Abstract
We propose and study a viral infection model with two nonlocal effects and a general incidence rate. First, the semigroup theory and the classical renewal process are adopted to compute the basic reproduction number as the spectral radius of the next-generation operator. It is shown that equals the principal eigenvalue of a linear operator associated with a positive eigenfunction. Then we obtain the existence of endemic steady states by Shauder fixed point theorem. A threshold dynamics is established by the approach of Lyapunov functionals. Roughly speaking, if , then the virus-free steady state is globally asymptotically stable; if , then the endemic steady state is globally attractive under some additional conditions on the incidence rate. Finally, the theoretical results are illustrated by numerical simulations based on a backward Euler method.
1. Introduction
Viruses are very tiny germs. Their presence in the body causes not only familiar infectious diseases (such as the common cold, flu, and warts) but also severe illnesses (such as HIV/AIDS [1–3], hepatitis B [4–7], hepatitis C [6, 7], and human T cell leukemia [8]). Because of the long infectious periods and difficulties in treating them, viral infections have been regarded as serious health problems and have brought heavy economic burden worldwide. The mechanism on a viral infection is quite complicated. Roughly, viruses invade target cells (healthy cells) and replicate in them and then replicated viruses are released. Mathematical modeling has been a very important and efficient way to better understand the evolution of viral infections and to evaluate antiviral drug therapies. In a typical compartmental viral infection model, there are three compartments for uninfected target cells (), infected cells (), and free virions ().
In recent years, spatial-structured models have played a crucial role in exploring viral dynamics. In most of the study (see, e.g., [5, 9–11]), uninfected cells and infected cells are assumed to be motionless, while virions can move freely. Obtained results include the existence of travelling waves [5] and asymptotical behavior [10, 11]. To the best of our knowledge, the spatial domain is either one-dimensional or infinitely dimensional. Usually a Laplacian operator with a diffusion coefficient is used to describe the random diffusion of each virion in the adjacent habitat (position). The diffusion term follows Fick’s law and this leads to systems coupling two ordinary differential equations with one parabolic equation.
Such mentioned diffusion is assumed such that the diffusive habitat is small. In fact, the motions of virions are always free and thus the limitations for the short diffusion are not reasonable. Thus, nonlocal diffusion accounting for the long-range diffusion effect has been proposed and extensively investigated. The convolution diffusion operator takes the formwhere and is a suitable Banach space. This means that a virion at position can affect another virion at position at the probability of . In [12], Garcia-Melian and Rossi assumed that represents the probability of skipping from location to . Then virions arrive at location from other places at rate . Based on peculiar features of the nonlocal diffusion operators, models in ecology [13–15], in epidemiology [16–19], and even in materials science [20, 21] have been investigated.
It is well known that incidence rates play a key role in understanding intrinsic mechanisms of viral infections. Though commonly used, the bilinear incidence (or called mass action) may not completely capture the viral dynamics. To overcome this deficiency, several forms of incidence rates have been proposed, which include the saturated incidence rate [22], Crowley-Martin functional response [23], and a general form of [8, 24].
Motivated by the aforementioned works, we propose a nonlocal diffusive viral infection model with a general incidence rate. To build the model, we still assume that there are three compartments involved in the viral infection for uninfected target cells, infected cells, and free virions. Their densities at time and position are denoted by , , and , respectively. Here is compact and connected with a smooth boundary and satisfies . The model to be studied is with the initial conditionHere is the created rate of uninfected cells at ; , , and are, respectively, the death rates of uninfected cells, infected cells, and virions at ; denotes the rate of new infections at time and location ; represents the transfer rate of virions from positions in to position with a kernel function while is the total transfer rate of virions from location to all the other locations. Note that if we take the kernel function , then system (2) becomes a classical within-host model with the nonlocal diffusion.
The main contribution of this paper has three aspects. Firstly, it is easy to see that the equation for has an integral term and thus the solution of system (2) lacks strong regularity. This implies that the semiflow generated by system (2) is not compact. To overcome this difficulty, we use Arzelà-Ascoli Theorem [25] to establish the asymptotic smoothness of the semiflow, which enables us to pass the dissipativity of system (2) from to . Secondly, the nonlocal diffusion further weakens the regularity of the solution and this requires nonroutine methods to deal with compactness and nonsupporting of the next generation operator. Finally, the nonlocal diffusion term enhances the difficulty in proving the global stability of the steady states. Inspired by the work of Thieme [26, Section 11], in order to construct a suitable Lyapunov functional, a nonnegative Borel measurable function should be picked to balance the nonlocal term. We explicitly identify such a Borel measurable function as instead of in an abstract form.
The organization of this paper is as follows. Section 2 gives the existence, uniqueness, and nonnegativity of solutions to system (2). Section 3 shows that the solution semiflow is asymptotically smooth by applying Arzelà-Ascoli Theorem. Section 4 focuses on the basic reproduction number of system (2) defined as the spectral radius of the next-generation operator and the relationship between and the spectral bound of a linear operator. Section 5 is devoted to the existence of endemic steady states by employing Shauder fixed theorem. We discuss the stability of the virus-free steady state in Section 6. The global behavior of system (2) including uniform persistence and global stability of endemic steady states is established in Sections 7 and 8. In Section 9, we carry out numerical experiments to validate the theoretical results. Section 10 concludes the paper with a succinct discussion.
2. Preliminaries
Let equip with the supremum norm Moreover, for , we denote and . Let and defineThen is also a Banach space equipped with the norm . Clearly, and are positive cones of and , respectively.
To study the asymptotic dynamics of system (2), we make the following assumptions.
Assumption 1. (i), , , , and are all strictly positive.(ii) is continuously differentiable with respect to the third and fourth variables and also satisfies the following:(ii-1) for all ;(ii-2)for all , is increasing in both and ;(ii-3)for all , the function is decreasing associated with the variable on ;(ii-4)for any there exists some such that(iii) and satisfies . Moreover, it is irreducible and symmetrical for all .(iv) and satisfy the balance condition
To continue the discussion, we define a linear operator and a nonlinear operator on byandfor and , respectively. Here is extended to through continuity.
Lemma 2. The operator defined by (8) generates a uniformly continuous semigroup on . Furthermore, for all .
Proof. We decompose the operator as , whereandfor and . We readily see that generates a strongly continuous and positive semigroup withfor and . Moreover, it follows from Assumption 1 that the operator is bounded. Then [27, Corollary VI 1.11] and [28, Theorem 1.2] combined ensure that the operator generates a positively continuous semigourp This completes the proof.
The following proposition gives the existence and uniqueness of solutions of system (2).
Proposition 3. For all , system (2) admits a unique classical solution with Moreover, the solution has the following properties: (i)If then .(ii)Let . If with in as , then(iii) for and .(iv)If , then for all .
Proof. [29, Lemma 3.1], together with Assumption 1, ensures that the operator defined by (9) is continuously Fréchet differentiable on . Therefore,Then from [30, Proposition 4.16] and [28, Theorems 1.2–1.5 in Chapter 6] assertions (i) and (ii) follow immediately.
We use by way of contradiction to prove (iii). We claim that there exists and such that . Notice that for all small enough . Hence, by the continuous dependence of the solution on the initial values, defineThus and with . But thenis a contradiction. This proves assertion (iii).
In order to establish (iv), note that for each positive there exists a sufficiently large such thatwhere is an open ball in with the center at and radius . We rewrite system (2) in the form of where is an indentity operator on and . Applying the method of variation of constant, we haveHence, for , .
In fact, as the following result shows, solutions exist globally with initial values in .
Proposition 4. Let . Then the solution of (2) with the initial value exists on .
Proof. Let be the maximal existence interval of the solution through . Define , for Then, for any , first, we add the first two equations of (2) to obtain where It follows that where . Next we integrate the third equation of system (2) on the set to obtain Solving this differential inequality, we get for . Lastly, it also follows from the last equation of (2) thatwhich givesIn summary, we have obtained thatBy (i) of Proposition 3, we get and this completes the proof.
By Propositions 3 and 4, a solution semiflow is defined byfor . Moreover, the existence of the solution of (2) is indeed global and the semiflow is bounded and dissipative. DefineThen it follows easily from the proof of Proposition 4 (actually using the same differential inequalities and the resulting inequalities in the proof) that is positively invariant and attracts all the bounded subsets of (2) in . Therefore, to study the asymptotic behavior of (2), we only need to focus on solutions with initial values in .
3. The Asymptotic Smoothness
A semiflow is asymptotically smooth if there exists a nonempty compact set attracting each forward invariant bounded closed set (see [31, Definition 2.25]), in other words, if the semiflow is asymptotically compact on every forward invariant bounded closed set.
Theorem 5. The semiflow defined by (27) is asymptotically compact and hence it is asymptotically smooth.
Proof. The proof is inspired by the ideal in [26, Section 4]. Assume that is any forward invariant bounded subset of . We show that is asymptotically compact on [32, pp. 28]. This needs to show that a sequence of solutions is equi-bounded. That is, there exists a sequence with as such that has a convergent subsequence in . Let us consider the translated solutions . Then By the Arzelà-Ascoli theorem, it suffices to show that , , and are equi-continuous. Then, for each , we obtainHere we have used the fact that is increasing with respect to and , where is small enough and satisfiesIntegrating the inequality (32) from to , we obtain By Assumption 1, for each bounded set , is uniformly continuous. The boundedness of , together with as , implies that as . This limitation is uniform for both and in compact subsets of . The equi-continuity of holds immediately for any .
Similarly, for , we can getArguing directly, we claim that is not equi-continuous for some . Then we can pick up a sequence such that as . Moreover, we choose a subsequence of satisfyingSince, for each , is equi-continuous, as . Applying Fatou’s lemma and Assumption 1, we get from (35) thatis a contradiction. This proves that is also equi-continuous for any .
Finally, again similarly as before, we can obtain By Assumption 1 and , we have as uniformly for . This proves the equi-continuity of and the hence the proof is complete.
4. The Basic Reproduction Number
This section is conducted for estimation of the basic reproduction number, which is defined as the expected numbers of secondary cases created by a typical infected individual among a completely susceptible population. Clearly, system (2) has a virus-free steady state , where for . Linearize (2) around in the disease invasion phase to obtain Solving them, we haveandrespectively. Plugging (40) into (41) yields Using change of variables gives us Therefore, following the approach of Diekmann et al. [33], we define the next-generation operator byfor and . Based on Assumption 1, the operator is well defined, continuous, and positive. In the following, we show that is compact and nonsupporting. Nonsupporting means that, for any and , there exists positive such that .
Proposition 6. Suppose that Assumption 1 holds. Then the next-generation operator defined by (44) is compact and nonsupporting.
Proof. Let be a bounded set. It follows from Assumption 1 that is bounded. To show that is compact, it suffices to show that is equi-continuous by Arzelà-Ascoli theorem. For , , and , Note that the uniform continuity by Assumption 1 and , where is a positive constant. We can easily see that is equi-continuous.
Now, we show that is nonsupporting. For any , by Proposition 3 and the monotonicity property of , we haveFrom the assumption on , it follows that for . Then, for any , we have (with ). This proves that is nonsupporting.
From the definition of the basic reproduction number by Diekmann [33], such value is defined bywhere represents the spectral radius of an operator. Hence, can be considered as a next-generation operator in [33–35].
Now, in order to clarify the relationship between the next-generation operator and a linear operator, we define :for and . From definitions of operators and , we immediately see that or , where is the identity operator on .
Theorem 7. , , and if and only if , , and , respectively, where denotes the spectral bound of .
Proof. First, suppose that . It follows from [36, Proposition 4.4] that there exists a positive function such that or . This implies that . Therefore, . On the other hand, suppose that . Then, with the help of the irreducibility of and [37, Theorem 2.2], we conclude that there exists an eigenfunction with respect to ; namely, . It follows that , which gives . This proves theta if and only if .
Now, we only show that if and only if as the proof of if and only if is similar. On one hand, let hold. Then Proposition 6 implies that there exists a positive eigenfunction associated with ; that is, . It follows that . Based on Assumption 1, we know that . It follows that . On the other hand, let . Following the above approach, we obtain the existence of a positive eigenfunction with respect to the eigenvalue ; that is, . Then , which implies that . This completes the proof.
5. Existence of Endemic Steady States
This section is conducted for the existence of endemic steady states of (2). Let be a feasible steady state and then it satisfies
We apply Shauder fixed point theorem to find solutions of (49). To overcome the difficulty caused by the nonlinear function , we make the following modifications on . For , definewhere . We focus on the following perturbation systems:where is a bounded and decreasing sequence in with as .
DefineSince is, and so is , increasing in , for each and there exists a unique such that
Lemma 8 ([26, Lemma 7.3]). For fixed , is continuous with respect to , uniformly for in bounded subsets of .
Theorem 9. Suppose that holds. Then system (2) admits at least one feasible endemic steady state with .
Proof. Solving the second equation of (51) yields . Then substitute this expression of into the third equation of (51) to getTherefore, we can define a map byfor and . Clearly, is continuous and nonincreasing by Assumption 1 and Lemma 8. Furthermore, for all ,Let be a closed nonnegative ball with the radius and the center at 0. If we pick up large enough , then maps into itself. Since is closed and convex, applying Shauder fixed theorem gives the existence of some such that . Note that . Then , , and satisfyAdding the first two equations of (57) giveswhere and Integrating the last equation of (57) on with respect to , together with , we have Consequently, is bounded. Thus is uniformly bounded. Furthermore, we can apply the similar argument as that in the proof of Proposition 6 to show that is equi-continuous. Thus it is precompact in . Then we can choose a subsequence, say itself, such that uniformly in as . Letting in (57), we see that is a steady state of system (2).
Now, we show that is an endemic steady state by showing that . By way of contradiction, assume that , which implies that , and uniformly for as and . For large enough , we know that for all . DefineThen . It follows thatNote that Lemma 8, together with Assumption 1, implies that the above inequality converges to zero as . From the compactness of and boundedness of , we can pick up a subsequence, again say itself, such thatfor some and . Set . Then as in . Proposition 6 ensures that . By Krein-Rutman theorem, there exists some and such that . HenceThis leads to a contradiction with .
6. Stability of the Virus-Free Steady State
The objective of this section is to establish the stability of the virus-free steady state .
Lemma 10. If , then for all .
Proof. Define . Noting that , we getSolving this equation, we have Because is nonnegative on by Assumption 1(ii) and , we conclude that, for all , .
Define bywhere and . We can separate into two operators and defined byandrespectively. Denote . Observe that actually acts on by
Lemma 11. If the problemhas a positive eigenvalue with a positive eigenfunction , then
Proof. Let , where represents the resolvent set of the operator . For any , let such thatThenandLet be a positive semigroup generated by the operator ThenWithout loss of the generality, letting and gives for each . By the definition of , we have Let Then the above equality can be rewritten:This means thatFollowing the approach in [38], we derive that the eigenvalue problem has a positive eigenvalue with a positive eigenvector for Since is positive, it follows that
Theorem 12. Suppose that . Then the virus-free steady state is locally asymptotically stable.
Proof. Linearizing system (2) around the virus-free steady state , we haveLet , , and be a solution of system (81). After substitution, we arrive atObserve that the first equation of system (82) decouples with the other two equations. Furthermore, the last two equations can be rewritten in the formwhere and are defined by (68) and (69), respectively. By [Theorem 2.2, [39]], the eigenvalue problem (83) has a principal eigenvalue with a unique positive eigenfunction. Lemma 11 implies that if . Otherwise, and then it follows from the first equation of (82) that . Therefore, local stability of the virus-free steady state follows immediately.
Next, we show the global attractivity of the virus-free steady state . To establish this result, we need the following Volterra-type: has the property as follows: for all and the quality holds if and only if . This function has been used very often to construct Lyapunov functionals (see, e.g., Yang et al. [40], Kuniya and Wang [29], and McCluskey [41] and the references therein).
Theorem 13. Suppose that and The virus-free steady state is globally asymptotically stable.
Proof. For , we consider the following Lyapunov functional:where , , , and is a weighted positive function to be determined later. Based on Lemma 10, is well defined.
Taking the derivative of along solutions of (2) with respect to , one arrives at Similarly, we getandTherefore, we can pick up and take a Lyapunov functional in the form ofwhere defined in (82) is a positive eigenvector function associated with eigenvalue . Then where is defined by (48). It follows from Theorem 7 that provided that . The equality holds if and only if and for each . Therefore, the largest invariant set in is . We employ the LaSalle invariant principle to conclude that is globally asymptotically stable.
7. Uniform Persistence
Denote LetAnd let be the positive orbit . From Theorem 13, we readily have the following result.
Lemma 14. For every , , where represents the omega limit set of
The semiflow is said to be persistent associated with if there exists an such thatfor any solution with .
Lemma 15. Let be any solution of system (2) with the initial value . Then for all .
Proof. Proposition 3(iii) has already asserted that for . Now we show that and for all . Moreover, since , we have . Then a similar argument to that for the proof of Proposition 3(iii) can also give for . Now, from the second equation of (2), it follows that, for ,
Lemma 15 implies that is a positive invariant of (2). Let be the restricted semiflow of on .
Lemma 16. Suppose that . Then system (2) is uniformly weakly persistent. That is to say that there exists a positive value such that, for all ,
Proof. For any , define byClearly, as . Note that since . Then, for small enough , has a positive principle eigenvalue with an associated eigenvector function .
By way of contradiction, assume that the conclusion fails. Then, for chosen above, there exists (will be denoted by for simplicity of notation) such thatThen, with a possible shift in time and by Lemma 15, we can assume that, for ,Then, by the monotonicity of in and , we getFrom Assumption 1 (ii-3), we obtainLet be the solution of the following auxiliary system:with . Then we have for . Let such that and . By comparison principle, we getfor . This contradicts with the fact that is bounded and hence the proof is complete.
With the help of Lemmas 15 and 16, we can establish the uniform persistence.
Theorem 17. If and , then system (2) is uniformly strongly persistent and hence the deduced semiflow has a global attractor in .
Proof. Define byClearly, . It follows from Lemma 15 that has the property that for if or with . Lemma 14 ensures that, for any , we have and hence there is no cycle in from to itself. Moreover, Lemma 16 tells us that , where is the stable manifold of . Applying [42, Theorem 3] gives a positive number such thatthat isSince is continuously differentiable and system (2) is dissipative, there exists a positive constant such thatUsing comparison principle again, one admitsThen, it follows from the second equation of system (2) thatTherefore,Letting completes the proof.
8. Global Attractivity of Endemic Steady State
This section is devoted to the uniqueness and the global stability of endemic steady states under and some additional condition on . The approach is Lyapunov functional method. Recall that the semiflow on is bounded. Then the following result, which follows directly from this and Theorem 17, is helpful to construct the Lyapunov functional.
Lemma 18. Suppose that is a solution and is an endemic steady state of system (2). Then there exist two positive numbers and such that
In order to establish the global attractivity of an endemic steady state, we impose the following hypothesis.
Assumption 19. Suppose that an endemic steady state of system (2) satisfiesfor all , and , , .
Assumption 19 is obviously satisfied iforwhere .
Theorem 20. Suppose that and . Under Assumption 19 and , the endemic steady state of system (2) is globally attractive.
Proof. Consider the Lyapunov functional defined bywhere is a strictly positive function to be defined later andwith and is defined as in the proof of Theorem 13. Lemmas 18 and 15 ensure that (, , ) are well defined for .
Differentiating along trajectories of system (2) yields Similarly, we getandRemember that . Then we substitute (117)–(119) into to obtainWe will rearrange the terms in (120) as follows.
First note thatNow the last term of (121) can be rewritten asFurthermore, the last term of equation (122) can be rearranged to beNext we can simplify the factor in the last term of (120) as In summary, we have foundIf , then the last but two term of (125) is equal to zero. Following [26, Proposition 11.1], we catch as a positive Borel function on such that Therefore,Setting and plugging (127) into (125), we arrive at By virtue of and Assumption 19, and the equality holds if and only ifReplacing in the first equation of system (2) by , we haveFrom Assumption 1, it is easy to see that for all Using the second equation of system (2), we conclude that the largest invariant set . Consequently, it follows from LaSalle Invariance Principle that the endemic steady state is globally attractive.
Remark 21. The existence and attractivity of endemic steady states (see Theorems 9 and 20) indicate that system (2) has a unique endemic steady state if the conditions of Theorem 20 hold.
9. Numerical Simulations
In this section, we perform numerical experiments to illustrate our main theoretical results and compare the effects of diffusion rate and incidence. For this purpose, we take and use the initial values in [29].Moreover, we fix the following parameters:The incidence rate takes the form wherewith varying. For the nonlocal effect, we setwhere is the diffusive coefficient. Firstly, we set , , and . Then . Figures 1(a) and 1(b) show that the density of free viruses approaches zero as goes to infinity.
(a)
(b)
(c)
(d)
Now, we enlarge to and get . In view of Theorem 20, the endemic steady state converges to a spatially positive endemic steady state (see Figures 1(c) and 1(d)).
Secondly, Figure 2(a) shows that increasing the diffusion rate enhances the infected risk and increases the final infected size. This means that controlling virions diffusion beats the viral replication. Figure 2(b) reflects that viral infection models with bilinear incidence rate have higher infected risk than ones with saturating incidence rates.
(a)
(b)
10. Discussion
In this paper, we firstly proposed a within-host viral infection model with nonlocal diffusion and nonlocal transmission. The model can be considered as a spatial generalization of that proposed by Nowark and Bangham [43], a continuous spatial model of Funk et al. [9]. We derived the next-generation operator and built the relationship between the basic reproduction number and the spectral bound of the operator (see Theorem 7). The asymptotic smoothness of the semiflow was established by Arzelà-Ascoli theorem in Section 3. The threshold dynamics of system (2) has been established by constructing suitable Lyapunov functionals. Biologically, whether or not the viral infection outbreaks is determined by the basic reproduction number
Compared with other within-host models with diffusion (discrete style [9] and continuous style (Laplace operator) [5]), our model can be considered as a generalization of the model proposed by Zhao and Ruan [18], where for the incidence they took the particular formIn [18], they adopted the semigroup method to investigate the asymptotical behavior of system (2). In this paper, we used functional analysis method together with the Lyapunov functional method to study the asymptotical stability of the system. We believe that the method used here can also be applicable to or be generalized to deal with other nonscalar systems with nonlocal diffusions.
Data Availability
The artificial data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This paper was partially supported by the National Natural Science Foundation of China (no. 61573016 and no. 61203228), Shanxi Scholarship Council of China (2015-094), Shanxi Natural Science Foundation of China (201801D121008, 201801D221160), Shanxi Scientific Data Sharing Platform for Animal Diseases, Startup Foundation for High-Level Personal of Shanxi University, and NSERC of Canada.