A PERIODIC ROSS-MACDONALD MODEL IN A PATCHY ENVIRONMENT

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number R0 and show that either the disease-free periodic solution is globally asymptotically stable if R0 ≤ 1 or the positive periodic solution is globally asymptotically stable if R0 > 1. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.

1. Introduction.Malaria, a widely prevalent vector-borne disease in tropical and subtropical areas, is caused by a parasite that is transmitted to humans and many other animals by the Anopheles mosquito.Once infected, people may experience a variety of symptoms, ranging from absent or very mild symptoms to severe complications and even death.It is one of the most deadly infectious diseases that causes major economic loss due to illness and death in humans.
The Ross-Macdonald model is the earliest and also simplest mathematical model describing malaria transmission between human and mosquito populations.It was initially proposed by Ross [31] in 1911 and later extended by Macdonald [23,24,25] in 1950s.The modeling framework is now widely used for malaria and some other mosquito-borne diseases.It captures the essential features of malaria transmission process, but ignores many factors of real-world ecology and epidemiology (see Ruan, Xiao and Beier [32]).
One omission in the classical Ross-Macdonald model is the temporal heterogeneity in the number of both populations and the human feeding rate of mosquitoes.In many nations like Niger, seasonal human migration has a long history, and destinations and reasons vary by community and ethnic group (Rain [29]).It is a common sense that there are more mosquitoes in the summer and fewer in the winter.The biological activity and geographic distribution of a malaria parasite and its vector are greatly influenced by climatic factors such as rainfall, temperature and humidity (Craig et al. [9], Gao et al. [12]).These influences can be investigated by assuming some parameters to be time dependent.In recent years, epidemic models with seasonal fluctuations have been proposed and explored by many researchers.A vector-borne model for epidemics of cutaneous leishmaniasis with a seasonally varying vector population and a distributed delay in humans was proposed and studied by Bacaër and Guernaoui [4].Dembele, Friedman and Yakubu [10] introduced a new malaria model with periodic mosquito birth and death rates.Bai and Zhou [6] developed a SIRS model for bacillary dysentery with periodic coefficients.Using an ODE model with periodic transmission rates, Zhang et al. [38] considered the transmission dynamics of rabies from dogs to humans in China.Liu, Zhao and Zhou [17] used a compartmental model with periodic infection rate and reactivation rate to describe the TB transmission in China.For a detailed discussion of seasonal infectious diseases, we refer the reader to two excellent survey papers (Altizer et al. [1], Grassly and Fraser [13]) and the introduction of a recent paper (Lou, Lou and Wu [19]).Periodicity mainly occurs in the contact rate, birth or death rate, vaccination rate, etc.
Another omission is the spatial movements of hosts between different patches.The migration of humans can influence disease spread in a complicated way (Gao and Ruan [11]).To have a better understanding of the disease dynamics, it is necessary to incorporate periodic variations and population dispersal into epidemic models.These two issues are considered in (Lou and Zhao [20]) via a periodic reaction-diffusion-advection model.In this paper, we will formulate a periodic Ross-Macdonald model in a patchy environment and establish the threshold dynamics of the model in Section 2 and Section 3, respectively.The last section gives some numerical simulations and a brief discussion of our main results and future research directions.
2. Model Formulation.Most mosquitoes can only travel a couple of kilometers throughout their lifetime (Costantini et al. [8] and Midega et al. [26]), so we assume no movement for vector populations between patches (see Auger et al. [3]).For simplicity, the human vital dynamics are ignored and the population model for vectors is the same as that studied in Smith, Dushoff and Mckenzie [33].Following the classical Ross-Macdonald model, we divide the adult female mosquito and human populations into two classes in each patch: susceptible and infectious.The total number of patches is p.We assume that the total populations of humans and mosquitoes at time t in patch i are H i (t) and V i (t), respectively.Let h i (t) and v i (t) denote the numbers of infectious humans and infectious mosquitoes in patch i, respectively.The interactions between humans and mosquitoes in patch i can be Human immigration rate from patch j to patch i for i = j −m ii (t) ≥ 0 Human emigration rate from patch i described by the following periodic system with non-negative initial conditions: ) ) ) The model parameters and their descriptions are listed in Table 1.To incorporate the seasonal mosquito dynamics and human migration, all time-dependent parameters in system (2.1) are continuous and periodic functions with the same period ω = 365 days.We assume that there is no death or birth during travel, so the emigration rate of humans in patch i, −m ii (t) ≥ 0, satisfies p j=1 m ji (t) = 0 for i = 1, . . ., p and t ∈ [0, ω].
For travel rates between two patches, we assume there must be some travelers between different patches at some specific seasons.Mathematically, we assume that the matrix ( ω 0 m ij (t)dt) p×p , which is related to the travel rates between different patches, is irreducible.The notation H(t) will mean (H 1 (t), . . ., H p (t)), with similar notations for other vectors.We will denote both zero value and zero vector by 0, but this should cause no confusion.The following theorem indicates that model (2.1) is mathematically and epidemiologically well-posed.
Theorem 2.1.For any initial value z in . ., p}, system (2.1) has a unique nonnegative bounded solution through z for all t ≥ 0.
Proof.Let G(t, z) be the vector field described by (2.1) with z(t) ∈ Γ.Then G(t, z) is continuous and Lipschitzian in z on each compact subset of R 1 × Γ.Clearly, G k (t, z) ≥ 0 whenever z ≥ 0 and z k = 0, k = 1, . . ., 4p.It follows from Theorem 5.2.1 in Smith [34] that there exists a unique nonnegative solution for system (2.1) through z ∈ Γ in its maximal interval of existence.The total numbers for hosts and  [35]) implies that N v (t) is ultimately bounded.Hence every solution of (2.1) exists globally.
3. Mathematical Analysis.In this section, we first show the existence of a unique disease-free periodic solution and then evaluate the basic reproduction number for the periodic system.By using the theory of monotone dynamical systems (Smith [34], Zhao [40]) and internally chain transitive sets (Hirsch, Smith and Zhao [15], Zhao [40]), we establish the global dynamics of the system.The following result is analogous to Lemma 1 in Cosner et al. [7] for the autonomous case.
Lemma 3.1.The human migration model (2.1a) with H i (0) ≥ 0 for i = 1, . . ., p and N h (0) > 0 has a unique positive ω-periodic solution H * (t) ≡ (H * 1 (t), . . ., H * p (t)), which is globally asymptotically stable.The mosquito growth model (2.1b) with V i (0) ≥ 0 for i = 1, . . ., p has a unique positive ω-periodic solution V * (t) ≡ (V * 1 (t), . . ., V * p (t)), which is globally asymptotically stable.Proof.Note that the travel rate matrix M (t) ≡ (m ij (t)) p×p has nonnegative offdiagonal entries and the matrix ( ω 0 m ij (t)dt) p×p is irreducible.Let φ(t) be the fundamental matrix of system (2.1a) which satisfies dφ(t)/dt = M (t)φ(t) and φ(0) = I.Then the pω period fundamental matrix φ(pω) is a strongly positive operator (Smith [34]).By using the pω period fundamental matrix φ(pω) in the proof of Theorem 1 in Aronsson and Kellogg [2], we see that the pω periodic system (2.1a) has a positive pω periodic solution H * (t), which is globally asymptotically stable with respect to all nonzero solutions.By using the global attractivity of the pω periodic solution, we can further show it is also ω periodic by using a similar argument in Weng and Zhao [37].Moreover, it is easy to see that . The i-th equation of (2.1b) has a unique positive periodic solution which is globally asymptotically stable.
Remark 3.1.We can relax the positivity assumption on ε i (t) to [ε i ] > 0 where or equivalently, ε i ( t) > 0 for some t ∈ [0, 2π).See Lemma 2.1 in Zhang and Teng [39] for results on a general nonautonomous system of the form The above result guarantees that system (2.1) admits a unique disease-free periodic solution . ., V * p (t), 0, . . ., 0, 0, . . ., 0).Biologically, both human and mosquito populations in each patch are seasonally forced due to periodically changing human migration and seasonal birth, death and biting rates of mosquitoes, respectively.Now we consider the asymptotically periodic system for malaria transmission In what follows, we use the definition of Bacaër and Guernaoui [4] (see also Bacaër [5]) and the general calculation method in Wang and Zhao [36] to evaluate the basic reproduction number R 0 for system (3.1).Then we analyze the threshold dynamics of system (3.1).Finally we study global dynamics for the whole system (2.1) by applying the theory of internally chain transitive sets (Hirsch, Smith and Zhao [15] and Zhao [40]).
Let Y (t, s), t ≥ s, be the evolution operator of the linear ω-periodic system That is, for each s ∈ R 1 , the 2p × 2p matrix Y (t, s) satisfies where I 2p is the 2p × 2p identity matrix.Let C ω be the ordered Banach space of all ω-periodic functions from R 1 to R 2p equipped with the maximum norm.We define a linear operator L : It is easy to verify that conditions (A1)-(A7) in Wang and Zhao [36] are satisfied.Therefore, the basic reproduction number of the periodic system (3.1) is then defined as R 0 := ρ(L), the spectral radius of L. The following lemma shows that the basic reproduction number R 0 is the threshold parameter for local stability of the diseasefree steady state.Lemma 3.2 (Theorem 2.2 in Wang and Zhao [36]).Let Φ F −V (t) and ρ(Φ F −V (ω)) be the monodromy matrix of system (3.2) and the spectral radius of Φ F −V (ω), respectively.The following statements are valid: Thus, the disease-free equilibrium P 0 of system (3.1) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.
Lou and Zhao [20] presented a global qualitative analysis for system (3.1) with a single patch.We will show that the system with multiple patches does not exhibit more complicated dynamics.
Theorem 3.1.System (3.1) admits a unique positive ω-periodic solution, denoted by , which is globally asymptotically stable with initial values in D 0 \{0} if R 0 > 1 and the disease-free equilibrium P 0 is globally asymptotically stable in Proof.To show the global asymptotic stability of P * (t) or P 0 , it suffices to verify that K(t, x) : R 1 + × D 0 → R 2p satisfies (A1)-(A3) in Theorem 3.1.2in Zhao [40].For every x = (h, v) ≥ 0 with h i = 0 or v i = 0, t ∈ R 1 + , we have By Lemma 3.3, the system (3.1) is cooperative in D t , and the ω-periodic semiflow is monotone.We can further prove that the pω-periodic semiflow is strongly monotone in IntD 0 , the interior of D 0 , due to the irreducibility of ( ω 0 m ij (t)dt) p×p .For each t ≥ 0, i = 1, . . ., p, there hold and for each x 0, α ∈ (0, 1).That is, K(t, •) is strictly subhomogeneous on R 2p + .Moreover, K(t, 0) ≡ 0 and all solutions are ultimately bounded.Therefore, by Theorem 3.1.2in Zhao [40] or Theorem 2.3.4 in Zhao [40] as applied to the pω solution map associated with (3.1), the proof is complete.
Using the theory of internally chain transitive sets (Hirsch, Smith and Zhao [15] or Zhao [40]), as argued in Lou and Zhao [22], we find that the disease either dies out or persists and stabilizes at a periodic state.Theorem 3.2.For system (2.1), if R 0 > 1 then there is a unique positive ωperiodic solution, denoted by E * (t) = (H * (t), V * (t), h * (t), v * (t)), which is globally asymptotically stable; if R 0 ≤ 1 then the disease-free periodic solution E 0 (t) is globally asymptotically stable.
According to Lemma 4.1(ii), we can show that the basic reproduction number for (3.1) is proportional to the biting rate, that is, R0 = kR 0 if βi (t) = kβ i (t) for k > 0 and 1 ≤ i ≤ p (see Lou and Zhao [21] or Liu and Zhao [18]).
The curves of the number of infected humans in patches 1 and 2, respectively, are blue solid and red dashed.(a) when there is no human movement, R 10 = 1.7571 > 1 and R 20 = 0.8470 < 1 indicate the disease persists in patch 1 but dies out in patch 2; (b) when human movement presents, R 0 = 1.3944 > 1 indicates the disease persists in both patches, and will stabilize in a seasonal pattern.
The basic reproduction number for the corresponding time-averaged autonomous system (see Wang and Zhao [36] and Zhang et al. [38]) is R0 = 1.2364.By Lemma 4.1 or the linearity of R 0 in the mosquito biting rates, we can multiply the biting rates by an appropriate constant such that R0 below 1 while R 0 above 1.Thus, the autonomous malaria model may underestimate disease severity in some transmission settings.
In order to investigate the effect of possible travel control on the malaria risk, we determine the disease risk when shifting the phases of the movement functions, while keeping their respective mean values and forcing strengths to be the same.With the same parameter values and initial data except that the migration rates m 12 (t) = 0.006 − 0.004 cos( 2πt 365 + φ 1 ) and m 21 (t) = 0.006 + 0.004 cos( 2πt 365 + φ 2 ), we evaluate the annual malaria prevalence over the two patches for φ 1 , φ 2 ∈ [0, 2π].Then Figure 2 presents the effect of phase shifting for movement functions on disease prevalence.The approximate range of malaria prevalence is from 17% to 23%.This result indicates that appropriate travel control can definitely reduce the disease risk, and therefore, an optimal travel control measure can be proposed.In this project, we have developed a compartmental model to address seasonal variations of malaria transmission among different regions by incorporating seasonal human movement and periodic changing in mosquito ecology.We then derive the basic reproduction number and establish the global dynamics of the model.Mathematically, we generalize the model and results presented in Cosner et al [7] and Auger et al [3] from a constant environment to a periodic environment.Biologically, our result gives a possible explanation to the fact that malaria incidence show seasonal peaks in most endemic areas (Roca-Feltrer et al. [30]).The numerical examples suggest that a better understanding of seasonal human migration can help us to prevent and control malaria transmission in a spatially heterogeneous environment.
It would be interesting to assess the impacts of climate change on the emerging and reemerging of mosquito-borne diseases, especially malaria.It is believed that global warming would cause climate change, while climatic factors such as precipitation, temperature, humidity and wind speed affect the biological activity and geographic distribution of the pathogen and its vector.It sometimes even affects human behavior through sea-level rise, more extreme weather events, etc.
The current and potential future impact of climate change on human health has attracted considerable attention in recent years.Many of the early studies (see Ostfeld [27] and the references cited therein) claimed that recent and future trends in climate warming were likely to increase the severity and global distribution of vector-borne diseases, while there is still substantial debate about the link between climate change and the spread of infectious diseases (Lafferty [16]).The controversy is partially due to the fact that although there is massive work using empirical-statistical models to explore the relationship between climatic factors and the distribution and prevalence of vector-borne diseases, there are very few studies incorporating climatic factors into mathematical models to describe disease transmission.
The research that comes closest to what we expect to do has been conducted by Parham and Michael in [28], where they used a system of delay differential equations to model malaria transmission under varying climatic and environmental conditions.Model parameters are assumed to be temperature-dependent or rainfall-dependent or both.Due to the complexity of their model, mathematical analysis was not possible and only numerical simulations were carried out.Furthermore, the basic reproduction number is also difficult to project through their model.We would like to extend our model to a mathematically tractable climate-based model in the future.

Figure 2 .
Figure 2. Effect of phase shift for movement rates between two patches on disease risk.

Table 1 .
The parameters in model (2.1) and their descriptions