homoclinic (Hom) curves divide the algebraic plane R0−k1 in four regions (Regions Ⅰ–Ⅳ). The dynamics of model system (2.1) is different in these four regions.
Figure 8.
(a) Bifurcation plot in R0−k1 Plane, (b)–(f) Phase Portrait in S−I−H space.
The model system (2.1), possess a unique endemic equilibrium at the BT point (R0,k1)=(0.979314,0.0.00161), which is saddle in nature and a disease-free equilibrium, Figure 8(b). For (R0,k1)≈(0.982,0.0015) (Region Ⅰ), the model system (2.1) exhibits only disease-free equilibrium, Figure 8(c). For (R0,k1)≈(0.991,0.00152) (Region Ⅱ), the proposed system (2.1) has two endemic equilibria from which one is stable focus, and another is saddle-node, Figure 8(d). When we consider the value of R0 and k1 from Region Ⅲ, i.e., (R0,k1)≈(0.991,0.00155), Figure 8(e) the model system (2.1) exhibits two endemic equilibria, one is saddle in nature and another is unstable, which surrounds itself with a stable limit cycle. Further, the system has one unstable equilibrium and one saddle-node when we choose the value of R0 and k1 from Region Ⅳ, i.e., (R0,k1)≈(0.988,0.00159), Figure 8(f).
Furthermore, when we choose k1=0.002 and generate the bifurcation plot in R0−α plane, the model system (2.1), showcase the generalized Hopf-bifurcation, Figure 9(a). This figure consists of two points (BT and GH) and three curves (violet, red and green). The points BT and GH sequentially represent the Bogdanov-Takens and generalized Hopf-bifurcation points. At the curve above GH point (green curve), model system (2.1) exhibits supercritical Hopf-bifurcation, and below GH point, subcritical bifurcation occurs. At the violet curve, the model system undergoes saddle-node bifurcation. The homoclinic curve generated from the BT point in S−I−Hb space is shown in Figure 9(b). The Figure 9(c) shows the phase portrait diagram of model system (2.1) for R0=1.05 and α=0.168. This value of R0 and α is chosen from Figure 9(a) (marked with red dot). From Figure 9(c), one can see that the model system exhibits one stable equilibrium, which surrounds itself with one stable (black curve) and one unstable (between red and blue curve) limitcycles.
Figure 9.
(a) Bifurcation plot in R0−α plane for model system (2.1), (b) BT-curve in S−I−Hb space, (c) Phase portrait in S−I plane for the value indicate with red dot in (a).
If we choose β=0.0003, k1=0.0007, ϕ=0.08 and ϕ0=0.04 and keep remaining parameter values same as in Table 1, the Eq (5.6) possess a unique positive real root, which ensure the existence of pair of purely imaginary roots for Eq (5.2). For the selected data τ0 is obtained 7.55 days. The bifurcation plot in τ−S−I space is show in Figure 10. The phase portraits in S−I−H space for τ=5 days (<τ0=7.55 days) and τ=12 days (>τ0=7.55 days) are sequentially shown in Figure 11(a), (b). This figure shows that the solution trajectories approaches to the equilibrium value when τ<τ0 and stable limitcycle obtained around unstable equilibrium, when τ>τ0.
Further, to show the different dynamical changes in the proposed model with time delay in increment of new hospital beds, we have choose β=0.0005, ν=0.1, ν1=0.5, α=0.09, ϕ=0.08, ϕ0=0.05 and the remaining parameter values are same as in Table 1. For these values of parameters, equation (5.6) has exactly two real positive roots i.e., ω+≈0.00722 and ω−≈0.00227 (ω+>ω−). Corresponding to these positive roots, the positive values of τ±m for k=1,2,3… is obtained. Thus, we have obtained τ+0=19.445 days, τ+1=93.399 days, τ+2=167.354 days, τ+3=241.308 days and so on and the values of τ−0=63.731, τ−1=195.497, τ−2=327.264, τ−3=459.0304 and so on. Therefore, using Theorem 10, one can observe that for the selected set of data the endemic equilibrium is stable if τ∈[0,19.445)⋃(63.7306,93.399) and unstable for (19.455,63.7306)⋃(93.399,∞). This dynamical change can be seen in bifurcation plot in Figure 12. The phase portraits in S−I−H space for τ=12, τ=40, τ=80 and τ=120 are sequentially displayed in Figure 13(a)–(d).
To reduce the prevalence of infectious diseases, it is imperative to arrange an adequate number of hospital beds. Still, it is not entirely clear how many hospital beds will be able to achieve the complete eradication of such diseases. Thus, the best way to arrange sufficient hospital beds is to increase the number of hospital beds with the increase in infected individuals. Keeping this goal in mind, this work deals with the four-dimensional nonlinear mathematical model in which the human population is divided into three classes, i.e., susceptible, infected, and hospitalized. The proposed model incorporates the assumption that susceptible individuals become infected via direct contact with infected individuals. Accordingly, it is assumed that the number of hospital beds will increase with an increasing number of individuals infected with the disease. Further, we also consider that the funds involved in the creation of new hospital beds limit the growth rate of newly created beds at a rate ϕ0.
We begin with a qualitative analysis of the deterministic model. To thoroughly analyze the dynamics of the proposed model with a slight change in considered parameters, we study its bifurcation behavior (i.e., transcritical, saddle-node, Hopf, and BT-bifurcation). From Section 4.1, it is found that whenever (d+θα+ν1k1(k1d+β∗ν1))[β∗(β∗Ad−ν)+k1dϕϕ0]<Ha, the system manifests transcritical bifurcation in the backward direction, which means the classical requirement to eliminate the disease from the population will contravene, and disease may persist even if R0<1. Also, the proposed (2.1) demonstrates saddle-node bifurcation, whenever k1≠βc(d+θα)d, and H∗b3≠H∗3. For the local representation of homoclinic bifurcation, we deduced the normal form around the endemic equilibrium E∗3. We have also obtained the periodic pattern for the model system (2.1). The model system (2.1) becomes unstable around the endemic equilibrium for high hospital occupancy rates (k1), resulting in multiple Hopf-bifurcations, the first being supercritical and the second is subcritical. Through subcritical Hopf-bifurcation, we have also dissected the possibility of multiple limitcycles and limitcycle bifurcation, Figure 5.
It is clear from the results obtained in this study that the increase in hospital beds becomes very important as the number of infected individuals increases. From Figure 1, it can be easily noted that the percentage of infected individuals and deaths due to the infection can be significantly reduced to 1.372 and 6.86 percent, respectively, when the increment rate of new hospital beds increased from 0.01 to 0.046. Thus, the increment of new hospital beds can reduce deaths caused by they infection.
In order to predict more realistic dynamics for the proposed system, we have also added a time delay in the increase of new beds. We have also discussed the stability behavior of equilibrium E∗1 and existence of Hopf bifurcation around it, in the presence of time delay. It is observed that the introduction of time delay changes the dynamics of the system as delay parameter crosses a critical threshold for one set of parameter values of τ (i.e., τ∈[0,τ0)). The periodic oscillations with increasing amplitude have been observed when we increase the value of the time delay above the threshold (τ0), i.e., the endemic equilibrium E∗1 becomes unstable. The system switches between stability and instability for another set of parameter values and eventually, it turns into unstable. The study reveals that to control an epidemic outbreak, the sufficient number of hospital beds are necessary. The delay in adding new beds to hospitals may destabilize the system and lead to stability switches via Hopf-bifurcation, which makes it difficult to predict and control the prevalence of infection. Hence, the addition of new hospital beds on time is necessary to control the transmission of infectious disease.
In the present paper, we have focused on the hospital bed's incrementation and its impact on emerged disease dynamics. For our model, we have assumed that the hospital beds are increased proportional to the number of infected individuals. However, limited resource availability may affect this increment. Thus, a separate differential equation could be included in the model to describe this limitation of resource availability. Meanwhile, it is essential to link the cost associated with the new hospital beds. These topics may be considered in future. The impact of other pharmaceutical interventions like vaccine, medicines, transport facilities of hospitals, etc. can also be studied in future.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.
Conflict of interest
There are no conflicts of interest disclosed by the authors.
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A. K. Misra, Jyoti Maurya, Mohammad Sajid. Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11628-11656. doi: 10.3934/mbe.2022541
A. K. Misra, Jyoti Maurya, Mohammad Sajid. Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11628-11656. doi: 10.3934/mbe.2022541
Transmission rate of individuals from susceptible class to infected class
2×10−3 person−1day−1
k1
Hospital bed occupancy rate
5×10−3 day−1
d
Natural mortality rate
0.01 day−1
ν
Self recovery rate
2.5×10−3 day−1
ν1
Hospital recovery rate
2×10−3 day−1
Ha
Total number of hospital beds
100
α
Disease induced mortality rate
0.2 day−1
ϕ
increment rate of new hospital beds
0.02 day−1
ϕ0
the rate at which hospital beds reduces
0.01 day−1
θ
Disease induced mortality coefficient of hospitalized individuals
0.1×10−3
Figure 1. The graphical representation of the change in percentage of disease-induced deaths (blue), hospitalized (yellow), infected (green), and susceptible (green) individuals for different increment rates of newly created hospital beds with k1=0.00002
Figure 2. Bifurcation plot in R0−I plane for model system (2.1). (a) forward bifurcation occurs when k1=0.0007, (b) backward bifurcation occurs for k1=0.0011. The green curve indicates equilibrium points that are stable, and the brown curve indicates equilibrium points that are unstable. The violet curves represent the time series plot of infected individuals generated for the value of basic reproduction number indicated with red dots
Figure 3. Surface plot of basic reproduction number (R0) and its threshold (Rc) with β=0.00016. The green surface represents basic reproduction number and blue surface represents its critical value Rc
Figure 4. Bifurcation plot in R0−I plane for model system (2.1). (a) k1=0.001212, (b) k1=0.00125. The green curve indicates equilibrium points that are stable, and the brown curve indicates equilibrium points that are unstable. The violet curves represent the time series plot of infected individuals, generated for the value of basic reproduction number indicated with red dots
Figure 5. Bifurcation plot for model system (2.1) when k1=0.0028 in (a) R0−I plane and (b) R0−S−I space. The green curve indicates equilibrium points (limitcycles) that are stable, and the brown curve indicates equilibrium points (limitcycles) that are unstable
Figure 6. Bifurcation plot for model system (2.1) when k1=0.0085 in (a) R0−I plane and (b) R0−S−I space. The green curve indicates equilibrium points (limitcycles) that are stable, and the brown curve indicates equilibrium points (limitcycles) that are unstable
Figure 7. (a) Bifurcation plot in R0−ϕ Plane, (b)–(f) Phase Portrait in S−I−H space
Figure 8. (a) Bifurcation plot in R0−k1 Plane, (b)–(f) Phase Portrait in S−I−H space
Figure 9. (a) Bifurcation plot in R0−α plane for model system (2.1), (b) BT-curve in S−I−Hb space, (c) Phase portrait in S−I plane for the value indicate with red dot in (a)
Figure 10. Bifurcation plot in τ−S−I space
Figure 11. Phase portrait in S−I−H space for (a)τ=5, (b) τ=12
Figure 12. Bifurcation plot in τ−I plane
Figure 13. Phase portrait in S−I−H space for (a) τ=12, (b) τ=40, (c) τ=80, (d) τ=120