We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.
Citation: Masoud Saade, Samiran Ghosh, Malay Banerjee, Vitaly Volpert. An epidemic model with time delays determined by the infectivity and disease durations[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 12864-12888. doi: 10.3934/mbe.2023574
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Abstract
We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.
1.
Introduction
In the last 15 years, US deaths due to the opioid epidemic have quadrupled from nearly 12,000 in 2002 to more than 47,000 in 2017 [1]. In October 2017, the US Department of Health and Human Services declared the opioid crisis a national public health emergency [2]. The increase in injection drug use and reduction of behavioral inhibition have also contributed to the spread of infectious diseases, particularly HIV [3]. The two epidemics – opioid and HIV – are intertwined and modeling them in tandem will lead to understanding their interdependence [4].
The HIV epidemic, which begun in 1981, has been modeled extensively both on immunological level and on epidemiological level. On within-host level typically models involve healthy and infected CD4 cells and the virus [5,6,7,8,9]. On the between-host scale, the very basic models include susceptible, infected and AIDS classes [10,11]. Multi-scale models of HIV, mostly of the nested type [12], have also received significant attention in the recent years [13,14,15,16].
Modeling of the opioid epidemic is more recent and picked up with the increasing importance of the substance abuse disorder. Early models [17] assume opioid use can be modeled similarly to infectious disease spread and that assumption has been used in current models of heroin use [18,19,20,21]. The transition to opioid use typically starts from prescription drug misuse and modeling of that has also been drawing attention lately [22].
Despite the importance of the HIV and the opioid epidemics and the clear interdependence between the two, relatively little modeling has been done at the interface of the two epidemics. The within-host modeling of the interplay between opioid and HIV has first drawn attention. Based on experiments in monkeys, Vaidya et al. [23,24,25,26] model the within-host dynamics of HIV and an opioid. The between host dynamics has been addressed even less. Duan et al. [27] models the two epidemics on population level and finds that the best control strategy should reduce the probability of opioid affected individuals getting HIV and should target the drug abuse epidemic. In this paper, we investigate the interplay of the two epidemics with a multi-scale model that incorporates both a within-host component and a between host component.
The sexual contacts leading to HIV are not homogeneous across the population. Typically, few individuals in the population partake in a lot of contacts, while most of the members of the population partake in few contacts. This type of heterogeneity of contacts in HIV transmission is modeled by scale-free networks. Modeling infectious disease dynamics on networks has been also drawing attention in the recent years, resulting in a book devoted to this topic [28]. We use here a network modeling approach introduced in [29] which gives a closed form model equations [30]. In this modeling framework of networks, ODE and age structured models have been investigated but the only multi-scale model on networks in closed form seems to be [16]. Here we expand the modeling framework developed in [16] to include the opioid epidemic thus considering for the first time a multi-scale network model of two diseases. This model is in closed form and we are able to perform analysis on it.
In Section 2 we introduce the multi-scale network model of HIV and opioid which consists of within-host component, between-host component and linking functions used to connect the two scales. In Section 3 we discuss the existence and stability of the disease-free equilibrium and compute an explicit form of the reproduction numbers. In Section 4 we focus on the existence and stability of the semi-trivial equilibria. In Section 4 we also compute the invasion numbers of the two epidemics. In Section 5 we perform simulations and consider different scenarios with differing parameter values. In Section 6 we summarize our results.
2.
The model
2.1. The within-host model
We modify a well-known within-host model of HIV by explicitly including the opioid drug concentration C(τ) and its impact on the average susceptibility of target cells. The model takes the following form.
Here T are the target cells, Ti are the infected target cells and Vi is the virus (HIV). Target cells are produced at rate s and cleared at rate dT. Infected cells die at a rate δi, releasing Nv viral particles at death. The clearance rate of the virus is denoted by c. Opioid is taken at doses Λ is degraded at rate dc. Infection rate of target cells by HIV in the presence of opioid is given by
k(C)=k0+k1C(τ)C0+C(τ),
where C0 is the half saturation constant, k0 is the transmission coefficient in the absence of opioid and k1 is a maximal increase in infection rate due to opioids.
A shortcoming of this model is that it does not consider multiple infection routes and drug resistant viral strains. This problem could have been remedied if a model such as the one formulated in [31] was used instead, as the base model. But because of the network structure, the system is already complicated, and addition of strains and infection route would further complicate it. We believe addition of strains and infection route would not give any different insights with the network structures.
2.2. The between-host model
To introduce the model, we define a complex network with size (number of nodes) N where each node is either occupied by an individual or vacant. The states for the epidemic transmission process on the network are divided into vacant state E, susceptible state S, opioid state U, infected state V, co-affected state i(τ,t) and AIDS state A. Nodes change states at rates to be introduced, and HIV transmission is governed by the network connections. A vacant state becomes susceptible state at the recruitment rate. Susceptible, opioid, infected, co-affected and AIDS states can change their state into a vacant state at natural death rate μ or at disease-induced death rates du, dv, di(τ), da, respectively. A susceptible state can be infected with HIV and change into an infected state, or can become opioid-dependent and change into opioid state. HIV and opioid states can get co-affected by adding the other disease. An opioid state or co-affected state can move to a susceptible or HIV-infected states respectively due to treatment denoted as δ. HIV-infected or co-affected states can move to the AIDS state at rates γv and γi(τ).
For an epidemic network, degree of a node is the number of contacts the node has with other nodes. We assume that the network contacts are HIV type contacts. That is an edge between any two nodes represents a potential for transmission of HIV, either through sexual contact or intravenous drug usage. For a network with maximal degree n, the average network degree is given by
<k>=n∑k=1kp(k),
where p(k) is the probability that a randomly chosen node has degree k. That is, <k> is a constant, when n is pre-specified. This is a standard notation for average degree. For further background, [32] is a good source for network science. Empirical studies suggest that many real-life HIV networks have scale-free degree distribution p(k)∼k−η, where 2<η<3[33,34]. The conditional probability p(j|k) that a node with degree k is connected to a node with degree j, is given by
p(j|k)=jp(j)<k>.
Basically, p(j|k) gives probability that an individual with k contacts is connected to an individual with j contacts. We assume contacts that lead to opioid addiction are homogeneous (transmission of opioid addiction can be between two nodes with the same probability). With the structure of a complex network and infection age, let Sk(t), Uk(t), Vk(t), Ak(t), be the number of susceptible, opioid-dependent, HIV infected and AIDS nodes respectively with degree k at time t for k∈{1,2,…,n}. Let ik(τ,t) be the density of co-affected nodes of degree k at time t and with infection age τ. Then we can formulate the following network multi-scale model of HIV and opioid epidemics.
The total population size of degree k is Nk(t)=Sk(t)+Uk(t)+Vk(t)+Ik(t) where Ik(t)=∫∞0ik(t,τ)dτ. The force of HIV infection, λv(t) takes into account the heterogeneous mixing in the network:
λv(t)=1<k>n∑k=1kp(k)λkv(t) where λkv(t)=βkv1Vk(t)+∫∞0βkv2(τ)ik(t,τ)dτNk(t).
(2.3)
Thus λkv(t) denotes the force of infection from a node with degree k, where βkv1 is the infection rate from Vk(t) (HIV-infected node with degree k) per effective contact, and similarly, βkv2(τ) is the time-since-infection dependent infection rate from ik(τ,t) (co-affected node with degree k) per effective contact. Then the force of infection λv(t) for the heterogeneous network model is obtained by summing over all the degrees of the network λkv(t) times the probability that the node with degree k is linked to the node with degree j. This is the average force of infection from each contact, and is therefore multiplied by k in the equations (2.2) for HIV transmission to nodes of degree k. The force of opioid addiction, λu(t) is then given by,
λu(t)=n∑k=1λku(t) where λku(t)=βuUk(t)+∫∞0ik(t,τ)dτNk(t).
(2.4)
Since the opioid contacts are homogeneous within the network, the force of addiction is obtained by summing over all the force of addictions of degree k, λku(t). The within-host model remains the same as in (2.1) and the linking functions are given below.
2.3. Linking functions
To link the within-host and between host models, we use data [35] to determine the form of the linking of the transmission coefficient βkv2(τ)[36] to the viral load. Fitting to the data [36], we obtain the following function for βkv2(τ):
βkv2(τ)=βk0Vri(τ)B+Vri(τ),
where r≈1. Further, we use the suggested functions in [37] to link the remaining τ-dependent rates:
di(τ)=d0(T(0)−T)+d1,γi(τ)=γ0(T(0)−T),σ(τ)=σ0,
where βk0,B,d0,d1,γ0,σ0, are constants. Disease-induced death rate di and transition to AIDS rate γi do not depend explicitly on the viral load because the viral load is high during the acute HIV phase but these rates are low during this same stage.
3.
Existence and stability of the disease free equilibrium
Equilibria are time-independent solutions of the system and often determine its long-term behavior. We find the equilibria of this system by setting the derivatives with respect to t equal to zero.
At the disease-free equilibrium Uk(t), Vk(t) and ik(t,τ) are zero for all k. So, the disease-free equilibrium is given by
ε0=(Λ1μ,0,0,0,⋯,Λnμ,0,0,0).
To determine the stability of the disease free equilibrium and to find the basic reproduction numbers of HIV infection and opioid addiction, denoted by Ru and Rv respectively, we linearize the system around the disease-free equilibrium. We take Sk(t)=S0k+xk(t), Uk(t)=uk(t), Vk(t)=vk(t), Nk(t)=N0k+nk(t) and ik(t,τ)=yk(t,τ). Then linearizing the system (2.2) takes the following form
Theorem 1.If max{Ru,Rv}<1 then the disease free equilibrium is locally asymptotically stable. If max{Ru,Rv}>1, then the disease-free equilibrium is unstable.
Then we notice that G1(0)=Ru, G2(0)=Rv, limλ→∞G1(λ)=0, limλ→∞G2(λ)=0.
We claim that if max{Ru,Rv}<1 then the disease free equilibrium is locally asymptotically stable, that is all the solutions of G1(λ)=1 and G2(λ)=1, have negative real parts. To show this, we proceed by way of contradiction. Suppose one of the equations G1(λ)=1 and G2(λ)=1 has a solution λ0 with ℜ(λ0)≥0. Then,
1=|G1(λ0)|≤|G1(ℜλ0)|≤|G1(0)|=Ru,
or
1=|G2(λ0)|≤|G2(ℜλ0)|≤|G2(0)|=Rv.
This is a contradiction. Hence ε0 is locally asymptotically stable when max{Ru,Rv}<1. If max{Ru,Rv}>1, let us assume, without loss of generality, Ru>1. Then G1(λ)=1 has a positive solution, λ∗. Thus, the disease free equilbrium is unstable.
4.
Semi-trivial (Boundary) equilibria
4.1. Existence of semi-trivial equilibria
In this section, we prove the existence and stability of the two boundary equilibria E∗1 and E∗2 corresponding to opioid addiction and HIV transmission in a single population respectively. To obtain E∗1 we let Sk(t)=S∗k1, Uk(t)=U∗k1 and Vk(t)=0 and ik(t,τ)=0 for all k, i.e., E∗1=(S∗11,U∗11,0,0,⋯,S∗n1,U∗n1,0,0). We get the following equations
Summing both sides of the equation from k=1 to k=n, and dividing by N∗k1 we get,
n∑k=1S∗k1N∗k1=μ+du+δβu=nRu
(4.2)
We know S∗k1+U∗k1N∗k1=1, and so
n∑k=1S∗k1+U∗k1N∗k1=n
(4.3)
i.e.,
n∑k=1U∗k1N∗k1=n(1−1Ru)
Plugging in the values in the first equation of (4.1) and solving for U∗k1 we get
U∗k1=ΛkμRu−1+μ+du=Λk(μ+du)(1−μ(1−Ru)(μ+du))
(4.4)
Now when Ru<1, ∑nk=1U∗k1N∗k1<0, which implies U∗k1<0 for at least one k. When Ru>1, U∗k1>0. Thus, E∗1 exists if and only if Ru>1.
To obtain E∗2 we let Sk(t)=S∗k2, Vk(t)=V∗k2 and Uk(t)=0 and ik(t,τ)=0 for all k, i.e., E∗2=(S∗12,0,V∗12,0,⋯,S∗n2,0,V∗n2,0). We get the following equations
λv(V∗2)=0 is a solution to (4.5) which gives the disease free equilibrium. Then for a boundary equilibrium, λv(V∗2)>0 is a root of f(λv), where
f(λv)=1<k>n∑k=1k2p(k)βkv1kλv(V∗2)+(μ+dv+γv)−1.
As λv increases f decreases. limλv→∞f(λv)=−1. But f(0)=Rv−1>0. Then if Rv>1, f(λv) has a unique zero, giving us a unique boundary equilibrium for the system.
4.2. Stability of boundary equilibria and invasion numbers
To find the invasion number of HIV and stability of E∗1 we first linearize the system (2.2) around E∗1. We set Sk(t)=xk(t)+S∗k1, Uk(t)=uk(t)+U∗k1, Vk(t)=vk(t), ik(t,τ)=yk(t,τ) and Nk(t)=nk(t)+N∗k1, the system for the perturbations becomes,
From the second equation of (4.2) we get yk(τ)=yk(0)π(τ)e−λτks, where π(τ) is as defined before. Suppose K=βuqu∑nk=1U∗k1N∗k1 and Q(λ)=δ[∫∞0σ(τ)π(τ)e−λτksdτ]. Then from the first and third equations of (4.2) we get,
∫∞0βjv2(τ)e−λτπ(τ)dτ=βj(λ). βj(λ) is bounded above by βj(0) and Q(λ) is bounded above by Q(0). Then Gvi(0)=R1vi and limλ→∞Gvi(λ)=0. Suppose (4.14) has a solution λ=x+iy with ℜ(λ)=x≥0 and R1vi<1. First we prove the following result.
where, C1=U∗j1N∗j1, C2=S∗j1N∗j1 and z=√(x+μ+dv+γv+K)2+y2. Since f′(z)<0, f(z) is a decreasing function. That is when z(0,0)≤z(x,y), f(z(0,0))≥f(z(x,y)). But f(z(0,0)) is just the right hand side of (4.16). Using (4.16) we can now state the following,
This is a contradiction. So (4.14) only has solutions with non-negative real parts when R1vi<1.
Now let us suppose, R1vi>1. It can be shown that Gvi(λ) is decreasing. Then since Gvi(0)=R1vi>1 and limλ→∞Gvi(λ)=0 for real and positive λ, (4.14) must have at least one positive root when Rvi>1. Now, from (4.1)–(4.3) we get,
S∗k1=nRuU∗k1n(1−1Ru)=U∗k1Ru−1,
i.e.,
U∗k1Ru−1+U∗k1N∗k1=U∗k1N∗k1(1Ru−1+1)=1.
Solving the equation we get,
U∗k1N∗k1=1−1Ru,S∗k1N∗k1=1Ru.
(4.18)
To find the remaining eigenvalues, satisfying the third, fourth and fifth equation of (4.11), yj(τ)=0 and vj=0 for all j=1,2,⋯,n. The first two equations then just reduce to
∑nj=1ujN∗j1=0 implies from (4.20) all uk would be zero, which would not be of interest. ∑nj=1ujN∗j1≠0 for non-equilibrium points, and cancelling the expression on both sides, then the characteristic equation becomes,
Simplifying the equation, we have λ2+bλ+c=0 where b=μ+βun>0 and c=(1−1Ru)(βun(μ+du)−(μ+du+δ)du)>0 since, βun>(μ+du+δ) when Ru>1. Hence this quadratic equation has only roots with negative real parts. Combining the work above we can conclude,
Theorem 2.The unique boundary equilibrium E∗1 is locally asymptotically stable if R1vi<1, and it is unstable if R1vi>1.
To find the invasion number of opioid addiction and stability of E∗2 we first linearize the system (2.2) around E∗2. We set Sk(t)=xk(t)+S∗k2, Uk(t)=uk(t), Vk(t)=vk(t)+V∗k2, ik(t,τ)=yk(t,τ) and Nk(t)=nk(t)+N∗k2. The system for the perturbations becomes,
We call R2ui the invasion reproduction number of opioid addiction. We claim that when R2ui<1 the boundary equilibrium E∗2 is locally asymptotically stable, that is all the roots of (4.30) have negative real parts. Suppose
This is a contradiction. Hence all roots of (4.30) have negative real parts when R2ui<1. Now let us suppose, R2ui>1. Then since G′ui(λ)<0 when λ>0, Gui(λ) is decreasing when λ>0. But we have, Gui(0)=R2ui>1 and limλ→∞Gui(λ)=0. Then (4.30) has at least one positive root when R2ui>1. If λ is not a solution of characteristic equation (4.30), we have uj=0, yj(0)=0, the remaining two equations of (4.25) then just reduce to
where T(λ)=1<k>∑nj=1jp(j)βjv1vjN∗j2(λ+μ+(dv+γv)V∗j2N∗j2). Since T(λ)=0 implies from (4.35), vk=0 for all k, T(λ)≠0 and we get the following characteristic equation,
This is a contradiction. So we can state the following theorem,
Theorem 3.The unique boundary equilibrium E∗2 is locally asymptotically stable if R2ui<1, and is unstable if R2ui>1.
5.
Numerical simulations
5.1. Numerical scheme and simulation
We present a numerical scheme for the immuno-epidemiological models (2.1) and (2.2). The within-host model, consisting of ordinary differential equations can be solved by a stiff ODE solver in MATLAB.
For the between-host model we introduce a finite-difference method. We discretize the domain
D={(t,τ):0≤t≤T,0≤τ≤A}
where A is a maximal infection age and time T<∞, a maximal time. We take Δt=Δτ, with ks=1, and so the points in age and line direction can be computed as,
τm=mΔttj=jΔt.
Setting M=[AΔt] and N=[TΔt], we obtain A=MΔt, T=NΔt. The numerical method computes approximations to the solution at the mesh points. We assume Sk(tj)=Sjk, Uk(tj)=Ujk, Vk(tj)=Vjk and ik(tj,τm)=ijm,k. We summarize the numerical method below,
To study the coexistence equilibrium analytically is not feasible for this model. So we take the help of simulations to predict the existence of coexistence equilibrium in a scale free network scenario. We consider specific parameter values for which R2ui and R1vi are greater than 1. The simulations suggest that the coexistence equilibrium exists and is stable. Given parameter values are constant, the invasion number of HIV, R1vi, seem to show dependence on the size of the network used. With the same parameter values given in Table 3, when the network contains 200 nodes, R1vi is close to 1.4, while with 300 nodes, R1vi increases to 2.9. The invasion number of opioid addiction R1vi remains stable near the same value 1.2 when size of the network is increased from 200 nodes to 300 nodes. This is to be expected since the spread of opioid has been considered homogeneous over the network. Simulations suggesting a coexistence equilibrium are shown in Figure 1.
Table 1.
List of parameters of the within-host model.
Figure 1.
Simulations with the network model. In this simulation the average degree is 7.63 and R1vi=1.2173, R2ui=1.1607. The number of nodes is 100. The maximal degree is 28 but there are no occupied nodes with degree 1, 2, 3.
5.2. Effects of the parametric values of qu, qv and δ
We define U(t) = ∑nk=1Uk(t) as the total opioid addicted population. Similarly we define V(t), I(t) and S(t) as the total HIV infected, total co-infected and total susceptible population respectively. The network utilized had 200 nodes. The parameters βu and βkv1 are estimated by the following formulas, βu=Ru(μ+du+δ); and βkv1=Rv(μ+dv+γv). Since the model we consider does not include treatment for HIV, we consider Rv=5.5. This estimate is an average value collected from [39], which gives an estimation of basic reproduction number of HIV in Rural South West Uganda. Estimated value of Ru according to [27] would be close to 1.1. Given the fact that a high percentage of US citizens mentioned it would be easy for someone to access opioids for illicit purposes, according to a poll conducted in 2018 (46 percent) and people who misuse opioids often get them from a family member or friend who has a prescription [40], we considered the Ru to be higher in range, around 3.25. The maximal degree n for the following simulations (except Figure 1) is 43, with number of nodes being 200.
The two parameters in (2.2) that are of particular interest are qu and qv, which determine how much one epidemics impacts the other. That is qv determines how likely opioid users are to get infected by HIV compared to non-users, and qu determines how likely HIV infected people are to become opioid addicted compared to non-infected people. In [27] the estimated value of the qv term equivalent was 94.5, and in [38], the estimated value of qv was 30.62. Both estimates suggest a high dependence effect of opioid usage on HIV infection. In our simulations we take the lesser estimate of qv=30.62. The total co-affected population due to varying values of qv is simulated, such as 0.5qv,qv,1.5qv,2qv,2.5qv,3qv. The estimated value for qu in [38] was below 1, but HIV-infected persons are more likely to have chronic pain, receive opioid analgesic treatment, receive higher doses of opioids, and to have substance use disorders and mental illness compared with the general population, putting them at increased risk for opioid use disorder [41]. So the simulations were done with the fitted value for qu along with double, five times and ten times the fitted value. While the total number of co-affected varies according to the network, the trend seems to be similar, with qv increasing, the total number of co-affected increases (Figure 2). A definite situation of interest is when qu=4.3, the maximum value seems to be achieved when qv is close to 60, not at the highest value of approximately 90. The simulation was repeated with these values for different network sizes and provided the same result.
Figure 2.
Figure shows total co-affected individuals for six different values of qv. Top Left: qu=0.86, Top Right: qu=1.72, Bottom Left: qu=4.3, Bottom Right: qu=8.6. The other parameters used are given in Table 3.
A second set of simulations were performed, taking the base value of qu=1.72. The four individual cases have all other parameters and network values same, only qv is varied, from 15,31,62 and 93 respectively. While some of the cases do have permutations, over all the trend is similar, and opposite to the simulations in Figure 2. That is qu increasing causes the total number of co-affected to decrease (Figure 3). Again we notice a situation of interest, when qv=60, the maximum value seems to be achieved when qu is close to 1.72, not at the lowest value of approximately 0.86. Repeated simulation with differing sized networks did not show changes in the results.
Figure 3.
Figure shows total co-affected individuals for six different values of qu, Top Left: qv=15, Top Right: qv=30, Bottom Left: qv=60, Bottom Right: qv=90. The other parameters used are given in Table 3.
Another parameter of interest is δ, which denotes the rate of recovery from addiction in the epidemiological model. The estimate for that in [27] is close to 0.033, while in [38] the estimate is approximately 0.11. If successful treatment is considered without subtracting the relapses, the rate would probably be close to 0.05[42]. To simulate for differing values of δ, we consider βu and βkv1 as constants, directly taking the values from Table 3. All the other parameter values are as mentioned in Table 3. We consider four differing situations for δ, with the value being 0.02, the fitted value 0.11, and target high values of 0.5 and δ=1. We also investigated different scenarios with differing network sizes, with number of nodes being 100, 300 and 500 respectively. Interestingly, irrespective of network size, the total number of co-affected people appeared to be higher with the value of δ increasing (Figure 4). One quite plausible explanation would be the higher recovery would decrease the number of opioid overdose deaths significantly, thereby increasing the co-affected prevalence.
Figure 4.
Figure shows total co-affected individuals for four different values of δ, Top Left: Network Size 100, Top Right: Network Size 300, Bottom: Network size 500. The other parameters used are given in Table 3.
We formulate a within-host model linked with a dynamic network HIV/opioid coinfection epidemiological model with demography, through epidemiological parameters. The system is described by ordinary differential equations coupled with partial differential equations in a nested fashion. The network multi-scale model here is an extension of the multi-scale model considered in [38]. The disease free equilibrium of the system always exists and is locally asymptotically stable when both the basic reproduction numbers of opioid and HIV, Ru and Rv are less than 1.
The boundary equilibrium E∗1 exists when Ru is more than 1 and E∗2 exist when Rv is more than 1. We define the invasion reproduction numbers R1vi and R2ui. The invasion reproduction number R2ui gives the reproduction of the opioid users when the population is at the equilibrium E∗2, that is, when HIV infection alone is at equilibrium in the single population. The invasion reproduction number R1vi gives the reproduction of the HIV infection at the equilibrium E∗1, that is when the opioid transmission alone is at equilibrium in the single population. When R1vi<1, E∗1 is locally stable and when R1vi>1, E∗1 is unstable. When R2ui<1, E∗2 is locally stable and when R2ui>1, E∗2 is unstable. The model is too complicated to compute or consider the stability of an endemic equilibrium, analytically, but simulations suggest that there is an interior equilibrium potentially under the condition that both invasion numbers are larger than one.
We use fitted parameters from [38], to perform simulations, to explore the effect of the change of the parameters qu, qv and δ. The parameters qu and qv represent the effect of one epidemic on the other, and we simulated for plausible values of qu and qv, to get estimates of co-affected prevalence. The estimate of qv, the likelihood of a heroin user to be infected with HIV, in [27] was approximately 94, and since a large number of opioid users progress to becoming heroin users, (Data from 2011 showed that an estimated 4 to 6 percent who misuse prescription opioids switch to heroin and about 80 percent of people who used heroin first misused prescription opioids) [43], chances of the qv estimate for all illicit opioid usage being close to the "heroin only" estimate is high. We notice that increase in qv causes the co-affected population to rise significantly, which indicates that control strategies focusing on reducing HIV infection among the opioid addicted population would be effective in decoupling the epidemics, corroborating with the conclusions in [27].
To the best of our knowledge, there have not been previous models with a parameter similar to qu, that provides an estimate for the effect of HIV infection on opioid use. The simulations from our model show that increase of qu in general causes the number of co-affected to decline. That is the prevalence of co-affected people declines sharply with the increase to more HIV infected people being addicted to opioids. This does corroborate real life data, since deaths due to overdose in the US per year were aproximately 120,000 in the year 2020, and the number increased 15 percent by 2021 [44], compared to the number of deaths due to HIV being around 18,500 in 2020 [45]. We noticed significant increase in the prevalence of co-affected individuals, when the parameter δ, representing the recovery from opioid usage was increased significantly. The sharp increase points to the idea that control measures should focus more on treatment of opioid use disorder, and that uncoupling the two epidemics is a priority to prevent loss of human lives. Counseling about the dangers of opioid addiction for HIV infected people must be provided, and similarly counseling about the dangers of getting infected with HIV should be provided to reported opioid addicts.
In summary, we have developed a novel multi-scale network model of HIV and opioid epidemics. We have analized the model and obtained conditions for HIV-only to persist or opioid-only to persist. Simulations suggest that the two epidemics can co-exist for some parameter values. Simulations further suggest that decreasing qv decreases the number of co-affected and may lead to decopuling the epidemics. Thus control measures targeted at reducing qv should be coupled with treatment of opioid affected individuals which is consistent with our previois findings.
Acknowledgments
Maia Martcheva was partially supported by NSF DMS-1951595. Necibe Tuncer was partially supported by NSF DMS-1951626.
Conflict of interest
The authors declare there is no conflict of interest.
Code availability
The code was written by one of the authors and is available on request by contacting Churni Gupta.
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Number of susceptible individuals at time t with degree k
Vk(t)
Number of HIV infected individuals at time t with degree k
Uk(t)
Number of opioid addicted individuals at time t with degree k
ik(t,τ)
Density of co-affected individuals with coinfection age τ at time t with degree k
Ak(t)
Number of individuals with AIDS at time t with degree k
Λk
Constant recruitment rate for nodes with degree k
βu
Transmission rate of opioid addiction
βv1
Transmission rate of HIV infection of HIV infected only individuals
βv2(τ)
Transmission rate of HIV infection of coinfected individuals
μ
Natural death rate
du
Death rate induced by opioid addiction
dv
Death rate induced by HIV infection
di(τ)
Death rate induced by coaffection at coaffection age τ
da
Death rate induced by AIDS
δ
Recovery rate from opioid addiction
σ(τ)
Recovery rate from coinfection to HIV only
qu
Increase coefficient of HIV infection due to opioid usage
qv
Increase coefficient of opioid usage due to HIV infection
γv
Rate of transition from HIV to AIDS
γi(τ)
Rate of transition from coinfection to AIDS
Parameter
Estimated Value
Units
βu
0.385676
1/time
βv1
0.0551
1/time
k0
0.00011046
1/time
B
15318.9
vRNA/ml
δ
0.118227
1/time
qu
0.867138
Unitless
qv
30.6189
Unitless
du
0.00817752
1/time
dv
0.0144092
1/time
da
1.2766e+11
1/time
d0
2.72895e-07
ml/(time×cells)
d1
3.4671e-06
1/time
γv
0.0223488
1/time
γ0
1.63927e-12
1/time
σ
0.000270006
Unitless
s
22843.6
CD4 count/(time×ml)
d
0.0766824
1/time
k1
2.02785e-05
vRNA/(CD4 count×time)
δi
0.725266
1/time
Nvδi
8465.63
vRNA/(CD4 count×time)
Figure 1. Left: dependence of Sf on β found analytically as solution of equation (3.11). Right: dependence of growth rates on β found analytically by formulas (3.13) (magenta) and (3.14) (cyan) for τ0=2, τ=6, N=105, S0=N−1
Figure 2. Numerical simulations of the delay model (3.6) and SEIRD model (2.20) for different countries and for the period of time from November 15, 2021 to May 15, 2022. (a) β=0.23, τ0=2, τ=6, ρ1=0.97, μ1=0.03, N=331.9×106, I−=130000, I0=257235; (b) β=0.276, τ0=3, τ=6, ρ1=0.97, μ1=0.03, N=67.75×106, I−=9000, I0=23557; (c) β=0.326, τ0=2, τ=4, ρ1=0.97, μ1=0.03, N=83.2×106, I−=14000, I0=19050; (d) β=0.596, τ0=1, τ=2, ρ1=0.97, μ1=0.03, N=143.4×106, I−=15000, I0=15450
Figure 3. Numerical simulations of the SEIRD model (2.20). Left: the values for disease transmission rates are from β=0.23 to β=0.281 (green curves), τ0=2, τ=6, ρ1=0.97, μ1=0.03, N=331.9×106. Right: β=0.4, λ=0.5, ρ0+μ0∈{0.25,0.2381,0.2222,0.2128}, N=331.9×106, I0=150000