ON LATENCIES IN MALARIA INFECTIONS AND THEIR IMPACT ON THE DISEASE DYNAMICS

In this paper, we modify the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions (P1(t) and P2(t)) to reflect the fact that the latencies differ from individuals to individuals. We justify the well-posedness of the new model, identify the basic reproduction number R0 for the model and analyze the dynamics of the model. We show that when R0 < 1, the disease free equilibrium E0 is globally asymptotically stable, meaning that the malaria disease will eventually die out; and if R0 > 1, E0 becomes unstable. When R0 > 1, we consider two specific forms for P1(t) and P2(t): (i) P1(t) and P2(t) are both exponential functions; (ii) P1(t) and P2(t) are both step functions. For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when R0 > 1 then the disease will persist; moreover if there is no recovery (γ1 = 0), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.

1. Introduction.Malaria is an infectious disease that is widely spread in tropical and subtropical regions for thousands of years and causes deaths in human beings.It is due to infection by one or more of a family of protozoa called Plasmodium, mainly consisting of four species: Plasmodium falciparum, Plasmodium vivax, Plasmodium malariae and Plasmodium ovale.The pathogen can parasite in the blood cells and other tissues of both human beings and mosquitoes.The infection between human beings and mosquitoes is through biting by female mosquitoes to human beings.Based on such a transmission mechanism, it was initially widely believed that the disease could be wiped out only by eradicating all vector mosquitoes, which turned out to be impossible in practice.
It was Ross [19] who firstly used a mathematical model to quantitatively investigated the spread of malaria.Ross' model was later further extended and studies by Macdonald [16,17,18], leading to the following system which has been referred to as the Ross-Macdonald model ( Here I 1 and I 2 represent the populations of the infectious classes of human beings and female mosquitoes respectively, N and M are the total populations of human beings and female mosquitoes respectively, which are assumed to be constants.The constant a is the mosquito biting rate; e 1 is the probability that a biting by an infective mosquito to a susceptible person will cause infection to the person; and e 2 is the probability that a biting by an susceptible mosquito to a infective human individual will cause infection to the mosquito.The parameters d 1 and d 2 are the death rates of infectious human beings and mosquitoes respectively.By analyzing this mathematical model, both Ross and Macdonald found that it is possible to eradicate the disease without killing all vector mosquitoes.Indeed, by looking at basic reproduction number for this model given by one knows that any measure(s) that can bring R 0 to a value less than 1 would eventually drive the disease to extinction, including controlling the mosquito population M to a sufficiently lower level.Obviously, the approach of mathematically modeling provides much insight into the spread of malaria, by which, effective means to control the disease can be suggested.For example, in addition to decreasing M to certain level (by spraying mosquito pesticides) which was Ross and Macdonald's finding, decreasing the biting rate (achievable by using mosquito nets) can also help eradicate malaria.The Ross-Macdonald model is mathematically tractable in the sense that long term solution behavior of the model system (1) can be fully determined by the combined parameter R 0 .Yet, it is biologically less accurate in the sense that many biological factors are omitted.One of the important factors is the latency in the transmission process.This can be seen from the life cycle of malaria parasite.The life cycle of the Plasmodium begins from a blood meal of female mosquito from human beings.After being bitten by an infected female mosquito, a person receives an inoculum of plasmodium parasite (sporozoites).About half an hour later, liver cells of the person are invaded by sporozoites.The reproduction of parasites (merozoites) occurs in liver cells again and again, releasing more free merozoites to infect more liver cells.The immature trophozoites, the name of the merozoites at this stage, become mature developing either in the sexual or asexual way.Those who undergo the asexual development will go to the erythrocytic cycle producing more immature trophozolie, while others grows to gametocytes in the sexual way waiting for going out to the body of a female mosquito via its biting.Once they are ingested into a female mosquito, the parasite gametocytes taken up in the blood will further differentiate into male or female gametes and then fuse in the mosquito gut.This produces an ookinete that penetrates the gut lining and produces an oocyst in the gut wall.When the oocyst ruptures, it releases sporozoites that migrate through the mosquito's body to the salivary glands, where they are then ready to infect a new human host.See, e.g.[2,22] for details on this topic.Both developments inside a mosquito and inside an human host described above take some time.
Some modellers have noticed the missing of latencies in the Ross-Macdonald model and have proposed replacements by delay differential equations, but most of these works only have incorporated a single delay denoting the latency of the parasite in mosquitoes, see, e.g., [2,3,14].Recently Ruan et al [20] modified the model (1) by adding two delays accounting for the latencies in mosquitoes and humans respectively, resulting in the following delayed and rescaled system where m = M/N , x = I 1 /N and y = I 2 /M , and the term e −d1τ1 (e −d2τ2 resp. ) accounts for the probability that an infected human host (mosquito resp.) can survive the latent period τ 1 (τ 2 resp.).Note that (3) is exactly the subsystem of the model proposed in Anderson and May [1, p.399], consisting of the infectious components only, which is decoupled from the full system there.However no analysis was done in [1].For this modified model, the basic reproduction number is adjusted to It is shown in [20] that when R 0 < 1, then the disease free equilibrium (0, 0) is stable; when R 0 ≥ 1, then (0, 0) is unstable and there is an endemic equilibrium (x * , y * ) which is locally asymptotically stable provided that the two delays are small and a 2 e 1 e 2 m < ae This condition is a mathematically technical one, and it does not seem to have a biological explanation.Numerical simulations indicate that solutions of (3) with initial values from the region [0, 1] × [0, 1] can go outside this region, causing a confusion since x(t) and y(t) are proportional variables.This confusion suggests a careful revisit to the model.Moreover, the latencies of the malaria parasite in mosquitoes may differ from individual to individual, and so do the latencies in humans.This requires some mechanism in the model to reflect such variances of the latencies.The goal of this paper is to derive a more general and more realistic model that incorporates not only the latencies of the malaria parasite in both mosquitoes and humans, but also the variances of the latencies.In Section 2, following the idea in [25], we will formulate a more general model with two probability functions P 1 (t) and P 2 (t) describing the latency distributions for humans and for mosquitoes respectively.In Section 3, we analyze our new model.Under some reasonable assumptions, we address the well-posedness, identify the basic reproduction number R 0 for the model, and prove that the disease free equilibrium is globally asymptotically stable if R 0 < 1.When R 0 > 1, the disease dynamics is more difficult to determine for general P 1 (t) and P 2 (t), hence we consider two specific cases for P 1 (t) and P 2 (t).In Sub-Section 3.2, we consider the case that P 1 (t) and P 2 (t) are both exponential functions, resulting in an ODE system; in Sub-Section 3.3, we take P 1 (t) and P 2 (t) as step functions, leading to a system of delay differential equations (DDE).In both cases, we are able to obtain results on the disease dynamics.In Section 4, we summarize our main results and give some remarks discussing the modelling issue.
2. Model formulation for general latency distributions.Denote the size of the population of human beings by N (t) and that of the female mosquitoes by M (t).Let S 1 (t) and I 1 (t) be, respectively, the sub-populations of the susceptible and infectious classes of human hosts and S 2 (t) and I 2 (t) be the respective sub-populations of the susceptible and infectious classes of female mosquitoes.As mentioned in the introduction, there is a complicated development process within a host as well as within a vector, causing a latency in each half of malaria life cycle.This requires introducing a third class of sub-population: latent (or exposed) class, consisting of those individuals who have been infected but are not infectious yet.Denote by L 1 (t) and L 2 (t) the sub-populations of the latent host and the latent female mosquito respectively.
We consider a simple demographic scenario by assuming constant natural birth rates and death rates for both humans and the mosquitoes, denoted respectively by b 1 , b 2 and d 1 , d 2 .As in the introduction, we use the constant a to denote the mosquito biting rate and let e 1 be the probability that a biting by an infectious mosquito to a susceptible person will cause infection to the person, and e 2 be the probability that a biting by a susceptible mosquito to an infectious human individual will cause infection to the mosquito.The malaria parasite only causes deaths in human beings but not in mosquitoes, and this suggests introducing a disease related death rate for human beings, denoted by d.Infected human beings may recover, either due to the functioning of the immune system or through a treatment including taking anti-malaria drugs such as Chloroquine, Quinine and Amodiaquine.Let γ 1 be the recovery rate which is assumed to be a constant.Now we introduce the latency distributions by following the idea in [25].Let P 1 (t) denote the probability (without taking death into account) that a latent host individual still remains in the latent class t time units after entering the latent class (i.e., being infected).and similarly, let P 2 (t) be the probability that a latent vector individual still remains in the latent class t time units after entering the latent class.It is biologically reasonable to assume that P 1 (t) and P 2 (t) possess the following properties: (H) : For i = 1, 2, P i : [0, ∞) → [0, 1] are non-increasing, piecewise continuous with possibly finitely many jumps and satisfy P i (0 + ) = 1, lim t→∞ P i (t) = 0 with ∞ 0 P i (u) du positive and finite.Assume that initially S 1 (0) > 0, I 1 (0) ≥ 0, S 2 (0) > 0, I 2 (0) ≥ 0 and L 1 (0) = L 2 (0) = 0. Then the equations governing the subpopulations are given by Since the emphasis of this paper is the impact of latencies, we will follow the existing models in [16,17,18,19,20] to assume constant total populations for both human beings and female vector mosquitoes, i.e., N (t) = N and M (t) = M both are constants.This can be achieved by, for example, assuming that (A1) Disease related deaths can be ignored (i.e., setting d = 0); (A2) The natural birth rates balance the natural death rates for both host and vector (i.e., b 1 = d 1 and b 2 = d 2 ).There may be other situations that can lead to constant populations, (e.g., a compensation to the disease caused deaths by immigration for human host).However for simplicity of discussion, we simply assume (A1) and (A2) in the rest of the paper.We point out that in many situations, N (t) and M (t) vary only slightly, and this also constitute a good scenario for approximating N (t) and M (t) by constants.With these assumptions, one only needs to work on four out of the six variables.We choose S 1 , I 1 , S 2 and I 2 for which the governing differential equations are derived as below.
Differentiating the L 1 and L 2 equations (in the sense of Riemann-Stieltjes integral) leads to Here and hereafter, D t means the derivative with respect to variable t.In the L 1 equation above, each term has its own biological meaning: the first term is the rate of new infections, the second term accounts for the rate at which the infected individuals move to the infectious class from the exposed class, and the third term is due to natural death.The terms in the L 2 equations are explained in the same way.Passing to the I 1 and I 2 equations and keeping the S 1 and S 2 equations ( 6) lead to the following reduced system Rescaling ( 8) by with the following obvious constraints: where m = M N represents the average mosquito number per person.
3. Mathmatical analysis of the model.By the theory for integro-differential equations in [15], one knows that for any given initial values S i (0) ≥ 0 and I i (0) ≥ 0, i = 1, 2, system (9) has a unique solution with (S 1 (t), I 1 (t), S 2 (t), I 2 (t)) satisfying the initial conditions.From the biological significance, we only need to consider system (9) in the set Indeed, one can easily show that the set Ω is positively invariant in the sense stated in the following lemma.
satisfying the initial conditions, which remains in Ω for all t ≥ 0.Moreover, if The proof of this Lemma is by a quite standard argument, namely, by using the variation-of-constant formula to individual equations as well as by way of contradiction, which is similar to that of Lemma 2.1 in [25].We omit it to save space.
Let Pi := lim Clearly, P1 (resp.P2 ) is the average time that an infected human being (resp.female mosquito) remains in the latent class before becoming infectious or dying (see [25]).By the properties of P i (u), one knows that Actually, P1 d 1 (resp.P2 d 2 ) is the probability that an infected host (resp.mosquito) will die during the latent period.Hence, Q 1 (resp.Q 2 ) represents the proportion of the exposed hosts (resp.vectors) that could survive the latent period, where Using Q i , i = 1, 2, the basic reproduction number for the model ( 9) can then be defined as accounting for the average number of secondary infections that a single infectious human being (female mosquito), once introduced into fully susceptible populations of mosquitoes and humans, is expected to cause to the humans (female mosquitoes) during the infection period.Here, due to the transmission nature of this vector-host disease, R 0 consists of two parts: m ae1 γ1+d1 •Q 1 accounts for how many new infectious mosquitoes an infectious human being can result in during his infection period and ae2 d2 • Q 2 explains how many new infectious human beings an infectious mosquito can lead to during its infection period.
Model system (9) has a disease free equilibrium E 0 , given by E 0 = (1, 0, 1, 0).In terms of the biological meaning of the basic reproduction number, R 0 = 1 should be a threshold value for the stability/instability of E 0 for the model (9), as is confirmed in the following Theorem.
Proof of Theorem 3.2.The linearization of ( 9) at E 0 is where Denote by X(z) the Laplace transform of X(t).Applying the Laplace transform to (12) yields where I d is the 4 × 4 identity matrix and C(z) is the Laplace transform of C(t), i.e., Thus, the stability of E 0 is determined by the roots of the characteristic equation det[z Expanding the determinant leads to where ) Since z = −d 1 and z = −d 2 are two negative real roots of ( 13), the stability of E 0 is fully determined by the roots of h(z) = 0, which is equivalent to Assume that R 0 < 1.Then Let z = x + iy.If x ≥ 0, then we have which contradicts (16).Therefore, the real part x must be negative, implying that E 0 is locally asymptotically stable if R 0 < 1.
Next, assume that R 0 > 1.To show that E 0 is unstable, it suffices to show that h(z) = 0 admits a positive real root.Considering z = x > 0 and let Note that T (x) is increasing in x ≥ 0 and On the other side, dξ dη is decreasing for x > 0, and Next, we show that E 0 is also globally attractive when R 0 < 1.To this end, we use the notations for a function defined for all large t.Let (S 1 (t), I 1 (t), S 2 (t), I 2 (t)) be a solution of (9) in Ω.By Lemma 3.1, we know that By the Fluctuation Lemma [11], there is a sequence t n with t n → ∞ as n → ∞ such that Rewrite the differential equation for I 1 (t) in (9) as Evaluating this equation at t = t n and letting n → ∞ on both sides of the resulting equation, we obtain By the Lebesgue -Fatou Lemma (see [23], P468), it follows that Similarly, we can establish the following: The two inequalities ( 22) and ( 23) imply that either I ∞ 1 and I ∞ 2 are both positive or both zero.We show that the former is impossible if R 0 < 1.Otherwise, ( 22) and ( 23) would yield Applying (25) and the theory of asymptotically autonomous systems (see, e.g., [4]) to the S i (t) equations in (9), we conclude that Thus, E 0 is globally attractive, and hence, globally asymptotically stable in Ω provided that R 0 < 1.The proof of the theorem is completed.
When R 0 > 1, for general functions P 1 (t) and P 2 (t), the dynamics of model ( 9) is difficult to determine.For example, even the important concept of endemic equilibrium remains a problem: for some choices of P 1 (t) and P 2 (t), model ( 9) may allow an endemic equilibrium while for others choices, it may not support an endemic equilibrium.To proceed further, we consider two special cases of P 1 (t) and P 2 (t), for which we are able to obtain some further information about the dynamics of (9).
3.1.Special Case I-An ODE system.In this section, we adopt P i (t) = e −εit , i = 1, 2 where ε 1 and ε 2 are positive constants.This means that the probabilities of infected hosts and vector remaining in the latent classes follow negatively exponentially distributions with mean exposed times being 1/ε 1 and 1/ε 2 respectively.In this case, model ( 9) reduces to the following system of ordinary differential equations: The two survival factors Q 1 and Q 2 are now given by Q i = εi εi+di , i = 1, 2, and accordingly, the basic reproduction number becomes When R 0 > 1, in addition to the disease free equilibrium E 0 which is unstable, (27) also admits an endemic equilibrium , where where are all positive constants.The following theorem shows that if γ 1 = 0, the global dynamics of system ( 27) is completely determined in terms of R 0 , which acts as a threshold in the global sense.
Proof of Theorem 3.3.To prove the global stability of E * , we consider the full model system associated with ( 27) by adding the latent classes: where, for the convenience of notation, we have introduced the new parameters β 12 = ae 1 m and β 21 = ae 2 .We will employ a Lyapunov function similar to those used in recent works [12,13,7,8].To this end, we set v 1 = β 21 S * 2 I * 1 and v 2 = β 12 S * 1 I * 2 and let where S * i and I * i , i = 1, 2, are given in (29), and Differentiating V (t) along any positive solution of (30) gives The third and fourth terms on the right side of last equality cancel out: The sum of the fifth and seventh terms can be rewritten as The sixth term vanishes since γ 1 = 0 is assumed.Thus, V (t) can be simplified as By the relation of arithmetic mean and geometric mean, we conclude that V (t) ≤ 0 with the equality holding if and only if By the Lyapunov-LaSalle Theorem, Ê * is globally asymptotically stable for (30).Back to (27), we conclude that E * is globally asymptotically stable for (27) among all positive solutions in Ω, completing the proof.
3.2.Special Case II-A DDE system.Consider step functions for P 1 (t) and P 2 (t): where τ 1 ≥ 0 and τ 2 ≥ 0 are constants.Although the latent period differs from individual to individual, choosing τ 1 and τ 2 as the respective average latencies for infected humans and infected female mosquitoes would make the above P 1 (t) and P 2 (t) reasonable approximations for the real situation.
With this pair of P 1 (t) and P 2 (t), the long term (e.g., for t ≥ max{τ 1 , τ 2 }) disease dynamics are governed by the following system of delay differential equations derived from (9): with Accordingly, Q i can be calculated as Q i = e −diτi , i = 1, 2, resulting in the following explicit formula for the basic reproduction number: For (34), when R 0 > 1, the components of the unique endemic equilibrium ) can be more explicitly expressed by ae1m(d1+γ1)d2D1 , where Because R 0 > 1, one can choose δ ∈ (0, 1) sufficiently small so that 1 − (1 − δ) 2 R 0 < 0, and hence, (41) has a root with positive real part.This means that positive solutions of (41) are unbounded.On the other hand, the comparison theorem for delay differential equations (see, e.g., Smith [21]) implies that I 1 (t) ≥ u 1 (t) and I 2 (t) ≥ u 2 (t) where (u 1 (t), u 2 (t)) is the positive solution of (41) with the initial function (φ 2 , φ 4 ) and hence is unbounded.This contradicts (37), and the contradiction completes the proof of the lemma.
Proof of Theorem 3.6.We use a Lyapunov functional to prove the theorem.Let ) ds.
The derivative of V along the trajectory of (34) is −ae 1 me −d1τ1 I 2 (t − τ 1 )S 1 (t − τ 1 ) Setting c 1 = ae 2 I * 1 S * 2 e −d2τ2 and c 2 = ae 1 mI * 2 S * 1 e −d1τ1 , and reorganizing the above formula, we obtain Now, by the relation between arithmetic and geometric means and the property of the function g(u) = 1 − u + ln u, we conclude that V ≤ 0 and V = 0 if and only if (S 1 , I 1 , S 2 , I 2 ) is at E * .It follows from the Lyapunov-LaSelle Theorem for DDEs (see [9]) that E * is globally asymptotically stable in X 0 + , completing the proof.

Conclusion and discussion.
In this paper, we have modified the classic Ross-Macdonald model for the disease dynamics of Malaria by incorporating latencies both in human beings and in the female mosquitoes.The novelty of our model is that we have introduced two general probability functions (P 1 (t) and P 2 (t)) to reflect the fact that the latencies of the malaria parasite differ from individuals to individuals in both humans and mosquitoes.We have justified the well-posedness of the new model, identified the basic reproduction number R 0 for the model and analyzed the dynamics of the model.We have shown, very naturally and as in most works on disease models, that when R 0 < 1, the disease free equilibrium E 0 is globally asymptotically stable, meaning that the disease will eventually die out; and if R 0 > 1, E 0 becomes unstable.When R 0 > 1, the dynamics of the model become more difficult for general P 1 (t) and P 2 (t), and this forces us to consider some specific functions.When P 1 (t) and P 2 (t) are both exponential functions, the model reduces to a system of ordinary differential equations; when P 1 (t) and P 2 (t) are both step functions, the long term disease dynamics are governed by a system of delay differential equations.In both cases, we are able to show that when R 0 > 1 then the disease will persist; moreover if there is no recovery (γ 1 = 0), then all admissible positive solutions will converge to the unique endemic equilibrium.
Our approach may provide a frame work for dynamics of other mosquito-borne diseases.Taking Dengue as an example, since this disease is caused by dengue virus (unlike malaria protozoa), the recovered human beings will carry immunity and hence will not return to the susceptible class, implying γ 1 = 0 in the corresponding model.Therefore, our approach (actually results) can be easily applied to the corresponding model(s) for dengue disease.
From the formula of the basic reproduction number R 0 for our model, we can see that it is indeed smaller than the one obtained by ignoring the latencies (i.e., setting Q 1 = 1 and Q 2 = 1).In other words, if the latencies are neglected in modelling the disease dynamics, the basic reproduction number will be over calculated, regardless of what forms of the latency probability functions P 1 (t) and P 2 (t) are adopted We point out that there is a mathematical theory for disease model which defines the basic reproduction number as the spectral radius of the so called next generation operator.Here in this paper, our R 0 is defined by the so called survival function (see,e.g., [10]).The difference lies in that " the survival function gives the total number of infections in the same class produced by a single infective of that class, while the next generation operator gives the mean number of new infections per infective in any class per generation.Value corresponding to the latter definition thus depend on the number of infective classes in the infection cycle " [10].Taking the the ODE model (27) as an example, using the next generation operator (matrix in this case) approach from [5,26], the basic reproduction number for (27) is defined as which is the square root of the formula in (28).Note that many researchers have used survival function scenario to define basic reproduction numbers for vectorborne diseases, see e.g., [2,10,20] and the references therein.Note that because the threshold value for the basic reproduction number is at 1, such a difference does not cause any mathematical problem in exploring the threshold property of vectorborne disease models.For a detailed discussion on this topic, we refer readers to [5,6,10,26].We conclude the paper by a remark that the way we have incorporated latencies in this paper may also help clarify the confusion for (1.3) mentioned in the introduction.Indeed, by adding τ 1 > 0 and τ 2 > 0 into the model, latent classes in both humans and mosquitoes are admitted and hence, the terms 1 − x(t − τ 1 ) and 1−y(t−τ 2 ) should be replaced by 1−l 1 (t−τ 1 )−x(t−τ 1 ) and 1−l 2 (t−τ 2 )−y(t−τ 2 ) respectively, where l 1 (t) is the proportion of the latent human beings and l 2 (t) is the proportion of the latent mosquitoes with both satisfying equations corresponding to (35).Since 1 − x(t − τ 1 ) is larger than 1 − l 1 (t − τ 1 ) − x(t − τ 1 ) and 1 − y(t − τ 2 ) is larger than 1 − l 2 (t − τ 2 ) − y(t − τ 2 ), this may explain why the solutions of (1.3) with initial values from [0, 1] × [0, 1] may go beyond this region.When only considering a discrete latency in mosquitoes, a similar situation is also discussed in Aron and May [2].

Figure 1 .
Figure 1.The transmission diagram of the host-vector SLIS model