Man Bites Mosquito: Understanding the Contribution of Human Movement to Vector-Borne Disease Dynamics

In metropolitan areas people travel frequently and extensively but often in highly structured commuting patterns. We investigate the role of this type of human movement in the epidemiology of vector-borne pathogens such as dengue. Analysis is based on a metapopulation model where mobile humans connect static mosquito subpopulations. We find that, due to frequency dependent biting, infection incidence in the human and mosquito populations is almost independent of the duration of contact. If the mosquito population is not uniformly distributed between patches the transmission potential of the pathogen at the metapopulation level, as summarized by the basic reproductive number, is determined by the size of the largest subpopulation and reduced by stronger connectivity. Global extinction of the pathogen is less likely when increased human movement enhances the rescue effect but, in contrast to classical theory, it is not minimized at an intermediate level of connectivity. We conclude that hubs and reservoirs of infection can be places people visit frequently but briefly and the relative importance of human and mosquito populations in maintaining the pathogen depends on the distribution of the mosquito population and the variability in human travel patterns. These results offer an insight in to the paradoxical observation of resurgent urban vector-borne disease despite increased investment in vector control and suggest that successful public health intervention may require a dual approach. Prospective studies can be used to identify areas with large mosquito populations that are also visited by a large fraction of the human population. Retrospective studies can be used to map recent movements of infected people, pinpointing the mosquito subpopulation from which they acquired the infection and others to which they may have transmitted it.


Distribution of mosquito population between patches
In order to introduce variation in the number of mosquitoes in each patch without changing the total number in the metapopulation we use the exponential distribution for x in the interval [0, ξ], F(x) = λe -λx , where λ controls the degree of skew. When λ is large the distribution is strongly skewed and the area under the curve on [0, ξ] is very close to 1. However, when λ is close to 0, the distribution is more uniform and the area under the curve defined on [0, ξ] may be much less than 1. Therefore we use the normalized function G(x) constructed by dividing F(x) by the total area under the curve on [0, ξ]: Given n patches, the interval [0, ξ] is divided into n equal subintervals by the partition x i = ξ(i -1)/n, i = 1…n + 1. The area under the curve G(x) in each subinterval gives the proportion of the vector population associated with that patch and multiplying by the total vector population size gives the actual number of vectors in the patch: Throughout this study ξ = 120.

Construction of equations for n patch model, no transit patch
The model integrates a SEIR framework for the host population and a SEI framework for the vector population [1] into a metapopulation structure in which distinct vector subpopulations are linked by host movement [2]. The total host population N h is divided into n subpopulations N h j where j = 1…n is the usual destination patch of that group. Each subpopulation is subdivided into a further n + 1 subpopulations N h ij where i = 0…n is the current patch of those individuals. Hosts in patch 0 leave at rate ρ and travel to their usual destination patch with probability (1 -δ ) + δ/n and one of the other n -1 patches each with probability δ/n. Hosts in patch i ≠ 0 leave at rate τ and return directly to patch 0. Each host subpopulation is subdivided according to infection status: susceptible (S h ij ), exposed (infected but not infectious, E h ij ), infectious (I h ij ) and recovered (R h ij ). Hosts of all classes die at constant rate µ h and are replaced with susceptible hosts. Infected hosts become infectious at rate ε h . Infectious hosts recover at rate γ. Recovered hosts have complete lifelong immunity to re-infection. All hosts continue to move at the same rate regardless of their infection status. Each vector subpopulation is subdivided into susceptible (S v i ), exposed (E v i ) and infectious (I v i ) classes. Vectors of all classes die at constant rate µ v and are replaced with susceptible vectors. Exposed vectors become infectious at rate ε v and remain in this class until they die. Vectors bite at rate β. So in patch i there are a total of βS v i bites by susceptible vectors. Considering the proportion of hosts in the patch that are infectious, the host-vector transmission rate is then: Following the same reasoning, in patch i, the vector-host transmission rate for hosts with normal destination patch j is: The complete system of ordinary differential equation is thus:

Derivation of approximate system for n patches, no transit patch
To simplify the model we apply a method previously used for an epidemiological metapopulation model with direct transmission [2]. We assume that, as long as the time spent away from patch 0 is relatively short, the timescale of the travel dynamics is much faster than the timescale of the epidemiological dynamics. Therefore we approximate the population size of each host type (S h ij , E h ij, I h ij, R h ij ) in each patch by assuming that they scale with the proportion of the total population expected to be in that patch at equilibrium. We then define S h j to be the total number of susceptible hosts with normal destination patch j, irrespective of their current location. E h j , I h j , R h j and N h j are defined similarly. So: Note that the definitions for the vector population S v j , E v j , I v j are unchanged and relate to the number of mosquitoes actually present in patch j. Also, the sum for the total population size here is over all patches i where that host type is currently present and is not the same as the sum that appears in the denominator of the transmission terms of the original equations, which is over all destination patches j. Taking the derivatives of the new variable, all the terms involving τ and ρ cancel, leaving: We now wish to eliminate the ij groupings in the host population by writing them in terms of the j groupings. In order to estimate the proportion of the population in each group we consider the disease free system, i.e. system (5) with I v i = 0 for all i. Then the susceptible population is equal to the entire population and the system is described by: Setting the derivatives equal to 0 gives: The I, E and R hosts groups simplify in the same way. Furthermore, we assume that the host travel dynamics are symmetric, so the number of hosts visiting each patch is the same for all patches: N/n. It follows that the total number of hosts of each group the same for all patches and so N ik Substituting all of these approximations back into the condensed system (7) gives the approximate system: This simplifies more than the direct transmission model considered by [2]. In that model it is necessary to estimate the probability of two hosts meeting in a patch. Here the vector facilitates the meeting indirectly so we only need to estimate the probability of a host meeting a vector and the vectors never leave the patch.

Basic reproductive number for model with one destination patch
Using the next generation method [3,4], the global reproductive number of the model with one destination patch is The first five terms appear in the classical R 0 for a host-vector system and give the expected number of transmission events before recovery or death if the initial infection is in patch 1 and the host population does not move. Quarantine of symptomatic infections would scale this quantity. The interpretation of these terms has been discussed extensively elsewhere [1,3]. The final term is the most relevant to this study. It takes into account the expected proportion of the infectious period an infected host actually spends in patch 1 [5].

Basic reproductive numbers for model with three destination patches
The global reproductive number for the model with three destination patches can be calculated numerically using the next generation method. In addition, we define the host reproductive number R 0 h j as the number of secondary host infections resulting from a single infected host with normal destination j. The vector reproductive number R 0 v j is defined similarly. As set out in detail below: As before, the terms compounded into Λ also appear in the classical R 0 for a host-vector system and here we focus instead on the terms involving δ. Beginning with R 0 h j , suppose there is initially one infected host that normally travels to patch j. The expected number of secondary infections in hosts that normally travel to patch j is: Here n(1 -δ ) 2 + 2δ(1 -δ)N j v corresponds to a transmission cycle entirely within patch j. The initial type j host infects vectors in patch j that go on to infect hosts of type j. The term (δ 2 /n)N v corresponds to a transmission cycle that only involves type j hosts, but occurs entirely outside of patch j. The initial type j host travels to a random patch and infects the local vectors which then infect other type j hosts that travel to this patch. The sum of these terms represents the local maintenance of disease within the host population associated with patch j. The expected number of secondary infections in hosts that normally travel to patches beyond j is: Here (n -2)δ(1 -δ)N j v corresponds to a transmission cycle in which patch j vectors spread the infection to other host types. The initial infected host infects vectors in patch j, which then infect hosts that are not of type j but are visiting patch j. The term (δ -δ 2 /n)N v corresponds to a transmission cycle in which the type j host spreads the infections to other patches. The initial infected host travels to patches other than j and infects the local vectors which then infect hosts of type other than j visiting that patch. The sum of these terms represents the spread of disease to host populations associated with other patches.
Turning to R 0 v j , suppose there is initially a single infected vector in patch j. Then the expected number of secondary vector infections within patch j is: Here n(1 -δ ) 2 + 2δ(1 -δ)N j v corresponds to a transmission cycle entirely contained within patch j. The term δ 2 /N j v corresponds to hosts of type other than j becoming infected while visiting patch j and either remaining in, or returning to, patch j to re-infect the local mosquito population. The sum of these terms represents the local maintenance of disease within the patch j vector population. The expected number of secondary vector infections in patches beyond j is: which simplifies to: In (17) the term 2δ(1 -(n -1)δ/n)(N v -N j v ) corresponds to a transmission cycle in which type j hosts spread the infection when, after being infected in patch j, they travel to a different patch and infect the local vector population. The term (n -2)δ 2 /n)(N v -N j v ) corresponds to a transmission cycle in which hosts of type other than j spread the infection by visiting patch j, becoming infected, and then traveling to a different patch. The sum of these terms represents the spread of disease to vector populations in other patches.
Basic reproductive number for model with n identical patches plus transit patch If we assume that the total vector population excluding the transit patch subpopulation is uniformly distributed between the n destination patches, so each patch contains N v /n vectors, and hosts always travel to the same destination patch, so δ = 0, then the basic reproductive number of the system found using the next generation method is: If N v A = 0 and vectors are absent from the transit patch, this reduces to the classical R 0 for a host-vector system with a host vector ratio of N v /N h .