Team:NAU-CHINA/CA Model

NAU-iGEM

Abstract

The ultimate goal of the project is to control the transmission of malaria by releasing mosquitoes carrying re-engineered Serratia marcescens. In order to guide this part of the implementation (indicate the optimal release strategy), we built a Cellular Automaton model to get the results of different release strategies. Moreover, based on the model mechanism, it can be extended to simulate a variety of vector-borne diseases in different regions in the world. In this article, taking Jiangsu Province, China as an example, we given some general results and a relatively good release strategy through qualitative analysis of 11 different release strategies. Under the optimal strategy, the quantity of re-engineered mosquitoes should be equivalent to the number of local mosquitoes, and the interval between the release sites should be approximately 1000 meters. In this way, we can make sure that malaria parasites in over 50% of mosquitoes can be effectively controlled when the number of mosquitoes reaches its peak.

1. Background

1.1 Background of the project

Our project enabled Serratia in the midgut of mosquitoes to secrete anti-malarial peptides by re-engineering it. And then the number of malaria parasites in the mosquito population can be controlled by releasing mosquitoes carrying Serratia which can be transmitted among mosquito populations. According to Wang’s research^[1], this kind of re-engineered bacterium can be maintained for three consecutive generations at least. In order to simulate the propagation of Serratia through mosquito populations when the project is mature enough, we assumed that re-engineered Serratia, with increased survivability, can persist for all subsequent generations within a year.

1.2 Background of mosquitoes

We took Anopheles sinensis in Jiangsu Province, China, as an example^[2]. Every year, these mosquitoes appear in May, disappear gradually in October, and the number of them peaks in July and August. The mosquitoes significantly gather in moist and cool areas, and most mosquitoes only move within 2 km^[3] of their birth range. Male mosquitoes can mate a lot of times, while female mosquitoes can only mate once and lay eggs several times after mating.

2. Model Establishment

2.1 Introduction to cellular automaton:

By setting a series of model construction rules, the cellular automaton^[4] can automatically simulate the diffusion of substances with a certain probability. In the dynamic system, cellular automaton can be divided into “homogeneous, periodic and chaotic structure” three types. In consideration of the aggregation and periodicity of insect activity and the complexity of boundary conditions caused by topography, the cellular automaton with periodic structure is obviously superior. In terms of algorithm complexity and accuracy, the cellular automaton also performs better than the partial differential equation.

2.2 The Establishment of Cellular Automaton

2.2.1 Symbols & Explanation

Variables	Explanation	Value
L_male	Average life expectancy of male mosquitoes	15 Days^[3]
L_female	Average life expectancy of female mosquitoes	45 Days^[3]
L_larva	The time larval hatching to adult	12 Days^[3]
N	Average number of female mosquitoes per spawn	880^[3]
P_grow	The probability that a larva hatches into an adult
p_sexinfect	The infection rate in mating process	0.9^*
p_larvainfect	The infection rate in hatching process	0.7^*
p_move	The probability that a mosquito move out the cellular
p_sex	Average mating rate between mosquitoes

Table 2.2.1(^* represents for estimating)

2.2.2 The Establishment of Environmental Cellular.

Considering the aggregation and periodicity of mosquito activity, we simplified the topographic map with (0,1)-Matrix by image segmentation algorithm, where 1 represents the highly attractive area and 0 represents the lowly attractive area.

Considering the range of mosquito activity and topographical features, we set the cellular size to 500 m × 500 m. In addition, considering the effective range, we set the number of cells to 100 × 100, thus the whole set of environmental cells stands for a square topographic map cover an area of 2500 square kilometers.

Since it is difficult to get an accurate topographic map, we use cellular evolution to generate a map randomly based on the topographic characteristics of temperate monsoon climate. Then the map is simplified to a (0,1)-Matrix of 100 × 100, where 39% are high attractive area. The matrix is shown in Fig 2.2.2 below.

Fig 2.2.2

2.2.3 The Establishment of Mosquito Cellular.

Considering the characteristics of the transmission of re-engineered bacteria in mosquito population, we divided the mosquitoes and larvae in every cell into 8 kinds, are shown in Table 2.2.3 below.

Male mosquito infected with Serratia

Male mosquito not infected with Serratia

Unmated female mosquito infected with Serratia

Unmated female mosquito not infected with Serratia

Mated female mosquito infected with Serratia

Mated female mosquito not infected with Serratia

Larva infected with Serratia

Larva not infected with Serratia

Table 2.2.3

There is a certain probability of cross-infecting of Serratia when male mosquitoes mating with female mosquitoes, and infected female mosquitoes will produce infected larva with a certain probability. This process is shown in Fig 2.2.3.1 and Fig 2.2.3.2. Fig 2.2.3.1 shows the process that female mosquitoes producing larvae. And Fig 2.2.3.2 shows the mating process of mosquitoes, in this process, there exist a certain probability of cross-infecting when infected male mosquitoes mating with uninfected female mosquitoes or uninfected male mosquitoes mating with infected female mosquitoes.

Fig 2.2.3.1(left) & Fig 2.2.3.2(right)

In the figures above, we denote these two male mosquitoes as A and B, and the corresponding numbers are a and b. Then we denote these unmated female mosquitoes as C and D, and the corresponding numbers are c and d. According to mass action principle, the probability of cross-infection can be computed by the following formula.

P1 = p_sexinfect × p_sex ×（ a × c ) ÷ ( ( a + b ) × ( c + d ) )

We assume that the re-engineered bacteria can transmit stably in the mosquito population, and it can produce enough anti-malarial peptides to effectively control the number of malaria parasites when the project mature enough. Therefore, the infection rate of Sarratia directly reflects the project's ability to control malaria. Currently, since these two parameters are difficult to determine, we made estimation and sensitivity analysis to explore their impact. Here we take p_sexinfect=0.9 and p_larvainfect=0.7 as an example.

In order to describe birth, death and mating, we introduce queues to store the age of mosquitoes in every cell dynamically. By updating the queue daily, it can increase the age of mosquitoes in every cell. And the transformation between different kinds of mosquitoes is realized by corresponding movement. Moreover, to further simplify the model, we assume that all the mosquitoes have the same life span. Thus, the position of the mosquito in the queue reflects its age. The mechanism of queue updating can be depicted in Fig 2.2.3.3 and Fig 2.2.3.4.

Fig 2.2.3.3(left) & Fig 2.2.3.4(right)

3. The Evolution of Cellular Automaton

1. In order to simulate the distribution of mosquitoes realistically, we abstract the square area into a three-dimensional figure, which means that when mosquitoes fly out of an area from a certain boundary, they will directly re-enter the area from the opposite boundary.

We randomly set the initial value of each cell between 0 and 1, and then the mosquito distribution before the evolution of the cellular automaton is shown as Fig 3.1 below.

Fig 3.1

2. All adult mosquitoes in the cell have a certain probability of p_move to migrate to the nearby lattices(cells). Mosquitoes are more likely to gravitate toward the "highly attractive" cells and stay away from the "lowly attractive" cells. Therefore, after the evolution of cellular automaton, the distribution of mosquitoes is shown as Fig 3.2below.

Fig 3.2

3. We control p_grow by introducing the function f(x) of environmental stress, to control the growing and declining of mosquito population. Then the probability P3 of larvae hatching into adults can be computed by the following formula.

P3 = P_grow × f(x)

The initial value of f(x) should be 1. When f(x) = 1 the population of mosquitoes remains stable. When f(x) is greater than 1, it means that the environment at that moment promotes the reproduction of mosquitoes. When f(x) is less than 1, it means that the environment inhibits the reproduction of mosquitoes.

4. Model Validation

1. To estimate p_move, the probability of mosquito moving out of the cell, we used traversal algorithm to find proper value of p_move, enabling the activity scope of 90% of mosquitoes not to exceed 2km within 20 days(assume that the average lifespan of mosquitoes is 20 days).

2. It is difficult to determine the value of the average mating rate p_sex of mosquitoes. In order to obtain this parameter, we make the cellular automaton evolve under an initial condition where the number of mosquitoes is a relative value. We know that the male-female ratio of mosquitoes should be 3:1, and the population size of mosquitoes should remain basically unchanged. We use traversal algorithm to find proper p_sex so that the evolution results converge to these two conditions as quickly as possible.

3. We obtained the curves of the number of mosquitoes in Jiangsu Province over time as Fig 4.1.

Fig 4.1 (from literature^[2])

4. Based on the curves above, the environmental stress function f(x) is fitted by genetic algorithm. The fitting result shows as Fig 4.2.

Fig 4.2

5. Numerical Simulation and Results Analysis

We obtain several results of releasing mosquitoes in Nanjing, China. Every year, these mosquitoes appear in May and disappear gradually in October, lasting about 160 days. The initial number of mosquitoes is 300,000. The total area of Nanjing city is about 2,500 square kilometers, and we simplified it into one large square consisting of 10,000 small square lattices.

We simulated 11 different types of release strategies, the changes of the infection rate of mosquitoes in every cell over time are displayed in the following 11 videos, and the total infection rates are described by the following pictures respectively:

1. release 300,000 mosquitoes into 10,000 cells

2. release 30,000 mosquitoes into 10,000 cells

3. release 300,000 mosquitoes into 5,000 cells

4. release 300,000 mosquitoes into 1,000 cells

5. release 300,000 mosquitoes into 500 cells

6. release 300,000 male mosquitoes into 10,000 cells

7. release 30,000 female mosquitoes into 10,000 cells

8. release 30,000 mosquitoes into 10,000 cells in 3 times

9. release 30,000 mosquitoes into 10,000 cells in 5 times

10. release 30,000 male mosquitoes into 10,000 cells in 3 times

11. release 30,000 female mosquitoes into 10,000 cells in 3 times

Through qualitative analysis, general results are as follows:

1.All types of strategies should be implemented when the total number of local mosquitoes is as small as possible.

2.The number of released mosquitoes should be at least 1/3 of the total number of local mosquitoes to have a significant lasting effect.

3.The sites of release should be as scattered as possible，at least cover 50% of the cells.

4.If only release female mosquitoes, the infection effect is obvious, the peak number of mosquitoes increased significantly.

5.If only release male mosquitoes, the infection effect is poor, however it does not cause a significant increase in the peak number.

6.The infection effect is optimal if release mosquitoes with a sex ratio of 1:1. In this way the peak number of mosquitoes in the region will not increase significantly, and the cost is lower.

7.Releasing mosquitoes many times will greatly increase investment, but the improvement is not obvious.

Therefore, the quantity of re-engineered mosquitoes should be equivalent to the number of local mosquitoes, and the sites of release should cover 50% of the cells. Under this strategy, we can control about 50% of the malaria parasites in mosquitoes when the mosquito population reaches its peak. When the number of mosquitoes declined, more than 70% of the malaria parasites in mosquitoes could be controlled.

6. Sensitivity Analysis

Because of p_sexinfect and p_larvainfect are difficult to obtain and have a great impact on the results. We carry out sensitivity analysis on both of them in order to guide our project to further optimize. Under the first release strategy, we simulate the infection rate by only changing p_sexinfect or p_larvainfect.

1. When p_sexinfect = 0.9, only changing p_larvainfect, the change of infection rate over time shows as Fig 6.1.

Fig 6.1

It can be found that p_larvainfect has a great influence on the final results.

2. When p_larvainfect = 0.7, only changing p_sexinfect, the change of infection rate over time shows as Fig 6.2.

Fig 6.2

The result shows that the model is not sensitive to P_sexinfect.

Therefore, the focus of follow-up projects should be on improving p_larvainfect.

3.In addition, considering the economic cost, we made a sensitivity analysis of the number of released mosquitoes. Fig 6.3 depicts the infection effect, only changing the number of released mosquitoes with other conditions, remain unchanged.

Fig 6.3

When the number of released mosquitoes is equivalent to the number of local mosquitoes, the effect is optimal. When the number is more than 1/3, the effect will gradually strengthen over time. However, when the number is less than 1/3, the effect will gradually decrease over time.

Therefore, considering the cost, the number of released mosquitoes should be equivalent to the number of local mosquitoes. If this is difficult to achieve, it should be more than one third of the number of local mosquitoes. Otherwise, the control effect on malaria would be negligible.

References

[1] Wang S , Dos-Santos, André L. A, Huang W , et al. Driving mosquito refractoriness to Plasmodium falciparum with engineered symbiotic bacteria[J]. Science, 2017, 357(6358):1399-1402.

[2] Wei-Ming W , Hua-Yun Z , Yao-Bao L . Comparison of seasonal fluctuation and nocturnal activity patterns of Anopheles sinensis in different regions of Jiangsu pronince[J]. China Tropical Medicine, 2013.

[3] http://www.cnki.com.cn/Article/CJFDTotal-KXDY201316030.htm

[4] Ceccherini‐Silberstein T, Coornaert M. Cellular Automata and Groups[M]. 2010.

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