The interplay of vaccination and vector control on small dengue networks

Dengue fever is a major public health issue affecting billions of people in over 100 countries across the globe. This challenge is growing as the invasive mosquito vectors, Aedes aegypti and Aedes albopictus, expand their distributions and increase their population sizes. Hence there is an increasing need to devise effective control methods that can contain dengue outbreaks. Here we construct an epidemiological model for virus transmission between vectors and hosts on a network of host populations distributed among city and town patches, and investigate disease control through vaccination and vector control using variants of the sterile insect technique (SIT). Analysis of the basic reproductive number and simulations indicate that host movement across this small network influences the severity of epidemics. Both vaccination and vector control strategies are investigated as methods of disease containment and our results indicate that these controls can be made more effective with mixed strategy solutions. We predict that reduced lethality through poor SIT methods or imperfectly efficacious vaccines will impact efforts to control disease spread. In particular, weakly efficacious vaccination strategies against multiple virus serotype diversity may be counter productive to disease control efforts. Even so, failings of one method may be mitigated by supplementing it with an alternative control strategy. Generally, our network approach encourages decision making to consider connected populations, to emphasise that successful control methods must effectively suppress dengue epidemics at this landscape scale.


Identify the infected subsystem
We begin by identifying which equations in the full set of ODEs describe new infections and changes in states of existing infections. As we are interested in the spread of a single serotype in this analysis, these equations are simplified by the elimination of secondary infection dynamics.
Ω i is the daily change in population size of patch i as a result of movement of individuals around the network. In a two patch network this must be equivalent to the loss of individuals from patch j, hence Ω i = −Ω j . Modifying equations (2) & (13) according to these criteria, (methods, section 2.4), we therefore have the change in infected hosts. I i , and infected vectors X i in patch i described by:

Linearize subsystem about infection free steady state
Next we assume that at the absolute start of the epidemic, the change in state of susceptible hosts is negligible, S i = N i , S j = N j . Furthermore demographic changes to the adult mosquito population are also negligible, dMi /dt = 0. As a result, equations (1)-(4) become functions only of other components included within the subsystem, with other state variables being reduced to constants. Hence,

Subsystem decomposition
Each equation in the subsystem has three components. The first corresponds to the rate of infections occuring within the focal patch, the second term models infections that are facilitated by host movement, and the final term accounts for mortality and recovery from infections. To keep track of these components in an easy manner, new notation is introduced here, where rateİ γδ corresponds to transmission events depending on patch δ that contribute to infections in γ. With this notation is becomes straightforward to decompose the infection subsystem into transmission rates and transitions rates to form the next generation matrix. For example, equation (5) can be rewritten as: Next we set y = (I i , I j , X i , X j ) , where indicates the transpose of this vector. Now we write the linearized infection subsystem in the form: Where T corresponds to transmissions, covering all epidemiological events giving rise to new infections, whilst Σ corresponds to all transitions, which encompasses all other rates responsible for changes in numbers of infections, namely, mortality and recovery rates. Infected states are referred to with indices α and β, with α, β ∈ 1, 2, 3, 4, such that T αβ represents the rate at which individuals in state β give rise to individuals in infected state α. Hence,

Calculate Next Generation Matrix
The Next Generation Matrix (NGM), K, is defined as K = −TΣ −1 . The biological interpretation for this is clear, because −Σ −1 αβ is the expected time that an individual in state β will spend in state α during their epidemiological life. Since Σ forms a diagonal matrix here, it is intuitive that this life span will equal 1 rate of recovery or death . Hence,

Split the Next Generation Matrix into components
Next the NGM is split into seperate matrices for rates of vector-to-host and host-to-vector transmission, represented by K V→H and K H→V . Hence, From this arrangement two R0s can be derived: R0 V →H and R0 H→V , by finding the dominant eigenvalue of matrices (15) & (16).

Calculating the dominant eigenvalue
The determinant is taken of the vector-to-host matrix minus the eigenvalue, λ, multiplied by identity matrix N.
The absolute largest solution to this quadratic equation gives the dominant eigenvalue, hence: By rearrangement an expression for λ can be found: Given the algebraic similarity between equations (15) & (16), for K V→H and K H→V respectively, it follows that by the preceeding derivation: These equations form the basis of the R0 analysis given in the main text.
Assumptions about host movement were analysed by taking different commuter behaviours into account, as demonstrated in figure 1. 'City-only' and 'town-only' control strategies were also compared when commuter flow between patches varied. So at a given level of control (2:1 insect release and 1% vaccination coverage) the effectiveness of controls on one way commute network types were compared to those generated when bidirectional commuting is permitted, with a commuter flow of 0.1 (figure 2).
As the one way plot (a) indicates, town-only control is more effective than city-only, with the difference increasing as more satellite patches are added to the network. This increase makes sense, as town-only control covers a larger proportion of the susceptible population for every additional town in the network. However, once bidirectional commuting is permitted, (b), this result is not so clear cut. Small networks benefit more from city-only controls under these conditions. The town-only method still becomes more effective as network size increases, but this method is less effective for bidirectional movement on networks.
One possible explanation for this is that bidirectional commuting reduces the city population size and increases the size of the towns relative to the one-way commute network. As a result, the ratio of vectors:hosts is higher in cities, biasing biting events in the city. Hence most infections in the bidirectional network are not occurring in towns, but the city instead, and control there can play a larger role in reducing infections across the network, particularly in smaller networks.
This does not hold for the four satellite network under town-only control. It is likely that the larger pool of town-dwelling susceptible individuals in larger networks may benefit even more from town-only controls, which offsets the high probability of individuals being bitten in the city. This city-biased biting effect is seen when just looking at DHF cases as well. Concordantly over the range of network sizes explored, city-only control is the most effective method for controlling these severe cases. One−way Bidirectional Figure 3: Secondary dengue control effectiveness differs by commute flow, with (a) displaying simulation results for a network using only one-way commuter behaviour, whilst (b) was generated using networks permitting bidirectional movements. All strategies become more effective as the networks increase in size, but the most effective strategy differs between (a) and (b).
From figure 3 it seems that commuter flow heavily influences where control measures will be most effective. In this case, more symmetry in commutes among patches brought about by bidirectional commutes makes biting events more common in the city, hence secondary infections disproportionately arise here, reducing the ability of town-only controls to target these cases.
Here we can also see that for a four satellite network under bidirectional commuting, town-only control is more effective at reducing primary infections, but city-only control is more effective at reducing secondary cases. This dichotomy in efficacy of control measures creates a trade-off when considering management of large networks. On the one hand, town-only control deals with a large number of mildly ill individuals, but city-only control will prevent a small number of people from developing very serious clinical syndromes.

Sensitivity Analysis
The model was examined using sensitivity analysis, where all aspects of the model were kept the same but one parameter was varied at a time. Vector parameters were increased by 5, 15, and 25%, and the subsequent increase in primary infections, as compared to a simulation where no parameters were augmented, was plotted. The results displayed in figure 4 show which parameters have most influence in the network.  Figure 5: Dramatic increases to secondary dengue in vector parameter stressed networks. As for figure 14, birth rate, larval development and vector longevity determine population dynamics of the vector, whilst bite rate and conversion rate of mature insects into dengue-carrying vectors determine disease transmissability.
Given how important commuter flow seems to be in altering the outcome of dengue fever epidemics and indeed the effectiveness of controls, sensitivity analysis was also applied to this ( figure   6). The movement coefficient is calculated as the product of connectivity and class movement terms, hence indicating the proportion of each patch that commutes. City patch infections are relatively resilient to changes in movement coefficient, whilst town infections are more sensitive to differences both in the magnitude and direction of commutes.
The city patch is relatively unaffected by the increase in commute traffic, but the level of infection in town patches varies considerably. For the one way commute type, (a), increased commuter flow around the network spreads the infection around the network leading to a greater number of infections in the towns. However this response becomes limited as the epidemic becomes constrained by other epidemiological factors (e.g. conversion rate of susceptible hosts to infected) more so than movement.
To an extent bidirectional movement shows a similar pattern, (b), however there is a difference between this line and the one-way network for intermediate commute values. This difference may be explained by fluctuating strength of town vector:host ratios in the bidirectional network, which varies as a function of the movement coefficient. This is also in keeping with the predictions made from R0 calculations, main text section 3, providing yet more evidence for the important role of commuting behaviour in altering the outcome of dengue outbreaks.
In terms of commute directionality, bidirectionality, with a commuter flow of 0.1, may actually mean that an unrealistic number of people enter a town daily from the city (in this particular example, town sizes fluctuate by a factor of five on a daily basis). So it may be that this effect may not occur so strongly in real systems. However, it is useful to illustrate that commuter dynamics may actually fall somewhere between one-way and bidirectional types explored here.
Hence bidirectional and one-way network results could be viewed as broad limits for describing how movement out of large, well-connected patches may influence epidemic dynamics.